Dual Eichler-Shimura maps for the modular curve
aa r X i v : . [ m a t h . N T ] F e b DUAL EICHLER-SHIMURA MAPS FOR THE MODULAR CURVE
JUAN ESTEBAN RODR´IGUEZ CAMARGO
Abstract.
We construct overconvergent dual Eichler-Shimura maps as an application of the Higher Coleman’stheory for the modular curve developed by Boxer-Pilloni. In the road, we redo the overconvergent Eichler-Shimura map of Andreatta-Iovita-Stevens. We give a new proof of Faltings Eichler-Shimura decompostion via apro´etale dual BGG theory. Finally, we prove that the overconvergent Poincar´e and Serre pairings are compatiblewith the overconvergent Eichler-Shimura morphisms.
Contents
1. Introduction 11.1. The main results 11.2. Outline of the document 3Acknowledgements 42. Preliminaries 42.1. The geometric setup 42.2. Sheaf theoretical preliminaries 72.3. Spectral theory over adic spaces 153. Overconvergent objects 213.1. Overconvergent neighbourhoods in the flag variety 213.2. Locally analytic functional spaces 233.3. Overconvergent modular symbols 253.4. Overconvergent modular forms 293.5. The dlog maps 314. Hecke Operators 344.1. The U p operator 344.2. Slope estimates of the U p -operator 395. Overconvergent Eichler-Shimura decomposition 405.1. Classical Eichler-Shimura decomposition 405.2. Overconvergent Eichler-Shimura decomposition 465.3. Compatibility with Poincar´e and Serre duality 47References 491. Introduction
Let p be a prime number, N ≥ p and n ≥
1. Let K be a finite extension of Q p , O K itsring of integers and G K its absolute Galois group. We let C p be the completion of an algebraic closure of K .Let Y ( N, p n ), Y ( N, p n ) and Y ( N, p n ) be the affine modular curves over K of level Γ ( N ) ∩ Γ( p n ), Γ ( N p n ),Γ ( N ) ∩ Γ ( p n ) respectively. We write X ( N, p n ), X ( N, p n ) and X ( N, p n ) for their natural compactificationsrespectively, cf. [DR73, KM85]. For simplicity we shall write Y and X for any of the affine and compactifiedmodular curves respectively. We see the modular curves as locally finite adic spaces over Spa( K, O K ) via itsanalytification [Hub96]. We have a generalized elliptic curve E defined over the compactified modular curves,its Tate module T p E is a Kummer-´etale Z p -local system over X . We denote by ω E the sheaf of invariantdifferentials of E over X .1.1. The main results.
One of the main interests of p -adic Hodge theory is to compare the ´etale cohomologytheory of analytic spaces with their coherent cohomology. The first to give such a relation in the case of modularcurves was Faltings [Fal87], he proved the following theorem Theorem 1.1.1.
Let k ≥ be an integer. There is a G K -equivariant isomorphism (1.1) H et ( Y K , Sym k T p E ) ⊗ Z p C p (1) = H ( X C p , ω − kE )( k + 1) ⊕ H ( X C p , ω k +2 E ) . We call (1.1) the Eichler-Shimura isomorphism. Later on, Faltings-Chai [FC90] proved that such a decom-position holds for more general Shimura varieties (Siegel varieties), and local systems arising from the de Rham cohomology. Then, via the B dR -comparison between the ´etale and the de Rham cohomology, one gets what iscalled the dual BGG decomposition of the ´etale cohomology. The idea behind the proof is the following: let B ⊂ GL be the opposite Borel of lower triangular matrices and F ℓ std = B \ GL the flag variety. Let M X be the GL -torsor of X trivializing the log de Rham cohomology H dR := H dR ( E/X ), the Hodge-Filtration of H dR provides a GL -equivariant period map π H : M X → F ℓ std . Then, one can pullback GL -equivariant vector Bundles over F ℓ std to vector bundles over X by taking F π ∗ H ( F ) / GL . It turns out that there is a way to pullback the BGG complexes of GL by π H , and one finds very nicequasi-isomorphisms with the de Rham complexes of local systems.Now, let B ⊂ GL be the Borel subgroup of upper triangular matrices and F ℓ = B \ GL . Let X ( N, p ∞ ) =lim ←− n X ( N, p n ) be the Scholze’s perfectoid modular curve [Sch15]. It can be understood as the moduli space of X trivializing the Tate module T p E . The perfectoid curve is endowed with a Hodge-Tate period map π HT : X ( N, p ∞ ) → F ℓ induced by the Hodge-Tate filtration of T p E ⊗ b Z p b O X , where b O X is the completed structure sheaf of X proket (cf. [Sch13a, DLLZ19]). Let K p ⊂ GL ( Z p ) be an open subgroup and suppose that X has p -level K p , so that X ( N, p ∞ ) → X is a Galois cover of group K p . For example if K p = Iw n = (cid:18) Z × p Z p p n Z p Z × p (cid:19) then X = X ( N, p n ).The map π HT induces the following functor:(1.2) { K p -eq. sheaves over F ℓ } { K p -eq. sheaves over X ( N, p ∞ ) } { Sheaves over X } F π ∗ HT F ( π ∗ HT F ) /K p The previous restricts to { algebraic B -rep. } { GL -eq. VB over F ℓ } { b O X -VB over X } . Using this functor and the ideas of the original work of Faltings-Chai, we give a new proof of Theorem 1.1.1 viaa pro´etale dual BGG decomposition, see Theorem 5.1.2.Furthermore, the functor (1.2) provides a good recipe to construct a wide class of sheaves over X . Forinstance, let St be the standard representation and consider it as a GL -equivariant VB over F ℓ . Then π ∗ HT St = T p E ⊗ b O X , and the sheaves b ω ± E = ω ± E ⊗ b O X are recovered from the B -filtration of St . We will apply thisstrategy to construct overconvergent sheaves over the modular curve. Indeed, we will recover the sheavesof locally analytic principal series and distributions of [AS08], the sheaves of overconvergent modular formsof [Col97,BP20], and the dlog map of [AIS15] from distributions to the overconvergent modular sheaf. Moreover,this approach also provides a dlog ∨ from the overconvergent modular sheaf to distributions, and we will beallowed to play the same game with locally analytic principal series instead. By taking pro-Kummer-´etalecohomology, we reconstruct the AIS overconvergent Eichler-Shimura map as well as ‘`ıts dual” in a purelygeometric way, commuting with U p -operators. We can therefore develop a finite slope theory and apply thetechniques of AIS to get, locally in the weight space, finite free interpolations of the small slope of the Eichler-Shimura decomposition.Let us sketch the main constructions of this paper. Let D = Spa( K h T i , O K h T i ) be the unit disk. Let T ⊂ GL be the diagonal torus, N ⊂ B and N ⊂ N ⊂ B the unipotent radicals. Let δ > n be a rationalnumber, and consider the δ -analytic neighbourhood of Iw n in GL I w n ( δ ) = (cid:18) Z × p (1 + p δ D ) Z p + p δ D p n Z + p δ D Z × p (1 + p δ D ) (cid:19) Given a Banach K -algebra R , and a ( δ − κ : T ( Z p ) → R × , we define the R -module of δ -analytic functions (or δ -analytic principal series) of weight κ to be A δκ = O ( I w n ( δ ))[ κ ] = { f : I w n ( δ ) → A ,anK b ⊗ R | f ( bg ) = κ ( b ) f ( g ) } In other words, A δκ is the R -Banach module of functions f : Iw n → R such thati. f ( bg ) = κ ( b ) f ( g ) andii. the restriction of f to Iw n ∩ N is δ -analyticWe write D δκ for the continuous dual of A δκ , and called it the R -module of δ -analytic distributions of weight κ .Both R -modules carry over a natural action of Iw n . We can define Iw n -equivariant constant topological sheavesover F ℓ in the obvious way: F ℓ × A δκ and F ℓ × D δκ endowed with the diagonal action. Then, the functor (1.2)pullbacks the previous sheaves to topological sheaves over X ( N, p n ). Moreover, the pullbacks of A δκ b ⊗ K O F ℓ UAL EICHLER-SHIMURA MAPS FOR THE MODULAR CURVE 3 and D δκ b ⊗ O F ℓ as b O X -sheaves are equal to O A δκ = A δκ b ⊗ K b O X and O D δκ = D δκ b ⊗ K b O X , this last is the same sheafof geometric distributions of [CHJ17].On the other hand, let W = { , w } be the Weyl group of GL . Let ǫ > n , we define the Iw n -invariantaffinoid neighbourhoods of w ∈ W in F ℓ : C n,w ( ǫ ) = B \ B w I w n ( ǫ ) . By the properties of π HT [Sch15, Theo III.3.18], π − ( C n,w ( ǫ )) is an affinoid perfectoid which descends to anaffinoid X w ( ǫ ) ⊂ X ( N, p n ). For ǫ > δ big enough, we have a reduction of ω E to an ´etale T ( Z p ) T (1 + p δ O X )torsor. Its completion in the pro-Kummer-´etale site can be seen as the pullback by π HT of the Iw n -equivariantsheaf over C n,w ( ǫ ) B \ ( B w I w n ( ǫ ) × T ( Z p ) T (1 + p δ O F ℓ )) , where we endow T ( Z p ) T (1 + p δ O ) with the B ∩ I w n ( ǫ ) action induced by the projection B → T . Theoverconvergent modular sheaf ω κE is defined as the w ( κ )-equivariant functions of this torsor. Then, we definedlog maps between the torsors (which still have a group theoretic interpretation) whose κ -equivariant functionsgive rise to b ω w ( κ ) E dlog ∨A −−−−→ O A δκ over X ( ǫ ) O A δκ dlog A −−−−→ b ω κE over X w ( ǫ ) , see Lemma 3.5.1. We highlight that the first map correspond to the B ( Z p )-equivariant injection κ ⊗ ֒ → A δκ ,while the second one corresponds to the B ( p n Z p )-equivariant surjection A δκ → w ( κ ) ⊗
1. Taking all this togetherwe obtain the following (cf. Propositions 3.5.1 and 3.5.2)
Proposition 1.1.1.
The maps dlog , dlog ∨ are G K and Hecke-equivariant (after a suitable Tate twist). If κ = ( k, ∈ X ∗ ( T ) is a dominant weight, then the dlog A morphisms are compatible with the maps Sym k T p E ⊗ b O X → b ω kE and b ω − kE → Sym k T p E ⊗ b O X . Let W be the weight space of T ( Z p ) and V ⊂ W an open affinoid subset. Let R = O ( V ) and κ un the universalcharacter over V . Taking pro-Kummer-´etale cohomology we obtain the following theorem (cf. Theorem 5.2.1) Theorem 1.1.2. (1)
Let X = X ( N, p n ) C p and α = (1 , − ∈ X ∗ ( T ) . There are G K and Hecke equivari-ant maps (1.3) H ,c,ǫ ( X, w ( κ un ))( κ un ) H proket ( X, O A δκ un ) H w ,ǫ ( X, κ un + α )( κ un ) . ES ∨A ES A whose composition is zero. (2) Let k ≥ and κ = ( k, ∈ V ( K ) . Fix h < k + 1 . The ≤ h -slope of (1.3) is, locally around κ , a shortexact sequence of finite free R -modules. (3) The maps of (1.3) are stable under specialization and are compatible with the classical Eichler-Shimuramorphisms.The cohomologies H w,ǫ ( X, κ ) and H w,c,ǫ ( X, κ ) are the overconvergent cohomologies with supports of [BP20]. Finally, using the fact that O A δκ and O D δκ are duals, we define an ad. hoc. pairing of interior cohomologies(1.4) h− , −i P : H proket,i ( X, O D δκ )(1)) × H proket,i ( X, O A δκ ) → R From [BP20] we also have a Serre-Pairing(1.5) h− , −i S : H w,c,ǫ ( X, − κ un ) × H ,cuspw,ǫ ( X, κ un + α ) → R. We have the following theorem (cf. Theorem 5.3.1)
Theorem 1.1.3.
Keep the previous notation. The pairings (1.4) and (1.5) are Hecke-equivariant and compatiblewith the overconvergent Eichler-Shimura maps. Moreover, locally around ( k, ∈ V ( K ) , the parings induceperfect pairings in the ≤ h -slope cohomology. Outline of the document.
The content of the paper is as follows. In Section 2 we recall very brieflythe representation theory of GL , particularly we make some conventions regarding the equivalence between B -algebraic representations and GL -equivariant VB over F ℓ . Next, we introduce the period sheaves of thepro-Kummer-´etale site of X [DLLZ18], later on we will be interested in the sheaf O C log = gr ( O B dR, log ). Weshow that O C log restricted to X ( N, p ∞ ) is a polynomial ring in one variable over b O X . Then, we define the bigpro-Kummer-´etale site of X as the “union” of the pro-Kummer-´etale sites of quasi-separated objects locally offinite type over X . We define the notion of uniform sheaf and uniform continuity, we prove that the ring ofcontinuous functions from b O X to b O X is a Tate algebra of one variable over b O X . Finally, we say some wordsregarding the spectral theory over adic spaces following [AIP18, Appendix B]. JUAN ESTEBAN RODR´IGUEZ CAMARGO
We define all the overconvergent objects in Section 3. First, we use the analytic Bruhat cells C w,n ( δ ) to definelocally analytic principal series and distributions. We define the torsors of overconvergent modular forms and thesheaves b ω κE . Then, we define the overconvergent modular symbols as the cohomology groups H proket ( X, O A δκ )and H proket ( X, O D δκ ). We define the overconvergent cohomologies with supports H w,c,ǫ ( X, κ ) and H w,ǫ ( X, κ ).We finish the section with the construction of the dlog maps of torsors and prove Proposition 1.1.1.In Section 4 we define the U p -operators of overconvergent modular symbols and overconvergent coherentcohomologies via finite flat correspondances. We recall why they are represented by compact operators in thecohomology complexes, and consequently that they admit finite slope decompositions.Finally, in Section 5 we prove the main Theorems 1.1.1, 1.1.2 and 1.1.3. We see how the Eichler-Shimuramaps are formally defined once we have constructed the overconvergent dlog maps (cf. Lemma 5.2.1). Acknowledgements.
None of this work could ever be possible without the support of my advisor VincentPilloni, I want to express my deep gratitude for the many hours he spent explaining to me the philosophybehind the overconvergent theories, and for introducing me to this lovely p -adic geometric world. I want tothank Professors Adrian Iovita and Fabrizio Andreatta for the very fruitful exchanges during the past year, andthe opportunity to give a talk on this subject in the Workshop of Higher Coleman Theory. Finally, I want tothank Joaquin Rodrigues, George Boxer and Damien Junger for the many discussions and the help to find outsome mistakes in the main proofs. This paper was written while the author was a PhD student at the ENS deLyon. 2. Preliminaries
The geometric setup.
Let p be a primer number, n ≥ N ≥ p ∤ N . Given apositive integer M we denote by Γ( M ), Γ ( M ) and Γ ( M ) the congruence subgroups of GL ( Z ). Let Y ( N, p n ), Y ( N, p n ) and Y ( N, p n ) be the affine modular curves over Q , seen as the moduli space parametrizing ellipticcurves with level structure Γ ( N ) ∩ Γ( p n ), Γ ( N ) ∩ Γ ( p n ) and Γ ( N ) ∩ Γ ( p n ) respectively. Let X ( N, p n ), X ( N, p n ) and X ( n, p n ) denote the compactified modular curves [KM85]. By an abuse of notation we willwrite in the same way the adic analytic spaces over Spa( Q p , Z p ) attached to the modular curves, see [Hub96].Let Y be an affine modular curve as above, X its compactification and D = X/Y the reduced boundarydivisor. Let
E/Y be the universal elliptic curve, it has an extension to a semi-abelian scheme over the compact-ified modular curve which we also denote by E [DR73]. Let e : X → E be the unit section. We let ω E = e ∗ Ω E denote the modular sheaf over X . Given k ∈ Z an integer we set ω kE := ω ⊗ kE .We will consider different Grothendieck topologies over adic spaces. Let ( K, K + ) be a non archimedeanaffinoid field of mixed characteristic, let W be a fs log adic space of finite type over Spa( K.K + ) [DLLZ19]. For? ∈ { an, et, ket, proet, proket } we denote by W ? the analytic, ´etale, Kummer ´etale, pro´etale and pro-Kummer-´etale sites of W respectively (see loc. cit. and [Hub96,Sch13a]). We write O W and O + W for the structural sheavesof W ? , if ? ∈ { proet, proket } we denote by b O W and b O + W their p -adic completions respectively. We endow themodular curve X with the log structure defined by the normal crossing divisors D (Example 2.3.17 of [DLLZ19]).Fiber products will always refer to fiber products of fs log adic spaces unless otherwise specified (Proposition2.3.27 of [DLLZ19]), if no log structure is given we endow the adic space with the trivial log structure. Theunderline space of a fiber product of fs log adic spaces is not in general the fiber product of the underlying adicspaces (Remark 2.2.30 of [DLLZ19]). However, this is the case if one of the spaces has the trivial log structure.Let E [ p n ] /Y denote the p n -torsion of the universal elliptic curve Y , it is ´etale local system which can benaturally extended to a Kummer ´etale local system over X (Theorem 4.6.1 of [DLLZ19]). Let T p E = lim ←− n E [ p n ]be the Tate module of E , it is a b Z p -local system of rank 2 on the pro-Kummer-´etale site of X . We let X ∞ = lim ←− n X ( N, p n )be the perfectoid modular curve of tame level Γ ( N ) [Sch15], it is a pro-Kummer-´etale Galois cover of X ( N, p n )of Galois group Iw n := (cid:18) Z × p Z p p n Z p Z × p (cid:19) trivializing T p E .2.1.1. Representation theory of GL . Let T ⊂ GL be the diagonal torus, B the Borel subgroup of uppertriangular matrices and N ⊂ B its unipotent radical. Let B and N be the opposite Borel and unipotentsubgroups. The Weyl group of GL acts on T by permutation of the diagonal components, we write W = { , w } and take any lift of w in N ( T ), say w = ( ). Let X ∗ ( T ) be the character group of T , it is identified with Z × Z via the isomorphism G m → T : ( t , t ) diag( t , t ).We write g , b and n for the Lie algebras of GL , B and N , we use the similar notation for the Lie algebrasof T , B and N . We see g as a GL -representation via the left adjoint action ( g, X ) gXg − for g ∈ GL and X ∈ g . Restricting to T we have a decomposition g = n ⊕ t ⊕ n , where T acts on n and n by the characters UAL EICHLER-SHIMURA MAPS FOR THE MODULAR CURVE 5 α + = (1 , −
1) and α − = ( − ,
1) respectively. We endow the character group of T with the partial order definedas: χ ≥ χ ′ if and only if χ = χ ′ + kα + for k ≥
0. Let det : GL → G m denote the determinant character ofGL , a character χ is said to be dominant if it is greater than some power of det. Let X ∗ ( T ) + ⊂ X ∗ ( T ) be thecone of dominant characters, it is in bijection with the pairs ( k , k ) such that k ≥ k .Let F be a finite extension of Q p and O F its ring of integers. Let S be a quasi-separated scheme of finitetype over Spec O F , S F denote its generic fiber and b S its completion along the special fiber. We can constructtwo adic spaces from S , namely S := b S × Spf O F Spa( F, O F ) and S an := S F × Spec F Spa( F, O F ) . There is always an open immersion S ֒ → S an which is an isomorphism if S/ O F is proper (Section 1.9 of [Hub96]).Thus, GL an is the analytic group whose ( R, R + ) points are GL an ( R, R + ) = GL ( R ) , and GL ⊂ GL an the open subgroup given by GL ( R, R + ) = lim −→ R ⊂ R + GL ( R ) . The limit is taken over all the subrings of definition R ⊂ R + . We have similar descriptions for the analytifica-tions of the algebraic subgroups of GL .Let F L = B \ GL be the Flag variety of GL , and let F ℓ denote its analytification. There is an identification F L = P induced by the right action on P :[ x : y ] (cid:18) a bc d (cid:19) = [ ax + cy : bx + dy ] , and taking [0 : 1] ∈ P as base point. As F L is proper, there are isomorphisms of adic spaces F ℓ = B\ GL = B an \ GL an . Let St denote the left standard representation of GL , there is a bijection between the set of dominantweights and the set of irreducible representations of GL modulo isomorphism as follows: X ∗ ( T ) + ←→ Irr-Rep GL κ = ( k , k ) V κ := Sym k − k St ⊗ det k . The representation V κ is the irreducible representation with highest weight κ = ( k , k ). Let V be a representa-tion of GL , we denote by V ∨ its contragradient representation: let f ∈ V ∨ and g ∈ GL , then ( gf )( v ) = f ( g − v )for v ∈ V . For irreducible representations we have isomorphisms(2.1) V ∨ κ ∼ = V − w ( κ ) = Sym k − k St ⊗ det − k . The representation V κ can be defined as a Borel induction as follows:(2.2) V κ = { f : GL → A | f ( bg ) = w ( k )( b ) f ( g ) for b ∈ B } . Indeed, let v ∨ ∈ V ∨ κ be a highest weight vector, and let h− , −i denote the natural pairing of V ∨ κ and V κ . Then v f v : g
7→ h v ∨ , gv i defines an isomorphism as in (2.2). We can describe V κ using the opposite Borel instead: V κ = { f : GL → A | f ( bg ) = κ ( b ) f ( g ) for b ∈ B } The isomorphism is provided by f L w f , where ( L w f )( g ) = f ( w g ).There is an equivalence of categories { Algebraic B -rep. } ⇆ { GL -equivariant VB over F L } (2.3) W
7→ W := GL × B WV := V| [1] ← [ V Where GL × B W is the quotient of GL × W by the left B -action: ( g, w ) ∼ ( bg, bw ), and [1] ∈ F L is the class of1 ∈ GL . Let κ = ( k , k ) ∈ X ∗ ( T ), we see κ as a character of B extending trivially the action to N . We denoteby L ( κ ) the GL -equivariant line bundle of F L constructed from w ( κ ) in the previous equivalence. Note thatif κ is dominant then V κ can be constructed as the global sections of L ( κ ). If W is the restriction of a GL -representation to B , then W is isomorphic to O dim WF L as a vector bundle over
F L , but not as a GL -equivariantone in general. In particular, the line bundle L (det) attached to det = (1 ,
1) is trivial which implies that theisomorphism class of L ( κ ) only depends on k − k . In fact, under the identification F L ∼ = P , the sheaf L ( κ )is isomorphic to the Serre twisted sheaf O ( k − k ). JUAN ESTEBAN RODR´IGUEZ CAMARGO
Pullbacks via the Hodge-Tate period map.
In [Sch15] Scholze defines a very special morphism of adicspaces called the Hodge-Tate period map π HT : X ∞ → F ℓ . This map encodes the geometric p -adic Hodge theory of the modular curves. Let us recall its definition (see[Sch15] and [Sch13b]). Let Y be an affine modular curve and π : E → Y the universal elliptic curve, we identify E with its dual E ∨ via the natural principal polarization (see [KM85]). For simplicity, we work at geometriclevel, i.e. after base change by the completion of an algebraic closure C p of Q p . Consider the constant sheaves µ p n of p n -th roots of unit over E , we have an exact sequence in the pro´etale site of E → µ p n → O × Y p n −→ O × Y → R π ∗ µ p n ∼ = Pic E [ p n ] = E ∨ [ p n ] ∼ = E [ p n ] . Taking inverse limits we get that R π ∗ b G m ∼ = T p E . On the other hand, by deformation theory we know that R π ∗ O E = Lie E ∨ ∼ = Lie E . The primitive comparison theorem of Scholze (Theorem 5.1 of [Sch13a]) providesan isomorphism T p E ⊗ b Z p b O Y = ( R π ∗ b G m ) ⊗ b Z p b O Y = R π ∗ ( b G m ⊗ b Z p b O E ) . Hence, we get a mapdlog ∨ : Lie E ⊗ O Y b O Y (1) = R π ∗ O E ⊗ O Y b O Y (1) → R π ∗ ( b G m ⊗ b Z p b O E ) = T p E ⊗ b Z p b O Y . Following the same construction with the dual elliptic curve and taking Tate duals one obtains a morphismdlog : T p E ⊗ b Z p b O Y → ω E ⊗ O Y b O Y . Write Lie E = ω − E . The Hodge-Tate filtration of the Tate module is the exact sequence(2.4) 0 → ω − E ⊗ O Y b O Y (1) dlog ∨ −−−→ T p E ⊗ b Z p b O Y dlog −−−→ ω E ⊗ O Y b O Y → . The previous maps extend naturally to the pro-Kummer-´etale site (see [AIS14]). Let Ψ univ : b Z p → T p E denotethe universal trivialization of the Tate module over the perfectoid modular curve X ∞ . The application π HT isdefined by the pullback of the exact sequence (2.4) via Ψ univ . Construction 2.1.
The Hodge-Tate period map is GL ( Q p )-equivariant and affine (Theorem III-3.18 of[Sch15]), this allows to define a functor from GL ( Q p )-equivariant sheaves over the Flag variety to pro-Kummer-´etale sheaves over the modular curves X (even from K p -equivariant sheaves over F ℓ to sheaves over X , where K p is the level at p of X ):Let K p ⊂ GL ( Z p ) be an open subgroup and X a compactified modular curve of p -level K p . Since p : X ∞ → X is a Galois cover of X of Galois group K p , there is an equivalence of categories between sheaves over X proet ,and K p -equivariant sheaves of X ∞ ,proet . Namely, Sh ( X proet ) ⇄ K -eq( Sh ( X ∞ ,proket )) F p ∗ F ( p ∗ G ) /K p ← [ G . Therefore, the π HT maps gives rise a functor π − : K p -eq( Sh ( F ℓ an )) → K p -eq( Sh ( X ∞ ,an )) → K p -eq( Sh ( X ∞ ,proket )) = Sh ( X proket ) , the first arrow is the usual inverse image functor in the analytic topoi of adic spaces, and the second is thepullback of the projection X ∞ ,proket → X ∞ ,an . If we restrict to sheaves of O F ℓ -modules we obtain a pullbackfunctor to sheaves of b O X -modules defined as π ∗ HT F := π − HT ( F ) ⊗ π − ( O F ℓ ) b O X . Example 2.1.1.
Let St be the standard representation of GL , and consider its B -filtration0 → (1 , Q → St → (0 , Q → . It induces after the functor (2.3) a short exact sequence over F ℓ (2.5) 0 → L (0 , → S t → L (1 , → O F ℓ . Then, by definition of π HT , the pullback of (2.5) is theHodge-Tate filtration (2.4). In particular, we have that π ∗ HT L (1 ,
0) = ω E ⊗ b O X and π ∗ HT L (0 ,
1) = ω − E ⊗ b O X (1).For κ = ( k , k ) ∈ X ∗ ( T ) define the coherent sheaf over X : ω κE := ω k − k E . With this notation we have ω κE ⊗ O X b O X ( k ) = π ∗ HT L ( κ ) . UAL EICHLER-SHIMURA MAPS FOR THE MODULAR CURVE 7
Sheaf theoretical preliminaries.
Period sheaves in the pro-Kummer-´etale site.
Let K be a finite extension of Q p , let O K denote the ringof integers of K and F its finite residue field. Let W be a smooth adic space over Spa( K, O K ) of dimension d .We define the following period sheaves over the pro´etale site of W , see Section 6 of [Sch13a].1) b O ♭, + W = lim ←− Φ b O + W and b O ♭ W = lim ←− Φ b O W with Φ : x x p .2) A inf = W ( b O ♭, + W ) and B inf = A inf [ p ]. There is a natural surjective map θ : B inf → b O W which locallyon perfectoid objects Spa( R, R + ) is the Fontaine map θ : W ( R ♭, + )[ 1 p ] → R. ker θ is locally pro´etale generated by a non zero divisor ξ .3) B + dR = lim ←− n B inf / (ker θ ) n , we endow this sheaf with the filtration induced by ker θ . We set B dR = B + dR [ ξ − ] endowed with the filtrationFil k B dR = X i + j = k ξ i Fil j B + dR .
4) Let O B + dR be the sheafification which sends an affinoid perfectoid Spa( R, R + ) = lim ←− i Spa( R i , R + i ) to thedirect limit of the ker ˜ θ -adic completion of( R + i b ⊗ W ( κ ) A inf ( R, R + ))[ 1 p ] , where ˜ θ is the map sending r i ⊗ α r i θ ( α ).Now suppose that W has a structure of fs log smooth scheme over Spa( K, O K ) given by a morphism ofsheaves of monoids α : M et → O W ,et (Definition 2.2.2 of [DLLZ19]). Let M ket : X ket → Sets be the presheaf ofmonoids which maps U M et,U ( U ), it is a sheaf after Proposition 4.3.4 of [DLLZ19]. Let ν : W proket → W ket be the projection from the pro-Kummer-´etale site to the Kummer ´etale site, M := ν − M ket and α : M → O W the given morphism of monoids, we set M + = α − ( O + W ). Let c M = M × α, b O W b O ×W and c M ♭ = lim ←− Φ c M . Thereis a natural map α ♭ : c M ♭ → b O ♭ W . To actually work with the log structure we need to introduce some periodsheaves involving M , c M and c M ♭ . We recall the construction of the sheaves O B + dR, log and O B dR, log in thepro-Kummer-´etale site of W (Section 2.2 of [DLLZ18]). Construction 2.2.
In the following we see M as a multiplicative monoid. Let B = O B inf [ M + × c M c M ♭ ] be themonoid algebra over O B inf , we denote by e log a ∈ B the element corresponding to a ∈ M + × c M c M ♭ . Let I bethe ideal generated by { α ( a ) ⊗ − (1 ⊗ [ α ♭ ( a )]) e log a } a , we set O B inf, log := B / I . Let ˜ θ log : O B inf, log → b O W bethe map extending θ which sends e log a
1. We define O B + dR, log as the sheafification of the presheaf sending alog affinoid perfectoid (Spa( R, R + ) , M ) = lim ←− i (Spa( R i , R + i ) , M ) modeled in a p -divisible monoid M = lim −→ i M i to the direct limit the (ker ˜ θ log )-adic completion of( R + i b ⊗ W ( F ) A inf ( R, R + )[ M i × c M ( R,R + ) c M ♭ ( R, R + )])[ 1 p ] / I . We set O B dR, log = O B + dR, log [ ξ − ] endowed with the filtrationFil k O B dR, log = X i + j = k ξ i Fil j O B + dR, log . Finally, we denote O C log ( • ) := gr • O B dR, log the graded ring, we shall often write O C log for O C log (0).We may extend the log connection ( d, δ ) : O W [ M ] → Ω W (log) to O B + dR, log as follows: for x = ( s ⊗ γ ) e log a ∈ O B inf, log we define ∇ ( x ) = γe log a ds +( s ⊗ γ ) δ ( a ) e log a . As ker θ log is generated by ker θ and the elements e log a − a ∈ M + × c M c M ♭ , one checks that ∇ (ker θ log ) k ⊂ ker θ k − ⊗ O W Ω W (log) ⊂ O W B inf, log ⊗ O W Ω W (log). Bycompleting with respect to ker θ log we obtain a log-connection ∇ log : O B + dR, log → O B + dR, log ⊗ O W Ω W (log)satisffying Griffiths trensversality. By inverting ξ we get a log-connection for O B dR, log . Hence, we have a log deRham complex(2.6) 0 → B + dR → O B + dR, log ∇ log −−−→ O B + dR, log ⊗ O W Ω W (log) → · · · ∇ log −−−→ O B + dR, log ⊗ O W Ω d W (log) → , similarly we have de Rham complexes for O B dR, log , O C log ( • ) and O C log . JUAN ESTEBAN RODR´IGUEZ CAMARGO
One of the most important features of (2.6) is that the Poincar´e lemma is satisfied (Corollary 2.4.2. of[DLLZ18]). It is a formal consequence of the fact that locally pro-Kummer-´etale the ring O B + dR, log looks as apower series ring of B + dR , cf. Proposition 2.3.15 of [DLLZ18] and Proposition 6.10 of [Sch13a].Let us recall how this last statement is proved for log adic spaces whose log structure is defined by a normalcrossing divisors. Let T = Spa( K h T ± i , O K h T ± i ) and D = Spa( K h S i , O K h S i ) be the analytic torus andclosed unit disc respectively, we endow D with the log structure defined by the divisor S = 0. Let e T =lim ←− n Spa( K h T ± pn i , O K h T ± pn i )) and e D = lim ←− n Spa( K h S pn i , O K h S pn i )) be the perfectoid torus and perfectoidlog closed unit disc respectively, we denote T ♭i = ( S pn i ) n and S ♭j = ( S pn j ) n . Definition 2.2.1. (1) A log affinoid space U = Spa( R , R +0 ) is modeled in a log toric chart if there existsan ´etale map f : U → T d − r × D r = Spa( K h T ± , . . . , T ± d − r , S , . . . , S r i , O K h T ± , . . . , T ± d − r , S , . . . , S r i )which factors through a finite composition of finite ´etale maps and rational localizations, and such thatthe log structure of U is the pullback of the log structure of T d − r × D r defined by S · · · S r = 0.(2) A fs log adic space W is said to be locally modeled in toric charts if there exists an ´etale cover byaffinoids which are modeled in toric charts. Lemma 2.2.1.
Let U = Spa( R , R +0 ) be a log adic space modeled in toric charts, and let e U = U × T d − r × D r e T d − r × e D r be the log perfectoid cover of U induced by the perfectoid toric chart. There exists an isomorphismof filtered B dR algebras B + dR | e U [[ X , · · · , X d − r , Y , · · · , Y r ]] → O B + dR, log | e U mapping X i ⊗ [ T ♭i ] − T i ⊗ and Y j e log[ S ♭j ] − .Proof. We denote θ log : B + dR | e U [[ X i , Y j ]] → b O | e U the quotient by the ideal ( ξ, X i , Y j ). Let { e j } rj =1 be the standardbasis of N r . Let β : p ∞ N r → B + dR | e U [[ X i , Y j ]] be the map p n e j (1 + Y j ) pn . As in the proof of Proposition 6.10of [Sch13a], we claim that that B + dR | e U [[ X i , Y j ]] has a unique structure of O e U -algebra v , and a unique morphismof monoids β : M + × M c M ♭ | e U → B + dR [[ X i , Y j ]] such that:i) The map v is compatible with θ log and v ( T i ) = [ T ♭i ] − X i .i) One has β ([ S ♭j ]) = β ( e j ) = Y j + 1.ii) The composition of β with θ log is the constant map 1.iii) For all a ∈ M × c M c M ♭ one has ν ( α ( a )) = [ α ♭ ( a )] β ( a ).Hence, one gets a morphism O B inf, log | e U → B + dR | e U [[ X i , Y j ]] compatible with θ log which after completion providesthe inverse O B + dR, log | e U → B + dR | e U [[ X i , Y j ]]. The proof of the claim follows the same steps as loc. cit. , see Lemma2.3.12 of [DLLZ18] for the details. (cid:3) The previous description in perfectoid log toric charts can be extended to log perfectoid charts with enoughramification in the boundary divisor:Let U = Spa( R , R +0 ) be as before. Let U ∞ = lim ←− n ∈ N Spa( R n , R + n ) be a perfectoid profinite-Kummer-´etalecover of U with finite surjective Kummer ´etale transition maps satisfying the following conditions:i) For n ≥ S j admit p n -th roots e S pn j in R + n modulo ( R + n ) × .ii) The map Spa( R n , R + n ) → T d − r × D r given by T i T i and S j ˜ S pn j is modeled in a log toric chart. Lemma 2.2.2.
Let U ∞ = Spa( R, R + ) = lim ←− n Spa( R n , R + n ) be as before. (1) For all ≤ j ≤ r there exists [ ˜ S ♭j ] = ( ˜ S ( n ) j ) n ∈ c M ♭ ( U ∞ ) ∩ R ♭, + such that ˜ S ( n ) j / ˜ S pn j ∈ ( R + ) × for all n ≥ . (2) The quotient S j / [ ˜ S ♭j ] is well defined in O B + dR, log ( U ∞ ) . (3) Let ˜ T i and γ j be any lift of T i ∈ b O + ( U ∞ ) and ˜ S (0) j /S j ∈ b O + ( U ∞ ) × in B + dR ( U ∞ ) respectively. Thereexists an isomorphism of filtered algebras B + dR | U ∞ [[ X , . . . , X d − r , Y , . . . , Y r ]] → O B + dR, log | U ∞ mapping X i ⊗ ˜ T i − T i ⊗ and Y i ( S j / [ ˜ S ♭j ]) γ j − .Proof. The condition i) and the fact that Φ : R + /p → R + /p is surjective imply that there exist a ( n ) j ∈ ( R + ) × for n ≥
0, such that:a) ( ˜ S p j a (0) j ) p ≡ S j ( mod pR + ) andb) ( ˜ S pn +1 j a ( n ) j ) p ≡ ˜ S pn j a ( n − j ( mod pR + ) for n ≥ UAL EICHLER-SHIMURA MAPS FOR THE MODULAR CURVE 9
Thus, ( e S pn j a ( n ) j ) n is an element in b O ♭, + ( U ∞ ), by setting ˜ S ( n ) j := lim k ( e S pn + k j a ( n + k ) j ) p k we get that ( ˜ S ( n ) j ) p =˜ S ( n − j and ˜ S ( n ) j / ˜ S pn j ∈ ( R + ) × . By property ii), we see that ( ˜ S ( n ) j ) n ∈ c M ♭ ( U ∞ ) ∩ R ♭, + and (1) is proved.To show (2) and (3) we can work locally pro-Kummer-´etale, let e T → T and e D → D be the perfectoid torusand unit disc, and let e U = U × T d − r × D r e T d − r × e D r . Then the fiber product U ∞ × U e U is a profinite-Kummer-´etalecover of U ∞ of Galois group Γ = Z dp . By Lemma 2.2.1 we get that S j [ ˜ S ♭j ] = S j [ S ♭j ] [ S ♭j ][ ˜ S ♭j ] = [ S ♭j ][ ˜ S ♭j ] e log[ S ♭j ] where [ S ♭j ] = ( S pn j ) n ∈ B + dR is a compatible system of p -th power roots of S j . Then, it suffices to check that[ S ♭j ] / [ ˜ S ♭j ] = ( ˜ S ( n ) j /S pn j ) n ∈ ( B + dR ( U ∞ × U e U )) × , but this is a consequence of condition i) and part (1) of thelemma. Furthermore, let [ T ♭i ] = ( T pn i ) n ∈ B + dR ( U ∞ × U e U ) be a compatible system of p -th power roots of T i ,then it is clear that ˜ T i / [ T ♭i ] ∈ ( B + dR ( U ∞ × U e U )) × and γ i ∈ B + dR ( U ∞ ) × . Part (3) then follows from Lemma 2.2.1and the definition of the map from the power series ring into O B + dR, log | U ∞ . (cid:3) Example 2.2.1.
The map of modular curves X ∞ → X ( N, p n ) satisfies the hypothesis of the previous lemmalocally analytic over X ( N, p n ), for n big enough. In particular, the restriction of O B + dR, log to X ∞ is locallyanalytic a power series ring over B + dR | X ∞ . Similarly, the restriction O C log | X ∞ is locally analytic a polynomialring over b O X ∞ .2.2.2. The big pro-Kummer-´etale site.
Let W be a quasi-separated fs log adic space locally of finite type overSpa( K, O K ). A geometric inconvenient with the pro-Kummer-´etale site of W is the restricted the representablesheaves are. For example, there is no object in W ket representing the structural sheaf O W . This problem can besolved in a natural way by defining the big pro-Kummer-´etale site of W , which is roughly speaking the union ofall the pro-Kummer-´etale sites of all fs log adic spaces locally of finite type over W . Let Rig/ W be the categoryof quasi-separated f s log adic spaces locally of finite type over W , we denote by ( Rig/ W ) ket the site whoseunderlying category is Rig/ W and whose covers are jointly surjective Kummer ´etale maps. Let Pro -( Rig/ W )be the Pro-category of Rig/ W , that is, the category whose objects are inverse systems {V i } i ∈ I of objects in Rig/ W and morphisms given by Hom( {V i } i ∈ I , {U j ∈ J } ) = lim ←− j ∈ J lim −→ i ∈ I Hom( V i , U j ) . An object V = { V i } i ∈ I has an attached topological space defined as |V| = lim ←− i ∈ I |V i | which is independent ofthe presentation of V . We refer to Definition 5.1.1. of [DLLZ19] for the definition of Kummer-´etale morphismand pro-Kummer-´etale morphism in Pro -( Rig/ W ). The following construction is identical to the definition ofthe pro´etale site for an adic space given in [Sch13a]. Definition 2.2.2.
We define (
Rig/ W ) proket as the full subcategory of Pro -( Rig/ W ) whose objects are inversesystems V = {V i } i ∈ I with an initial object V i , such that the map V → V i is pro-Kummer-´etale. The coveringsof ( Rig/ W ) proket are given by jointly surjective pro-Kummer-´etale maps. Remark . An object
V ∈ ( Rig/ W ) proket is pro-Kummer-´etale over an adic space V ∈ Rig/ W . In particular,it belongs to ( V ) proket . As fiber products of objects in ( Rig/ W ) ket exist (Proposition 2.3.27. of [DLLZ19]), andpullback of Kummer ´etale morphisms are Kummer ´etale (Proposition 4.1.14. of loc. cit. ), the site ( Rig/ W ) proket has all the fiber products. Furthermore, due to the nature of the covers and the definition of fiber products,it can be shown that cohomology in the big pro-Kummer-´etale site is the same as in the small pro-Kummer-´etale site, see Lemma [Sta20, Tag 03YY]. We have that representable presheaves on ( Rig/ W ) ket are sheaves(Proposition 4.3.5. of [DLLZ19]). As a consequence we get that representable presheaves on ( Rig/ W ) proket aresheaves as well (cf. Proposition 5.1.6. of [DLLZ19]). Definition 2.2.3. (1) We define the structural sheaf O W ,ket : ( Rig/ W ) ket → Sets by sending V O V ( V ).Let ν : ( Rig/ W ) proket → ( Rig/ W ) ket be the natural projection of sites, we define O W = ν − ( O W ,ket ).Similarly, we define O + W ,ket and O + W .(2) We set b O + W = lim ←− n O + W /p n and b O W = b O + W [ p ]. Remark . The definition of the completed sheaves b O W and b O + W is just the “disjoint union” of the completedsheaves restricted to the small sites. We could also consider other period sheaves over the big site, howeverthere are some bad behaviours if we want to study the B + dR -algebra structure of O B + dR . In fact, the relativedimension of O B + dR over B + dR is not constant! Example 2.2.2. (1) The sheaf O + W is represented by D W := W × K D where D is endowed with the triviallog structure. Similarly, the sheaf O W is represented by A ,an W := W × K A ,anK where A ,anK is endowedwith the trivial log structure.(2) Let L be a pro-Kummer-´etale b Z p -local system over W . Consider the natural morphism of sites µ :( Rig/ W ) proket → W proket . Then the sheaf µ − L is a pro-Kummer-´etale b Z p -local system which we stilldenote by L .(3) Let F be a coherent sheaf over W an , we can extend F to ( Rig/ W ) an by setting for U ∈ ( Rig/ W ) an F | U an := p ∗ F = p − ( F ) ⊗ p − ( O W ,an ) O U,an , where p : U → W is the structural map. Thus F is a rigid O W ,an -module in the sense that for any map f : U → U ′ of objects in ( Rig/ W ) an we have F | U an = f ∗ F | U ′ an . If F is a vector bundle over W an theinduced rigid big sheaf over ( Rig/ W ) an is representable. Similarly, we can pullback the coherent sheaf F to other big sites by tensoring with the structural sheaf O W .Similarly as with the pro´etale site, we have a basis by perfectoid objects in the big pro-Kummer-´etale site. Proposition 2.2.1 ( [DLLZ19, Prop. 5.3.12.]) . Let W be a quasi-separated adic space locally of finite type over K . There exists a basis of ( Rig/ W ) proket consisting in log affinoid perfectoid objects. In Section 2.3 we will extend certain kind of topological sheaves of O W or b O W -modules to the big sites. Inthe definition of overconvergent modular forms and overconvergent modular symbols, we shall work with torsorsover analytic neighbourhoods of p -adic Lie groups, to extend the constructions over the big sites one has toextend the torsors in a natural way. Let G be an analytic group locally of finite type over Spa( K, O K ). For? ∈ { an, et, ket, proet, proket } we see G as a group object in ( Rig/ W ) ? , and we identify the group with thesheaf it defines. We denote by G| W ? its restriction to a sheaf of groups on the small site W ? . Definition 2.2.4.
Let µ : ( Rig/ W ) ? → W ? be the natural projection of sites, we keep the above notation.(1) Let T be a G| W ? -torsor, let U ∈ ( Rig/ W ) ? and f : U → W be the structural map, we define f ∗ T := G| U ? × f − ( G| W ? ) f − ( T ) . (2) Let T be as in (1), we denote by µ ∗ T the extension of T to a G -torsor in the big site given by µ ∗ T | U ? = f ∗ T , for f : U → W an object in ( Rig/ W ) ? . By an abuse of notation we write T instead of µ ∗ T . Remark . (1) Note that the definition of the G -torsor µ ∗ T above only depends on T and G . In par-ticular, we can make the same extension whenever we have a group object in the big site and a torsorin the small site. For example, if G is an affine analytic group we may consider its completion b G as agroup object in ( Rig/ W ) proket , cf. Definition 2.2.9.(2) The torsor µ ∗ T satisfies a similar rigid property as the coherent extension of Example 2.2.2 (3), namelyfor g : U → U ′ an arrow in ( Rig/ W ) ? we have T | U ? = G| U ? × g − ( G| U ′ ? ) T | U ′ ? . Proposition 2.2.2.
Let ( A, A + ) be an affinoid K -algebra topologically of finite type, let ( B, B + ) be a finite´etale ( A, A + ) -algebra and ( R, R + ) an affinoid ( B, B + ) -algebra topologically of finite type. Let (( R, R + ) , δ ) dea descent datum as affinoid algebra. Then there exists an affinoid ( A, A + ) -algebra topologically of finite type ( S, S + ) such that ( S, S + ) ⊗ ( A,A + ) ( B, B + ) = ( R, R + ) , i.e. the descent datum is effective.Proof. Recall that (
B, B + ) being finite ´etale over ( A, A + ) means that B is finite ´etale over A and that B + isthe integral closure of A + in B . By extending B we may assume that B/A is a Galois extension of Galois groupΓ, in particular Γ leaves B + invariant. Let b ∈ B + and let p b ( X ) = X n + a n − X n − + · · · + a be its characteristic polynomial. The coefficients a i are symmetric integral polynomials in γ ( b ) with γ ∈ Γ.Hence, a i ∈ B + ∩ A = A + . This assertion is true for any Galois extension of affinoid rings.As B is finite over A , we have for any affinoid algebra ( C, C + ) over ( A, A + ) that C b ⊗ A B = C ⊗ A B . Let S = eq( R ⇒ R ⊗ A B )be the equalizer of the descent datum, it is a closed subalgebra of R satisfying S ⊗ A B = R by faithfullt flatdescent. Define S + := S ∩ R + , we claim that ( S, S + ) is topologically of finite type over A and that R + is theintegral closure of S + in R . As a consequence we have that( S, S + ) ⊗ ( A,A + ) ( B, B + ) = ( R, R + )and the descent datum (( R, R + ) , δ ) is effective. UAL EICHLER-SHIMURA MAPS FOR THE MODULAR CURVE 11
Let Y , · · · Y m ∈ R + be topological generators of R over A such that R + is the integral closure of A + h Y , . . . , Y m i .Let p i ( X ) = X n + s i,n − X n − + · · · + s i, be the characteristic polynomial of Y i for i = 1 , . . . , m . we have already seen that s i,j ∈ S + . Let ˜ S + := A + h s i,j i ⊂ S + be the Tate algebra over A + . it is enough to show that R + is the integral closure of ˜ S + . It iseasy to see that S = ˜ S + [1 /p ], then ˜ S + is an open subring of S contained in S + . Moreover, Y i is integral over˜ S + for all i , therefore A + h Y , . . . , Y m i is integral over ˜ S + and so is R + . This shows that S + is the integralclosure of ˜ S + and that ( S, S + ) is of finite type over ( A, A + ). (cid:3) Corollary 2.2.1.
Let W be an adic space locally of finite type over K . Let { U i } i ∈ I be an ´etale cover of W whose elements factor as maps U i → V i ⊂ W with U i → V i finite ´etale and V i ⊂ W an open subspace. Then anaffine descent datum of finite type ( U , δ ) over { U i } i ∈ I is effective.Proof. Without lose of generality we may assume that V i is affinoid. By Proposition 2.2.2 the affinoid U × W U i descent to an affinoid V i of finite type over V i , as { V i } i ∈ I is a covering in the analytic topology, we can gluethe affinoids V i to an space V affine and of finite type over W . It is clear that V × W U i = U| U i which finish thecorollary. (cid:3) Definition 2.2.5. (1) An adic space X is said to be a quasi-Stein space if it can be written as a countableincreasing union of affinoids X n = Spa( C n , C + n ) by open immersions with C n +1 dense in C n .(2) A morphism p : X → W is called locally quasi-Stein if there exists an analytic cover of W by affinoids { U i } i ∈ I such that p − ( U i ) is a quasi-Stein space.(3) Let Y = ∪ Y n be another quasi-stein space, a morphism of quasi-stein spaces f : X → Y is said torespect the quasi-stein presentation if it maps X n → Y n . We say that f is a quasi-stein isomorphism ifit maps X n isomorphically onto Y n .Let G be a quasi-stein analytic group over K , and let T be an ´etale torsor over W . Definition 2.2.6.
We say that T admits a quasi-Stein trivialization if there exists an ´etale cover { U i } i ∈ I trivializing T such that the descent isomorphism G| U i × W U j p ∗ −→ T | U i × W U j ( p ∗ ) − −−−−→ G| U i × W U j is quasi-Stein. Proposition 2.2.3.
Let T be as before. Let { U i } i ∈ I be an ´etale covering of W which is a quasi-Stein trivial-ization of T . Assume that U i factors as U i → V i ⊂ W with U i → V i finite ´etale and V i ⊂ W an open affinoidsubspace. Then the sheaf µ ∗ F over ( Rig/ W ) et is representable by a locally quasi-Stein adic space over W .Proof. Since we can glue adic spaces in the analytic topology, we may assume that W = Spa( A, A + ) is affinoidand that U → W is a finite ´etale cover which is a quasi-Stein trivialization of T . Let G = ∪ n G n be a quasi-Steinpresentation of G such that the descent morphism G × U p ∗ −→ T | U ( p ∗ ) − −−−−→ U × G is a quasi-Stein isomorphism. Then the composition ( p ∗ ) − ◦ p ∗ gives a descent data for G n × U . By Proposition2.2.2 the descent data is effective and there exists an affinoid adic space V n of finite type over W such that V n × W U = G n × U , by descent of properties of morphisms [Con06] the natural map V n → V n +1 is an openimmersion, and the torsor T is represented by ∪ n V n . By extending U to a Galois cover of W , and by an argumentof traces one shows that O ( V n +1 ) is dense in O ( V n ) proving that T = ∪ n V n is a quasi-Stein presentation. (cid:3) Continuous H om -sheaf. Definition 2.2.7 ( H om sheaves) . Let F and G be two sheaves over ( Rig/ W ) proket . We define the internalpresheaf of homomorphisms from F to G by H om( F , G )( U ) := Hom U ( F | U , G | U )where ·| U is the restriction to ( Rig/U ) proket . Similar for the other big sites. Remark . Morphisms of sheaves over any site can be glued, therefore the pre-sheaf H om ( F , G ) is actuallya sheaf. Lemma 2.2.3 (Pushforward of a localization) . Let F be a sheaf over ( Rig/ W ) proet and W ′ ∈ ( Rig/ W ) proet an object with structural morphisms π : W ′ → W . Then H om ( W ′ , F ) = π ∗ ( F | W ′ ) . Proof.
Let U ∈ ( Rig/ W ). Then H om ( W ′ , F )( U ) = Hom U ( W ′ × W U, F | U )= F | U ( W ′ × W U )= F ( W ′ × W U )= π ∗ ( F | W ′ )( U ) . (cid:3) Example 2.2.3.
Let W ′ ∈ ( Rig/X ) proket and let π : ( Rig/ W ′ ) proket → ( Rig/X ) proket be the natural map ofsites, then H om( W ′ , D W ) = π ∗ O + W ′ , H om( W ′ , A ,an W ) = π ∗ O W ′ , H om( W ′ , b O (+) W ) = π ∗ b O (+) W ′ . We will need to work with topological sheaves later on. Let us give a definition
Definition 2.2.8 (Topological sheaves) . Let W be a quasi-separated adic space locally of finite type over K .Let B be a basis of ( Rig/ W ) proket consisting on log affinoid perfectoid objects, we have an isomorphism of topoi Sh ( B ) ∼ = Sh (( Rig/ W )). Let T op Haus be the category of Hausdorff topological spaces.(1) A Hausdorff topological sheaf on (
Rig/ W ) is a functor F : ( Rig/ W ) opproket → T op Haus such that for any object U ∈ ( Rig/ W ) proket and any cover { U i } i ∈ I , the diagram F ( U ) → Y i F ( U i ) ⇒ Y i,j F ( U i × U U j )is an equalizer in T op Haus . Equivalently, a Hausdorff topological sheaf it is a functor F : B op → T op Haus satisfying the following conditions:(i) Given
U, V ∈ B we have F ( U F V ) = F ( U ) × F ( V ).(ii) Let V ∈ B , and let { V j → V i → V } be a truncated finite hypercover of lenght 2. Then thediagram F ( U ) → Y i F ( V i ) ⇒ Y j F ( V j )is an equalizer in T op Haus .(2) Let F and G be two Hausdorff topological sheaves on B . A continuous morphism of sheaves f : F → G is a morphism of sheaves f such that for all V ∈ B the map f ( V ) : F ( V ) → G ( V ) is continuous.(3) We say that a subsheaf F ⊂ G of a Hausdorff topological sheaf G is dense if it is when evaluating atall V ∈ B (Note that B may depend on F and G ).(4) Let F and G be two Hausdorff topological sheaves. The pre-sheaf of continuous internal homomorphismsis defined as H om ( F , G )( V ) = Hom U ( F | V , G | V )where Hom V ( F | V , G V ) denote the set of continuous morphisms from F | V to G | V .We will make use of uniform continuity to extend maps of sheaves to some completions. Let us introducethe notion of uniform sheaf Definition 2.2.9.
We keep the previous notation(1) A Hausdorff topological sheaf F is said to be a uniform sheaf if for all V ∈ B the space F ( V ) is auniform space (see [Bou71, Chapter II]).(2) Let F and G be uniform sheaves. A morphism f : F → G is uniformly continuous if f ( V ) is uniformlycontinuous for all V ∈ B .(3) We say that a uniform sheaf G is complete if for every V ∈ B the uniform space G ( V ) is complete. Example 2.2.4.
Let U = Spa( R, R + ) be a log affinoid perfectoid adic space over W . Then b O + X ( U ) = R + is naturally a metrizable topological ring when endowed with its p -adic topology. Let { V j → V i → U } be atruncated hypercover in B with V i = Spa( S i , S + i ) and V j = Spa( S ′ j , S ′ + j ). Then the diagram R + → Y i S + i ⇒ Y j S + j is an equalizer of topological rings. Indeed, as b O + X is a sheaf for the pro-´etale topology it is an equalizer of theunderlying rings. Since there is no p -torsion, we see that the p -adic topology of R + is the induced topology UAL EICHLER-SHIMURA MAPS FOR THE MODULAR CURVE 13 of the finite product Q S + i . The sheaf b O + X is complete with respect to this topology. Furthermore, writing U = lim ←− Spa(
R, R + i ) as an inverse limit of objects in ( Rig/X ) ket we have that O + X ( U ) = lim −→ i R + i , this implies that O + X ⊂ b O + X is a dense subring. Lemma 2.2.4 (Uniform extension) . Let F and G be uniform sheaves over B . Let F ′ ⊂ F be a dense subsheafand suppose that G is complete. Let f : F ′ → G be a uniformly continuous map, then f extends uniquely to auniformly continuous map ˜ f : F → G .Proof. Given a uniform space X we denote by X its topological completion. Let U ∈ B , by uniform continuity f ( U ) extends uniquely to a uniformly continuous function f ( U ) : F ′ ( U ) → G ( U )Since F ′ is dense in F , we have F ( U ) ⊂ F ′ ( U ). We define ˜ f ( U ) as the composition ˜ f ( U ) : F ( U ) → F ′ ( U ) → G ( U ). Given a fixed U , ˜ f ( U ) is the unique function extending f ( U ). Let V /U be an object in B living over U ,then both compositions F ( U ) Res UV −−−→ F ( V ) ˜ f ( V ) −−−→ G ( V ) F ( U ) ˜ f ( U ) −−−→ G ( U ) Res UV −−−→ G ( V )extend the arrow Res UV ◦ f ( U ) : F ′ ( U ) → G ( V ). Since F ′ ( U ) is dense in F ( U ) this implies Res UV ◦ ˜ f ( U ) =˜ f ( V ) ◦ Res UV and ˜ f is a uniformly continuous morphism of sheaves extending f . Uniqueness is clear. (cid:3) Proposition 2.2.4.
Let W ′ ∈ ( Rig/ W ) proket , then every continuous function of sheaves f : b O + W ′ → b O + W ′ isuniformly continuous. Therefore H om ( b O + W ′ , b O + W ′ ) = H om( O + W ′ , b O + W ′ ) = b O + W ′ h T i . Where the last sheaf is defined as b O + W ′ h T i (Spa( R, R + )) := R + h T i endowed with the p -adic topology, for Spa(
R, R + ) a log affinoid perfectoid.Proof. By Lemma 2.2.4, to get the first equality it suffices to show that all the morphisms f : O + W ′ → b O W ′ are uniformly continuous. So it is enough to prove the second equality. Note that O + W ′ is represented by D W ′ = W ′ × K D K , let π : D W ′ → W ′ be the projection. Lemma 2.2.3 says that H om( D W ′ , b O + W ′ ) = π ∗ b O + D W′ . Take V = Spa( R, R + ) / W ′ a log affinoid perfectoid space, then π ∗ ( b O + D W′ )( V ) = b O + ( V × D K ) . Let V = lim ←− i Spa( R i , R + i ) be a pro-Kummer ´etale presentation, then V × D K = lim ←− i Spa( R i h T i , R + i h T i ). Thisimplies b O + ( V × D K ) = lim ←− n (lim −→ i ( R + i /p n )[ T ]) = lim ←− n ( R + /p n [ T ]) = R + h T i proving the proposition. (cid:3) Product of sites.
Let K be a finite extension of Q p and O K its ring of integers, Let Y and V be two adicspaces locally of finite type over Spa( K, O K ). Definition 2.2.10.
Let ♣ , ♠ ∈ { an, et, ket, proet, proket } . We define the product site Y ♣ × V ♠ as the categorywhose objects are the pairs ( U, W ) ∈ Y ♣ × V ♠ , and morphisms the pairs of maps ( f, g ) : ( U , W ) → ( U , W )where f : U → U and g : W → W are morphisms in Y ♣ and V ♠ respectively. We endow Y ♣ × V ♠ withthe topology such that { ( U i , W i ) → ( U, W ) } is a covering if and only if the maps U i → U are of type ♣ , resp. W i → W are of type ♠ , and {| U i | × | W i | → | U | × | W |} is a topological cover. We define in a similar way thesites ( Rig/Y ) ♣ × V ♠ , and ( Rig/Y ) ♣ × ( Rig/V ) ♠ . Remark . Let (cid:7) ∈ { an, et, ket, proet, proket } and suppose that the property of maps (cid:7) implies ♣ and ♠ (e.g. (cid:7) = proket ). Then there is a natural morphism of sites( Y × K V ) (cid:7) → Y ♣ × V ♠ , similar for the big sites. Let p ≥ F and abelian sheaf over Y ♣ × V ♠ . Let p and p denote the natural projections of Y ♣ × V ♠ onto Y ♣ and V ♠ respectively. Consider the presheaf of V ♠ with values in AbSh ( Y ♣ ) defined by R p F : V ′ R p p , ∗ F | Y ♣ × V ′♠ . Let R p F = ( R p F ) denote its sheafification. Given U ∈ Y ♣ and p ≥
0, let H p ( U ♣ , R q F ) : V ♠ → Ab be thesheafification of V ′ H p ( U ♣ , R q F ( V ′ )) . Note that as H ( U ♣ , − ) : AbSh ( Y ♣ ) → Ab is left exact, we have H ( U ♣ , R p F )( V ′ ) = H ( U ♣ , R p F ( V ′ )) . Lemma 2.2.5.
There exists a Leray spectral sequence of abelian sheaves over V ♠ E p,q = H p ( Y ♣ , R q F ) ⇒ R p + q p , ∗ F Proof.
The sheaf R p + q p , ∗ F is the sheafification of the presheaf V ′ H p + q ( Y ♣ × V ′♠ , F ) . We have a Leray spectral sequence E p,q ( V ′ ) = H p ( Y ♣ , R q p , ∗ F | Y ♣ × V ′♠ ) ⇒ H p + q ( Y ♣ × V ′♠ , F ) . Let E p,qr be the sheafification of V ′ E p,qr ( V ′ ). Since sheafification is an exact functor, we have a Leray spectralsequence { E p,qr } converging to R p + q p , ∗ F . Then, it is enough to show that E p,q := ( H p ( Y ♣ , R q F )) = H p ( Y ♣ , R q F ) . Let U • be an hypercover of Y , in Y ♣ and let G ∈ Sh ( Y ♣ ). We denote by C ( U • , G ) the ˇCech complex attachedto U • and G , and by ˇ H p ( U • , G ) = H p ( C ( U • , G )). From [Sta20, Tag 01H0] we know that H p ( Y ♣ , G ) = lim −→ U • ˇ H p ( U • , G ) , where U • runs over all the hypercovers of Y . Fix U • an hypercover as above, consider the presheaves over V with values in complexes of abelian groups: C ( U • , R q F ) : V C ( U • , R q p , ∗ F | Y ♣ × V ′♠ )(2.7) C ( U • , R q F ) : V ′ C ( U • , R q F ( V ′ )) . (2.8)As H ( U n , − ) is left exact for all n , the presheaf (2.8) is actually a complex of sheaves. Furthermore, it is thesheafification of (2.7). Let C • be a complex of sheaves over V ♠ , we denote by H p ( C • ) and H p ( C • ) its p -thcohomology as presheaves and sheaves respectively, note that H p ( C • ) = ( H p ( C • )) . We have H p ( Y ♣ , R q F ) = ( H p ( Y ♣ , R q F )) = (lim −→ U • ˇ H p ( U • , R q F )) = lim −→ U • ( H p ( C ( U • , R q F ))) = lim −→ U • H p ( C ( U • , R q F ))= lim −→ U • ( H p ( C ( U • , R q F ))) = ( H p ( Y ♣ , R q F )) which finish the proof. (cid:3) Proposition 2.2.5.
Keep the preceding notation. Let F an abelian sheaf on Y proket × V an . Suppose thatthere exists pro-Kummer-´etale covering { U i → Y } i ∈ I such that for any open affinoid V ′ ⊂ V , the sheaf p , ∗ F | U i,proket ×V ′ an admits a d´evisage in finitely many steps to F p by colimits and countable cofiltered limitssatisfying the Mittag-Leffler condition. Then R q F = 0 for q > and R p p , ∗ F is the sheafification of V ′ H p ( Y proket , p , ∗ F | Y proket × V ′ an ) . Proof.
By Lemma 2.2.5 it is enough to show R q F = 0 for q >
0. Let x ∈ V , then(2.9) R q F x = lim −→ x ∈ V ′ R q p , ∗ F | Y proket × V ′ an . The problem is local on Y proket , thus we can assume that Y is quasi-compact and that there exists a pro-finiteKummer-´etale cover U → Y of Y such that p , ∗ F | U proket × V ′ an admits a d´evisage to F p as in the statement,for all affinoid x ∈ V ′ ⊂ V . Moreover, by further localizing U we may assume that H q ( U proket , F p ) = 0 for UAL EICHLER-SHIMURA MAPS FOR THE MODULAR CURVE 15 q >
0. (cf. Theorem [DLLZ19, Prop. 5.3.13]). As U is quasi-compact, a d´evisage argument implies that H q ( U proket , p , ∗ F | U proket × V ′ an ) = 0 for all V ′ ⊂ V affinoid.To prove the vanishing of (2.9) it is enough to show thatlim −→ x ∈ V ′ H q ( U proket × V ′ an , F ) = 0 . Let U ≤ q • → U be a q -truncated hypercover of U , we denote by U • its extension to a hypercover via the coskeleton.We write ˇ H q ( U ≤ q • , G ) := ˇ H q ( U • , G ). Then H q ( U proket , p , ∗ F | U proket × V ′ an ) = lim −→ U ≤ q • → U ˇ H q ( U ≤ q • , p , ∗ F | U proket × V ′ an )= lim −→ U ≤ q • → U ˇ H q ( U ≤ q • × V ′ , F ) , where U ≤ q • runs over all the q -truncated hypercovers of U in Y proket . On the other hand, let x ∈ V ′ be anaffinoid. We have H q ( U proket × V ′ an , F ) = lim −→ ( U × V ′ ) ≤ q • ˇ H q (( U × V ′ ) ≤ q • , F )where ( U × V ′ ) ≤ q • runs over all the q -truncated hypercovers of U × V ′ in Y proket × V an . As U is quasicompact,given ( U × V ′ ) ≤ q • as before, there exists a refinement of ( U × V ′ ) ≤ q • locally around x ∈ V ′ of the form U ≤ q • × V ′′ where U ≤ q • is a q -truncated hypercover of U , and x ∈ V ′′ ⊂ V ′ an open affinoid. Therefore, we deduce that0 = lim −→ x ∈ V ′ H q ( U proket , p , ∗ F | U proket × V ′ an ) = lim −→ x ∈ V ′ lim −→ U ≤ q • → U ˇ H q ( U ≤ q • × V ′ , F )= lim −→ x ∈ V ′ lim −→ ( U × V ′ ) ≤ q • ˇ H q (( U × V ′ ) ≤ q • , F )= lim −→ x ∈ V ′ H q ( U proket × V ′ an , F )which proves the proposition. (cid:3) Spectral theory over adic spaces.
In the study of overconvergent theories over Shimura varieties,the main objects involved have representations as complexes of topological vector spaces. A very pleasantapproximation to their spectral theory these objects is via Banach sheaves and the finite slope theory of Coleman[Col97], systematised by Buzzard [Buz07]. We will follow Appendix B of [AIP18] and Section 2 of [GB20].Let ( K, O K ) be a non-archimedean field of mixed characteristic (0 , p ) and rank 1 valuation. Let W be areduced adic space locally of finite type over K .2.3.1. Banach sheaves.
Definition 2.3.1. (1) Let A be a Tate algebra topologically of finite type over K . A projective Banach A -module is a topological A -module V which is a direct sumand of a ON Banach module b ⊕ i ∈ I A forsome index set I .(2) A Banach sheaf over W is a topological sheaf F satisfying:i. for all quasi-compact open subset U ⊂ W , F ( U ) is a Banach O W ( U )-module.ii. There exists an affinoid covering W = S i U i such that for all i and any open affinoid V ⊂ U , thenatural map O W ( V ) b ⊗ O W ( U i ) F ( U i ) → F ( V )is a topological isomorphism. In other words, F | U i is attached to their global sections.(3) A Banach sheaf F over W is called projective if there exists a cover as in (2) such that F ( U i ) is aprojective Banach O W ( U i )-module.We are mainly interested in the finite slope part of compact operators acting in overconvergent cohomologies.We need the notion of a compact operator in projective Banach sheaves: Definition 2.3.2. (1) Let (
A, A + ) be a reduced affinoid K -algebra of topologically of finite type, and let M and N be projective Banach A -modules. A continous map φ : M → N is said to be relativelycompact with respect to ( A, A + ) (or simply compact if the context is clear), if there are open andbounded A + -submodules M + ⊂ M and N + ⊂ N such that φ ( M + ) ⊂ N + , and for all n ∈ N the imageof M + φ −→ N + → N + /p n N + is a finite A + /p n A + -module. Since A is reduced, this property is independent of the subring of definition A of A . (2) Let F and G be two projective Banach sheaves on W . Let φ : F → G be a continuous morphism of O W -modules. We say that φ is compact if there exists an affinoid cover W = S i U i as in Definition2.3.1 (2) for both F and G , such that φ ( U i ) : F ( U i ) → G ( U i )is a compact map of O W ( U i )-modules.To compute slope stimates of compact operators one needs to work with open bounded submodules ofprojective Banach modules. The kind of open and bounded subsheafs considered in higher Coleman theoryappears as sheaves of O + W et -modules which, locally ´etale on W , are orthonormal sums of copies of O + W et . Evenin classical Coleman theory, the sheaf of bounded overconvergent modular forms (cf. Definition 3.4.3) is a ONdirect sum of copies of O + W et after the finite ´etale localization induced by the inclusion of the arithmetic groupsΓ ( p n ) ⊂ Γ ( p n ). Definition 2.3.3 (Definition 2.26 of [GB20]) . Let F be a projective Banach sheaf over W , we see F as an´etale sheaf via its natural coherent extension. An integral structure on F is an O + W et -subsheaf F + ⊂ F suchthat i. F + ⊗ O K K = F andii. there is an ´etale cover { U i } i of W by affinoids such that F + ( U i ) is the p -adic completion of a free O + W ( U i )-module, and the canonical map F + ( U i ) b ⊗ O + W ( U i ) O + U i → F + | U i is an isomorphism. Lemma 2.3.1 ( [GB20, Lemma 2.28]) . Let F be a projective Banach sheaf and F + an integral structure on F . Then for any quasi-compact ´etale map U → W , F + ( U ) is an open and bounded submodule of F ( U ) . Inparticular, the restriction of F + to the analytic site W an is an open and bounded subsheaf of F . The following lemma allows to compare the cohomology groups of a projective Banach sheaf and an integralstructure
Lemma 2.3.2 ( [GB20, Lemma 2.30]) . Let W be an affinoid smooth adic space over K , and let F be a projectiveBanach sheaf over W associated to its global sections. Let F + be an integral structure of F . There exists N > such that for all i > the cohomology groups H i ( W an , F + ) are annihilated by p N . Spectral theory.
Let’s recall the spectral theory of adic spaces of [AIP18]. Let Spa(
A, A + ) be a reducedaffinoid space of finite type over Spa( K, O K ) . Let Ban( A ) denote the category of Banach A -modules, it is aquasi-abelain category, a sequence V ′ → V → V ′′ of A -Banach spaces being exact if and only if the naturalmap V ′ / ker f → ker g is an isomorphism of Banach A -modules. A morphism of A -Banach spaces f : V ′ → V is strict if it identifies V ′ / ker f onto a closed subspace of V . A bounded complex of projective Banach A -modules is said to be aperfect complex. We define the derived category D b (Ban( A )) as the localization of the homotopy category ofBounded Banach A -modules with respect to acyclic bounded complexes of Banach A -modules, cf. [Urb11]. Definition 2.3.4. (1) A formal series F = P n ≥ a n T n ∈ A [[ T ]] is Fredholm if:i. a = 1,ii. for all r ∈ Z , a n p rn → n → ∞ .(2) The eigenvariety V ( F ) is the closed subspace of A A defined as the glueing of the affinoid subspaces { V n ( F ) ⊂ Spa( A h p n T i , A + h p n T i ) } n ∈ N where V n ( F ) = Spa( A h p n T i / ( F ) , ( A h p n T i ) + )with ( A h p n T i ) + being the integral closure of A + h p n T i in A h p n T i / ( F ).The notion of Fredholm series can be extended to general adic spaces, as a glueing of Fredholm series overaffinoids. The eigenvariety V ( F ) has the following properties: Theorem 2.3.1 ( [AIP18, Th´eor`eme B.1]) . The morphism ω : V ( F ) → Spa(
A, A + ) is locally quasi-finite, flatand partially proper. For all x ∈ V ( F ) and ω ( x ) ∈ Spa(
A, A + ) , there exists a neighbourhood V of x in V ( F ) containing { x } , and a neighbourhood U of w ( x ) in Spa(
A, A + ) such that ω ( V ) ⊂ U and ω | V : V → U is finiteflat. the previous theorem is an enhancement to adic spaces of the Coleman’s finite slope decomposition for rigidspaces via resolvants. Indeed, one has the following result UAL EICHLER-SHIMURA MAPS FOR THE MODULAR CURVE 17
Corollary 2.3.1 ( [AIP18, Corollary B.1]) . Let x ∈ V ( F ) , U and V as above. Suppose further that V =Spa( C, C + ) and U = Spa( B, B + ) , and that V is of constant rang d over U . Then the Fredholm series F has afactorization with coefficients in B : F = QP where Q = 1 + b T + · · · + b d T d ∈ B [ T ] , b d ∈ B × , S is a Fredholmseries with coefficients in B prime to Q and C = B [ T ] / ( Q ( T )) . The Fredholm series we will be considering arise as Fredholm determinants of compact operators of projectiveBanach sheaves. In the following we assume that W is a reduced adic space locally of finite type over Spa( K, O K ), F is a projective Banach sheaf over W , and φ : F → F is a compact operator. Lemma 2.3.3.
There exists a series F ( T ) ∈ O W ( W )[[ T ]] such that for any open affinoid cover { U i } i ∈ I as inDefinition 2.3.2 (2) one has F ( T )( U i ) = det(1 − T φ ) ∈ O W ( U i )[[ T ]] . Moreover, for any quasi-compact U ⊂ W the series F ( T )( U ) is a Fredholm series.Proof. Let us first recall how det(1 − T φ ) is computed for a compact operator φ : V → V of a projective Banach A -module. Let V ′ be a projective Banach A -module such that ˜ V = V ⊕ V ′ is an ON Banach A -module withbasis { e j } j ∈ J . Let ˜ φ : ˜ V → ˜ V be the composition˜ V → V φ −→ V → ˜ V , then ˜ φ is a compact operator of ˜ V . Given a finite subset F ⊂ J , let φ F denote the composition of ˜ φ with theorthogonal projection ˜ V → L j ∈ F Ae j . Thendet(1 − T φ ) = lim F ⊂ J det(1 − T φ F ) . Since Im ˜ φ ⊂ V and ˜ φ | V = φ , the power series above only depends on ( V, φ ).Let { U i } i ∈ I be a cover of W as in the statement., i.e. such that F | U i is attached to their global sections and φ ( U i ) : F ( U i ) → F ( U i ) is a compact operator for all i ∈ I . For any affinoid subset V ⊂ U i we have thatdet(1 − T φ ( U i )) | V = det(1 − T φ ( V )) . Thus, the series det(1 − T φ ( U i )) glue to a power series F ( T ) ∈ O W ( W )[[ T ]] such that F ( T ) | U i = det(1 − T φ ( U i )).Let U ⊂ W be a quasi-compact open subset and let U = U ′ ∪ · · · U ′ s be an affinoid covering by affinoidsrefining { U i ∩ U } i ∈ I . Then we have an exact diagram of topological rings O W ( U ) → s Y k =1 O W ( U ′ k ) ⇒ s Y k,t =1 O W ( U ′ k ∩ U ′ t ) . Since F ( T ) | U ′ k is a Fredholm series for all k = 1 , . . . , s , one gets that F ( T ) | U is a Fredholm series by controllinguniformly the coefficients with the finite cover. (cid:3) Corollary 2.3.2.
With the previous notation, the Fredholm series F ( T ) = det(1 − T φ ) defines an eigenvariety V ( F , φ ) over W satisfying the same properties of Theorem 2.3.1. We can construct the spectral sheaf of non zero eigenvectors of ( F , φ ) as a coherent sheaf over V ( F , φ ) asfollows: Theorem 2.3.2 ( [AIP18, Theorem B.2]) . There is a functor from the category of projective Banach sheavesover W endowed with a compact operator to the category of coherent modules over A W whose support is quasi-finite and partially proper over W .Proof. Let F be a projective Banach sheaf over W and let φ be a compact operator, let F ( T ) ∈ O W ( W )[[ T ]]be the Fredholm series of ( F , φ ). Let x ∈ V ( F , φ ) and y = ω ( x ) ∈ W , by Theorem 2.3.1 there exists anaffinoid neighbourhood x ∈ V ⊂ V ( F , φ ) containing x , and an affinoid neighbourhood y ∈ U ⊂ W such thatthe restriction ω : V → U is finite flat of constant rank d . Assume that F | U is attached to its global sections.Corollary 2.3.1 says that F | U has a factorization F = QP where Q is a Fredholm polynomial of degree d andinvertible leading coefficient, and P is a Fredholm polynomial prime to Q . Let Q ∗ ( T ) = T d Q (1 /T ), by theRiesz theory in families developed in Section A of [Col97], there exists a unique direct sum decomposition F ( U ) = M ( Q ) ⊕ N ( Q )where M ( Q ) is a projective O W ( U )-module locally of rank d , such that Q ∗ ( φ ) acts by zero on M ( Q ) andinvertible on N ( Q ). Hence, M ( Q ) has a natural structure of finite O V ( V )-module and defines a coherent sheafover V , they glue to a coherent sheaf F fs over V ( F , φ ). We also denote F fs the pushforward to A W , then werecover V ( F , φ ) = Supp( F fs ).Let G be another projective Banach sheaf over W endowed with a compact operator ψ . Let f : F → G be acontinuous map of Banach sheaves commuting with the compact operators. It is enough to define the map f fs : F fs → G fs locally around V ( F , φ ) ∩ V ( G , ψ ) as it can be extended by zero to A W . Let x ∈ V ( F , φ ) ∩ V ( G , ψ )and let ω ( x ) ∈ U ⊂ ω ( V ( F , φ )) and ω ( x ) ∈ U ⊂ ω ( V ( G , ψ )) be as in Corollary 2.3.1, set U = U ∩ U . Let { x } ⊂ V ⊂ V ( F , φ ) and { x } ⊂ V ⊂ V ( G , ψ ) be affinoid neighbourhoods finite flat over U of constant rank d and e respectively, set V = V ∩ V . Let F ( T ) = Q P and F ( T ) = Q P be the factorization as the corollaryof the Fredholm determinants of φ and ψ respectively. Then we have decompositions F | U = M ( Q ) ⊕ N ( P ) and G | U = M ( Q ) ⊕ N ( P ) . Finally, choose an quasi-compact open subset ˜ V ⊂ A W such that ˜ V ∩ V ( F , φ ) ∩ V ( G , ψ ) = V . Then we define f fs | V as the composition M ( Q ) → F ( U ) f −→ G ( U ) → M ( Q ) , where the first arrow is the inclusion and the last arrow the orthogonal projection. The map f is well defineand does not depend on the factorization of F locally around ω ( x ). Indeed, as F fs is finite flat over W andthis last is reduced, we can check this at the fibers of ω : A W → W . The claim follows from the fact that f maps λ -eigenspaces of F y to λ -eigenspaces of G y for any constant λ ∈ k ( y ), for y ∈ W fixed. Thus, we can gluethe f fs | V to a map f fs : F fs → G fs in a natural way. (cid:3) Corollary 2.3.3.
The functor ( F , φ ) F fs is exact.Proof. By the proof of the previous theorem, one reduces to the same statement over an affinoid field W =Spa( C, C + ). The corollary follows from the spectral theory of a ON Banach space over a field. (cid:3) Remark . When W = Spa( A, A + ), the restriction F ≤ n of F fs to the affinoid W ×
Spa( K h p n T i , O K h p n T i )is called the ≤ n -slope part of F .Spectral theory over projective Banach sheaves is not enough for our purposes. Indeed, the overconvergent´etale cohomologies and the overconvergent analytic cohomologies we are interested in do not have a naturalstructure of Banach sheaves in general. Instead, the cohomology complexes are represented by perfect Banachcomplexes, or (co)limits of them. We use the previous theorem to define eigenvarieties and finite slope sheavesin this situation Definition 2.3.5.
Let W be a reduced adic space locally of finite type over Spa( K, O K ).(1) A perfect complex of Banach sheaves over W is a bounded complex C • whose modules are projectiveBanach sheaves. A morphism of perfect Banach complexes φ • : C • → D • is compact if φ n : C n → D n is compact for all n .(2) Let C • be a perfect complex of Banach sheaves over W and φ • a compact endomorphism of C • .We define the eigenvariety attached to ( H i ( C • ) , φ ) as follows: Let C • ,fs be the bounded complex ofcoherent A W -modules provided by Theorem 2.3.2. Then V ( H i ( C • ) , φ ) is defined as the support of thesheaf H i ( C • ,fs ). Proposition 2.3.1 (cf. Lemma 2.2.8 of [Urb11]) . Let ( C • , φ ) and ( D • , ψ ) be two perfect complexes of Banachmodules endowed with compact endomorphisms as above. Let f : C • → D • be a morphism of Banach complexescommuting with φ and ψ up to homotopy. Then there is a well defined map in cohomology H i ( f fs ) : H i ( C • ,fs ) → H i ( D • ,fs ) , depending only on the homotopy class of f .Proof. Let x ∈ Supp( H i ( C • ,fs )) ∩ H i ( D • ,fs ). As in the Theorem 2.3.2, we may find an open affinoid x ∈ ˜ V ⊂ A W containing { x } , and an open affinoid ̟ ( x ) ∈ U ⊂ W such that ˜ V ∩ Supp( C • ,fs )) and ˜ V ∩ Supp( D • ,fs ) are finiteflat over U . Let F C and F D be the Fredholm series of C • and D • (i.e, the product of Fredholm series of theterms). Then we have factorizations F C = Q C P D and F D = Q C P D as in the theorem, and decompositions ofcomplexes C • | U = M ( Q C ) ⊕ N ( P C ) , D • | U = M ( Q D ) ⊕ N ( P D ) . Then, we define H i ( f sp ) | e V as the i -th cohomology of the composition M ( Q C ) → C • f −→ D • → M ( Q D ) . The fact that f commutes with φ and ψ up to homotopy implies that H i ( f sp ) | e V glue to a unique map ofcoherent sheaves over A W \{ } , independent of e V and U . (cid:3) Definition 2.3.6. (1) A perfect complex of compact type over W (resp. a perfect nuclear Fr´echet complex)is a countable direct system of perfect Banach complexes (resp. a countable inverse system) C • = { C • n } n ∈ N such thati. The complexes C • n are uniformly bounded,ii. The transition maps C • n → C • n +1 (resp. C • n +1 → C • n ) are compact injections with dense image. UAL EICHLER-SHIMURA MAPS FOR THE MODULAR CURVE 19 (2) A compact endomorphism of a perfect complex of compact type over W (resp. a perfect nuclear Fr´echetcomplex) is a compatible system of endomorphisms φ n : C • n → C • n such that there is a factorization C • n C • n C • n +1 C • n +1 φ n φ n +1 resp. C • n +1 C • n +1 C • n C • nφ n +1 φ n We denote C • by lim −→ n C • n (resp. lim ←− n C • n ). Proposition 2.3.2.
Let C • = { C n } n ∈ N be a perfect complex compact type or a perfect nuclear F´echet complexover W . Let φ : C • → C • be a compact endomorphism. Then the natural maps C • ,fsn → C • ,fsn +1 (resp. C • ,fsn +1 → C • ,fsn ) are isomorphisms. We also write C • ,fs = lim −→ n C • ,fsn (resp. C • ,fs = lim ←− n C • ,fsn ).Proof. We prove the case of direct systems, the projective limit case is shown in a similar way. Note that themaps φ n : C • n → C • n are compact. We have induced maps of complexes φ fsn : C • ,fsn → C • ,fsn − → C • ,fsn . But φ fsn is an isomorphism of C • ,fsn , then C • ,fsn − → C • ,nn is an isomorphism. (cid:3) Spectral theory in non-Banach spaces.
In the definition of overconvergent modular symbols we also con-sider different kind of functional spaces which are not Banach, Fr´echet or a compact direct limit Banach spaces.Nevertheless, spectral theory, and more precisely finite slope theory, still makes sense in topological vectorspaces endowed with a compact operator which factor through a Banach space.
Definition 2.3.7.
Let V be a Hausdorff topological K -vector space. A spectral operator on V is a continuous K -linear endomorphism φ such thati. φ factors through a Banach space V and a continuous injective K -linear map V → V .ii. The restriction of φ to V is a potent compact operator (i.e. φ n is compact for some n ≥ V, φ ) a spectral space.
Proposition 2.3.3.
Let ( V, φ ) a spectral space. Let V and V be two (not necessarily closed) Banach subspacesthrough which φ factors, and φ | V i is potent compact. Then V fs = V fs .Proof. We may assume that φ | V is compact. Consider the natural map V ⊕ V → V and let V be its coimage.It is enough to prove the case V ⊂ V . But then we have a factorization V φ −→ V → V . The propositionfollows since φ fs is an isomorphism. (cid:3) The previous proposition says that the finite slope part of (
V, φ ) is actually well defined, and does not dependon the Banach space V . Let ( A, A + ) be a reduced affinoid K -algebra. Definition 2.3.8.
Let V be a Hausdorff topologial A -module. A spectral operator on V is a continuous A -linearendomorphism φ such thati. φ factors through a (not necessarily closed) projective Banach A -submodule V ⊂ V .ii. The restriction of φ to V is a potent compact operator.We call ( V, φ ) a spectral datum and V a Banach subspace of definition.The analogous of Proposition 2.3.3 holds for families: Proposition 2.3.4.
Let ( V, φ ) be a spectral datum and V , V Banach subspaces of definition. Then there is anatural isomorphism V fs = V fs of sheaves over A A,A + ) \{ } Proof.
The map φ : V → V induces a map of sheaves φ fs : V fs → V fs . Moreover, we also have φ fs : V fs → V fs and their composition corresponds to φ ,fs : V i → V i which is an isomorphism. Therefore, the naturalisomorphism is given by V fs φ fs −−→ V fs φ fs, − −−−−→ V fs . (cid:3) Definition 2.3.9.
A spectral complex of A -modules is a bounded complex of Hausdorff topological A -modules C • endow with a endomorphism φ such that ( C n , φ ) is a spectral datum for all n . We say that ( C • , φ ) is astrong spectral complex if φ factors trough a (not necessarily closed) perfect Banach subcomplex C ′• ⊂ C • suchthat φ | C • , ′ is compact. We call ( C • , φ ) a perfect Banach complex of definition. Lemma 2.3.4.
Let ( V, φ V ) and ( W, φ W ) be two spectral datums. Let f : V → W be a continuous A -linear mapcommuting with the action of φ . Then there is a natural map f sp : V sp → W sp of sheaves over A A,A + ) \{ } . Proof.
Let V ⊂ V and W ⊂ W be Banach subspaces of definition, then φ W ◦ f maps V to W and commuteswith the action of φ . Then the natural map is given by φ sp, − W ◦ ( φ W ◦ f ) sp : V sp → W sp . (cid:3) Let ( C • , φ ) be a spectral complex of A -modules. The previous lemma shows that it has an attached finiteslope complex C • ,fs over A A,A + ) \{ } . Moreover, if ( C • , φ ) is a strong spectral complex and ( C • , φ ) is aperfect Banach complex of definition, we have that C • ,fs = C • ,fs . The analogous of Proposition 2.3.1 also holdsin the situation of spectral datums by reducing to the Banach case. Definition 2.3.10.
Let W be an adic space locally of finite type over Spa( K, O K ). A spectral sheaf of O W -modules over W is a topological O W -module F , endowed with an endomorphism φ : F → F satisfying thefollowing conditionsi. There exists an open affinoid cover { U i } of W , and projective Banach sheaves G i over U i such that theendomorphism φ factors through φ : F | U i → G → F | U i . ii. The composition G i → F | U i φ −→ G i is potent compact.A spectral complex over W is a bounded sheaf of topological O W -modules C • endowed with an endomorphism φ such that ( C n , φ n ) is an spectral sheaf for all n . We say that C • is a strong spectral complex if there existsan affinoid cover as in i) above such that φ | U i factors through a perfect Banach complex D • i such that thecomposition D • i → C | • U i φ −→ D • i is potent compact.2.3.4. Cohomology with supports.
Definition 2.3.11.
Let X be a topological space, Z ⊂ X a closed subset and U = X \ Z . We define the functorof sections with supports in Z AbSh ( X ) −→ Ab F Γ Z ( X, F )where Γ Z ( X, F ) = { s ∈ F ( X ) | Supp s ⊂ Z } . In other words, Γ Z ( X, F ) = ker(Γ( X, F ) → Γ( U, F )) . The functor Γ Z ( X, − ) is left exact, we denote its derived functor by R Γ Z ( X, − ). Proposition 2.3.5.
Keep the previous notation. There is a natural distinguished triangle R Γ S ( X, F ) → R Γ( X, F ) → R Γ( U, F ) + −→ . Proof.
Let j : U → X and ι : Z → X denote the open and closed immersions. Given a topological space Y let Z Y denote the constant sheaf of Z on Y . Consider the proper direct image j ! Z U , it is constructed as thesheafification of j ! Z U ( V ) = ( Z if V ⊂ U R Γ( X, − ) = R Hom( Z X , − ), R Γ( U, − ) = R Hom( j ! Z U , − ) and R Γ Z ( X, − ) = R Hom( ι ∗ Z Z , − ). Moreover, there is a short exact sequence0 → j ! Z U → Z X → ι ∗ Z Z → . By taking R Hom one obtains the desired distinguished triangle. (cid:3)
The formation of cohomology with supports is functorial in X and Z : Lemma 2.3.5.
Let X and Z ⊂ X a closed subset. The following holds (1) Let X ′ be topological spaces, and f : X ′ → X a continuous function. Denote Z ′ := f − ( Z ) . Then thereis a natural transformation R Γ Z ( X, − ) → R Γ Z ′ ( X, − )(2) Let V ⊂ X be an open subset containing Z . Then the natural map R Γ Z ( X, − ) → R Γ Z ( U, − ) is anisomorphism. (3) Let W ⊂ Z be a closed subset. There exists a natural correstriction map Cor : R Γ W ( X, − ) → R Γ Z ( X, − ) . Proof. (1) Let ι : Z → X and ι ′ : Z ′ → X denote the closed immersions. Then f ∗ ι ∗ Z Z = ι ′∗ Z Z ′ , whichimplies what we want. UAL EICHLER-SHIMURA MAPS FOR THE MODULAR CURVE 21 (2) Note that both functors Γ Z ( V, − ) and Γ Z ( X, − ) are equal. Indeed, given a sheaf F over X and asection s ∈ Γ Z ( V, − ), it can be extended by 0 to a section in Γ Z ( X, − ). This implies the equality forderived functors.(3) Let ι : W → X also denote the closed immersion. Part (3) follows from the fact that there is a naturalmap ι ∗ Z W → ι ∗ Z Z . (cid:3) Overconvergent objects
There are two different notions of overconvergence we consider in this document. Namely, the theory ofoverconvergent modular symbols, and the theory of overconvergent modular forms. The former studies ´etalecohomology of locally analytic principal series of overconvergent neighbourhoods of the Borel B ( Z p ) in GL ( Z p ),i.e. the Iwahori subgroups Iw n . The second studies quasi-coherent locally analytic principal series in overcon-vergent neighbourhoods of the ordinary locus of X . We exploit the representability of the structural sheaf ofthe big site ( Rig/X ) proket to define the overconvergent sheaves as spaces of functions of analytic torsors.We keep the notation of Section 2.1. We let C p denote the completion of an algebraic closure of Q p and O C p its ring of integers.3.1. Overconvergent neighbourhoods in the flag variety.
Let n ≥ n ⊂ GL ( Z p )the compact open subgroup Iw n = (cid:18) Z × p Z p p n Z p Z × p (cid:19) . Given δ ∈ Q , we denote by p δ ∈ C p an element with p -adic valuation | p δ | = p − δ . We always assume that p δ is algebraic over Q p . Definition 3.1.1.
Let δ ≥ n be a rational number.(1) We define the analytic Iwahori open subgroup of GL I w n ( δ ) = (cid:18) Z × p (1 + p δ D ) Z p + p δ D p n Z p + p δ D Z × p (1 + p δ D ) (cid:19) (2) We define the complete analytic Iwahori open subgroup c I w n ( δ ) := Z × p (1 + p δ b O + ) Z p + p δ b O + p n Z p + p δ b O + Z × p (1 + p δ b O + ) ! Remark . (1) The sheaf on groups c I w n ( δ ) is a complete uniform sheaf (cf. Definition 2.2.9). The sheafrepresented by I w n ( δ ) is dense in c I w n ( δ ).(2) The Iwahori subgroups {I w n ( δ ) } δ ≥ n form a basis of strict Iw n -invariant neighbourhoods of Iw n ⊂ GL .Let m ≥ δ > N δ = (cid:18) p δ D (cid:19) , N m = (cid:18) p m Z p (cid:19) , T δ = (cid:18) p δ D
00 1 + p δ D (cid:19) , T m = (cid:18) p m Z p
00 1 + p m Z p (cid:19) . We define N δ and N m in a similar way. We denote by b T δ , b N δ and b N δ the completions in ( Rig/X ) proket of T δ , N δ and N δ respectively. Finally, we also write T = T = T ( Z p ), N = N = N ( Z p ) and N = N = N ( Z p ). Remark . Even though the analytic groups above are described using an element in C p , they are welldefined analytic groups over Q p as the analytic parameter δ is rational. Lemma 3.1.1.
Let δ ≥ n be a rational number. There is an Iwahori decomposition I w n ( δ ) = ( N n N δ ) × ( T T δ ) × ( N N δ ) . The same holds for c I w n ( δ ) .Proof. It suffices to prove the equality at (
R, R + )-points, with ( R, R + ) a uniform affinoid ( Q p , Z p )-algebra. Bydefinition we have I w n ( δ ) = (cid:18) Z × p (1 + p δ D ) Z p + p δ D p n Z p + p δ D Z × p (1 + p δ D ) (cid:19) then I w n ( δ )( R, R + ) = (cid:18) Z × p (1 + p δ R + ) Z p + p δ R + p n Z p + p δ R + Z × p (1 + p δ R + ) (cid:19) Let g ∈ I w n ( δ )( R, R + ). Writing g = (cid:18) x (cid:19) (cid:18) x x (cid:19) (cid:18) x (cid:19) and solving the equations one finds x ∈ p n Z p + p δ R + , x and x ∈ Z × p (1 + p δ R + ), and x ∈ Z p + p δ R + . Asimilar argument works for c I w n ( δ ). (cid:3) Let F ℓ = B an \GL an = B\GL be the analytic flag variety. Also write F ℓ std = B\GL . Definition 3.1.2.
Let W = { , w } be the Weyl group of GL and δ ≥ n . Let w ∈ W , we define the δ -analytic w -Bruhat cell of F ℓ (resp. F ℓ std ) to be C w,n ( δ ) := B\B w I w n ( δ ) (cid:0) resp. C wn ( δ ) = B\B w I w n ( δ ) (cid:1) . By the Iwahori decomposition of Lemma 3.1.1 one deduces isomorphisms C ,n ( δ ) ∼ = N n N δ and C w ,n ( δ ) ∼ = N N δ . Similarly we have as subspaces of F ℓ std C n ( δ ) = N N δ and C w n ( δ ) = N n N δ . To define analytic principal series and distributions one has to consider (left) torsors over the analytic Bruhatcells. We set
T C w,n ( δ ) := N \N w I w n ( δ ) and T C wn ( δ ) := N \N w I w n ( ǫ ), they are trivial T T δ -torsors over C w,n ( δ )and C wn ( δ ) respectively. Let ˜ F ℓ = N \G and ˜ F ℓ std = N \G , they are left T -torsors over the Flag varieties F ℓ and F ℓ std respectively. We have commutative diagrams T C w,n ( δ ) ˜ F ℓ T C wn ( δ ) ˜ F ℓ std C w,n ( δ ) F ℓ C wn ( δ ) F ℓ std equivariant for the action of I w n ( δ ).In order to study the Hecke action in the overconvergent cohomology, we first have to understand how the freeabelian group Λ := { diag( p a , p b ) | a, b ∈ Z } acts on the analytic Bruhat cells. It suffices to understand the actionby the elements ̟ := diag(1 , p ) and z = diag( p, p ) since ( ̟, z ) form a basis of Λ. Let Λ + = { t ∈ Λ : t − N t ⊂ N } be the semigroup generated by ̟ and z ± (resp. Λ − the semigroup generated by ̟ − and z ± ). Clearly theaction of z by right multiplication is trivial, it is left to understand the behaviour of ̟ .We let Σ ± n := Iw n Λ ± Iw n be the multiplicative semigroup of GL ( Q p ) generated by Iw n and Λ ± . We havethe following theorem Theorem 3.1.1. (1)
The map diag( p a , p b ) b − a induces a multiplicative total order in Λ / h z i such that Λ + = { t ∈ Λ | t ≥ } . (2) The canonical map Λ ± → Iw n \ Σ ± n / Iw n is a bijection. (3) The map α : Σ ± n → Λ ± defined by composing Σ → Iw n \ Σ ± n / Iw n with the inverse of (2) is a multiplica-tive homomorphism.Proof. The proof is exactly the same as for the Iwahori group Iw in [AS08, Theorem 2.5.3]. The only detailwhich must be checked is that Σ ± n is stable by multiplication, this follows from the Bruhat-Tits decompositionof G ( Q p ), and that for t ∈ Λ + we have tN n t − ⊂ N n and t − N t ⊂ N . (cid:3) Let δ ≥ n , we can see T C wn ( δ ) as the quotient(3.1) T C wn ( δ ) = ( N an Λ) \ ( N an Λ w I w n ( δ )) . Indeed, I w n ( δ ) ∩ ( N an Λ) = N n N δ and I w n ( δ ) ∩ N an Λ = N δ . The following lemma describes the dynamics of ̟ Lemma 3.1.2.
Let ̟ = diag(1 , p ) ∈ Λ + and δ ≥ n + 1 . We have (1) T C n ( δ ) ̟ Iw n = T C n ( δ + 1) , T C w n ( δ ) ̟ Iw n = T C w n ( δ − , and (2) C n ( δ ) ̟ Iw n = C n ( δ + 1) , C w n ( δ ) ̟ Iw n = C w n − ( δ − .Proof. It follows from the equality (3.1) and the computation (cid:18) p − (cid:19) (cid:18) a bc d (cid:19) (cid:18) p (cid:19) = (cid:18) a pbp − c d (cid:19) . (cid:3) UAL EICHLER-SHIMURA MAPS FOR THE MODULAR CURVE 23
By (3.1) and Lemma 3.1, we deduce a similar dynamics for the analytic Bruhat cells of ˜ F ℓ and F ℓ replacing1 ! w .3.2. Locally analytic functional spaces.
Let T = T ( Z p ). Let W = Spf Z p [[ T ]] be the weight space. Wedenote by W its adic generic fiber over Spa( Q p , Z p ). The space W is described in ( R, R + )-points by W ( R, R + ) = [ R ⊂ R + Hom cont ( T, R × )where R runs over all the subrings of definition of R contained in R + .To construct sheaves from locally analytic principal series we will need the following proposition Proposition 3.2.1 ( [AIP15, Proposition 2.2.2]) . Let
V ⊂ W T be an open subset contained in a quasi-compactsubspace of W T . Then there exists an integer r V ∈ (0 , ∞ ) ∩ Q such that the universal character κ V : T → O ( V ) × extends to an analytic character (3.2) κ V : V × T T r V → G m Recall that Λ ⊂ T ( Q p ) is the submonoid generated by diag( p, p ) and diag(1 , p ). The analytic character (3.2)can be extended to Λ T T r V by imposing κ V | Λ ≡ V ⊂ W T an open affinoid (say a polydisk), and we take δ ≥ max { r V , n } . We take K ⊂ C p a finite extension of Q p containing an element p δ of valuation δ . We see W and X as adic spaces overSpa( K, O K ). Consider the T T δ -torsor g : T C n ( δ ) → C n ( δ ) . Let κ denote either a character k ∈ V ( K ) or κ V . We let R be either K or O W ( V ) according to κ , and let R + ⊂ R be O K or O + W ( V ) respectively. Definition 3.2.1. (1) The space of δ -analytic functions of weight κ over the big cell is the R -module A δκ = ( A δ V := Γ( V × C n ( δ ) , g ∗ ( O V×T C n ( δ ) )[ κ V ]) if κ = κ V ,A δk := Γ( C n ( δ ) , g ∗ ( O T C n ( δ ) )[ k ]) if κ = k ∈ V ( K ) . Where for a character κ and a T T δ -equivariant sheaf F , the sheaf F [ κ ] consists on the κ -equivariantvectors of F . We let A δ, + κ ⊂ A δκ be the subsheaf replacing O by O + above. We endow A δκ with the p -adic topology where A δ, + κ is an open and bounded R + -submodule. Thus, the space A δκ is a Banach R -module.(2) Dually, the space of (bounded) δ -analytic distributions of weight κ over the big cell is the R + -module D δ, + κ = Hom R + ( A δ, + κ , R + ) . We set D δκ := D δ, + κ [ p ]. In other words, we have D δκ = Hom R ( A δκ , R ) . We endow D δκ with the weak topology. Then, D δ, + κ is the weak dual of A δ, + κ and an open boundedsubset of D δκ . Lemma 3.2.1.
For any open affinoid V ′ ⊂ V we have (1) A δ, + V ′ = A δ, + V b ⊗ R + O + W ( V ′ ) , (2) D δ, + V ′ = D δ, + V b ⊗ R + O + W ( V ′ ) .Moreover, the formation V ′ ⊂ A δ V ′ defines a projective Banach sheaf over V , its weak continuous dual beingthe sheaf V ′ D δ V ′ .Proof. The T T δ -torsor T C n ( δ ) → C n ( δ ) is trivial and induces an isomorphism C n ( δ ) ∼ = N N δ . Let { e i } i ∈ I be abasis of O + ( N N δ ) as a ON Banach O K -module. Then for any affinoid V ′ ⊂ V we have A δ V ′ ∼ = Γ( V ′ × N N δ , O + ) = O + ( V ′ ) b ⊗ O K O + ( N N δ ) . The lemma follows. (cid:3)
Form (3.1) we have the description
T C n ( δ ) = ( N an Λ) \ ( N an Λ w I w n ( δ )). Then the multiplicative monoid Σ + n of Theorem 3.1.1 acts on A δκ by left multiplication, where A δ, + κ is a Σ + n -stable open bounded R + -submodule.The space D δκ carries an action of Σ − n in the obvious way. Lemma 3.2.2.
Let ̟ = diag(1 , p ) . The action of ̟ on A δ +1 , + κ factors through A δ, + κ . Dually, the action of ̟ − on D δ, + κ factors through D δ +1 , + κ . In particular, ̟ (resp. ̟ − ) is a compact R + -linear operator. Proof.
By Lemma 3.1.2 we know that C n ( δ ) ̟ Iw n = C n ( δ + 1). Then, given f ∈ A δ +1 , + κ and x ∈ C n ( δ ) we have( ̟f )( x ) = f ( x̟ ) = κ ( ̟ ) f ( ̟ − x̟ ) = f ( ̟ − x̟ ) . This shows that ( ̟f ) can be extended to a function g ̟f ∈ A δ, + κ defined by g ̟f ( x ) := f ( ̟ − x̟ ).Dually, let µ ∈ D δ, + κ and f ∈ f ∈ A δ +1 , + κ . We define ^ ̟ − µ by ^ ̟ − µ ( f ) = µ ( g ̟f ).Then ^ ̟ − µ ∈ D δ +1 , + κ is such that its image to D δ, + κ is ̟ − µ . Furthermore, we know that C n ( δ ) is a strictneighbourhood of C n ( δ + 1). Hence the restriction map is compact which implies that ̟ is compact. (cid:3) We want A δκ and D δκ to define topological sheaves over ( Rig/X ) proket or ( Rig/X ) proket × V an . We use theIw n -Galois cover X ∞ → X to define the sheaves. We only have to check that the Iw n -action on A δκ and D δκ iscontinuous. We need some lemmas. Lemma 3.2.3.
Let δ > , then N δ N ⊂ N T δ N δ , N N δ ⊂ N δ T δ N . Proof.
It suffices to prove the inclusions at the level of points. A direct computation shows that (cid:18) p δ (cid:19) (cid:18) y (cid:19) = (cid:18) y (1 + p δ xy ) − (cid:19) (cid:18) p δ xy
00 1 − p δ xy (1 + p δ xy ) − (cid:19) (cid:18) p δ x (1 + p δ xy ) − (cid:19) which proves the first inclusion. The second is proved in a similar way. (cid:3) Lemma 3.2.4.
Let H = Spa( A, A + ) be an affinoid adic analytic group, and X = Spa( R, R + ) an affinoid adicspace topologically of finite type over Spa( K, O K ) . Let Θ :
H × X → X be an action of H over X . Then forall N > there exists a neighbourhood ∈ U ⊂ H such that for all g ∈ U , x ∈ X and f ∈ O + ( X ) we have | f ( x ) − f ( gx ) | ≤ | p | N .Proof. As O + ( X ) = R + is topologically of finite type over O K , it suffices to prove the proposition for a single f ∈ R + . Let Θ ∗ : R + → ( A + b ⊗ O K R + ) + be the pullback of the multiplication map. Let V ⊂ H × X be the openaffinoid subspace defined by the equation | ⊗ f − Θ ∗ ( f ) | ≤ | p | N . As V contains 1 × X and this is a quasi-compact closed subset of H × X , there exists 1 ∈ U f ⊂ H such that U f × X ⊂ V . Therefore, for all g ∈ U f and x ∈ X we have | f ( x ) − f ( gx ) | ≤ | p | N . (cid:3) Proposition 3.2.2.
Let s ≥ be an integer. There exists a positive integer r such that Γ( p s + r ) acts triviallyon A δ, + κ /p s A δ, + κ .Proof. By Lemma 3.2.3 it is enough to show that there exists r > T s + r N s + r acts trivially on A δ, + κ /p s A δ, + κ . Let f ∈ A δ, + κ and g = tn ∈ T N , then ( gf )( x ) = κ ( t ) f ( t − xtn ) for all x ∈ C n ( δ ). Since κ iscontinuous, we can find r >> κ ( t ) ≡ p s for all t ∈ T r . Applying Lemma 3.2.4, one finds r >> f ( t − xtn ) ≡ f ( x ) mod p s for all x ∈ C n ( ǫ ) and tn ∈ T r N r . Taking r = max { r , r } weobtain the proposition. (cid:3) Corollary 3.2.1.
The action of Iw n on A δκ and D δκ is continuous. Thus, A δ V defines a ON Banach b O V -moduleon the site ( Rig/ X ) proket × V an , trivialized in X ∞ . The weak dual of A δ V is the topological sheaf D δ V .Proof. With the notation of Proposition 3.2.2, we already know that Iw n acting on A δ, + κ /p s A δ, + κ factors throughIw n / Γ( p s + t ). Let { e i } i ∈ I be a ON O K -basis of O + ( N N δ ). Then A δ, + κ ∼ = dM i ∈ I R + e i and there is an exhaustive increasing family of finite subsets { J ⊂ I } such that L j ∈ J ( R + /p s ) e i is stable byIw n . This shows that the action of Iw n on D δ, + κ endowed with the weak topology is continuous, and so is theaction over D δκ . (cid:3) To work with the ´etale cohomology of A δκ it is useful to work with certain weak completion. For that wedefine the following filtrations Definition 3.2.2.
We let Fil s A δ, + κ and Fil s D δ, + κ to be the kernel of A δ, + κ → A δ +1 , + κ /p s A δ +1 , + κ and D δ, + κ → D δ − , + κ /p s D δ − , + κ respectively. We denote b A δ, + κ = lim ←− A δ, + κ / Fil s , endowed with the inverse limit topology. Wewrite b A δκ = b A δ, + κ [ p ]. We introduce the space of completely discontinuous distributions D c,δ, + κ as the strongcontinuous dual of b A δ, + κ . Then D c,δ, + κ ⊂ D δ, + κ and the map D δ +1 , + κ → D δ, + κ factors through D c,δ, + κ . UAL EICHLER-SHIMURA MAPS FOR THE MODULAR CURVE 25
Remark . The transition maps A δ, + κ → A δ +1 , + κ and D δ, + κ → D δ − , + κ are compact. Then, the quotients A δ, + κ / Fil s and D δ, + κ / Fil s are finite ( R + /p s )[Σ ± n ]-modules. Moreover, b A δ, + κ is the completion of the image of A δ, + κ in A δ +1 , + κ , and D δ, + κ = lim ←− n D δ, + κ / Fil s as topological R + -modules. From Corollary 3.2.1 we deduce that D δ, + κ ∼ = Y i ∈ I R + e ∨ i , b A δ, + κ ∼ = Y i ∈ I R + e i and D c,δ, + κ ∼ = dM i ∈ I R + e ∨ i . We continue with the relation with algebraic representations of GL . We have the Iw n -equivariant diagramwith horizontal open immersions(3.3) C , + n ( δ ) ˜ F ℓ std C n ( δ ) F ℓ std . g g Definition 3.2.3.
Let k be an dominant weight of GL . We denote by V + k the GL ( O K )-module V + k = Γ( F ℓ std , g ∗ O +˜ F ℓ std [ k ]) , that is, the irreducible representation of highest weight k . We write V k = V + k [ p ]. Proposition 3.2.3. (1)
There is a natural Iw n -equivariant inclusion V + k → A δ, + k for all δ ≥ n . (2) There is a natural Iw n -equivariant map D δ, + k → V + , ∨ k = V + − w ( k ) which becomes surjective after inverting p .Proof. From diagram (3.3) we obtain a natural Iw n -equivariant map V + k → A δ, + k which is clearly an inclusion.The map D δ, + k → V + , ∨ k is defined by taking duals. (cid:3) In order to make the maps of Proposition 3.2.3 Σ ± n -equivariant we have to normalize the action of Λ + on V + k . Namely, we let t ∈ Λ + act on V + k by ( tf )( x ) := k ( t ) f ( xt ). Thus, within this normalization, the abovemaps V + k → A δ, + k and D δ, + k → V + − w ( k ) are Σ ± n -equivariant.From the proof of Lemma 3.2.1 one deduces the compatibility of A δκ and D δκ with specialization to V ( K ) Lemma 3.2.5.
Let k ∈ V ( K ) be a closed point defined by the regular ideal J k . Then we have exact sequences J k A δ, + κ V A δ, + κ V A δ, + k J k D δ, + κ V D δ, + κ V D δ, + k . The same holds true for b A δ, + κ and D c,δ, + κ .Remark . Let W = Spf( Z p [[ Z × p ]]) and W = Spa( Q p , Z p ) × Spf Z p Spf( Z p [[ Z × p ]]) its adic generic fiber. LetSt be the standard representation of GL . In [AIS15] the space W is used as a weight space, and the irreduciblerepresentations of interest are Sym s St for s ∈ N . The two approaches are equivalent, the only difference is thatworking with W takes into consideration twists of the determinant.More explicitely, given δ ≥ n and η a ( δ − Z × p , we define A δ, + η = { f : Z × p × Z p → R + | f | × Z p is δ -analytic , and f ( tx ) = η ( t ) f ( x ) for t ∈ Z × p } Then A δ, + η is a ON R + -module endowed with a left Iw n -action. Let κ = ( κ , κ ) be a ( δ − T . Then there is a natural isomorphism of Iw n -modules A δ, + κ = A δ, + κ − κ ⊗ (det) κ . Overconvergent modular symbols.
We keep the notation of the previous section. In the following wesee the analytic modules defined above as topological sheaves on the sites (
Rig/X ) proket × V an or ( Rig/X ) proket according to the character κ . Definition 3.3.1. (1) Let k ∈ V ( K ) be a character. The δ -analytic modular symbols of weight k aredefined as the cohomology groups H proket ( X C p , D δk ) and H proket ( X C p , b A δk ) . (2) Let p : ( X C p ) proket × V an → V an be the projection. The p -adic families of r -analytic modular symbolsof weight κ V are defined as the sheaves over V an R p , ∗ D δ V and R p , ∗ b A δ V . By Proposition 2.2.5 they are the shefification of the pre-sheaves V ′ H proket ( X C p , p , ∗ b A δ V ′ ) and V ′ H proket ( X C p , p , ∗ D δ V ′ )respectively. Namely, they are the sheaves over V an H proket ( X C p , b A δ V ) and H proket ( X C p , b D δ V )of Section 2.2.4.Let’s consider the complete analytic Iwahori group c I w n ( δ ) = Z × p (1 + p δ b O + ) Z p + p δ b O + p n Z p + p δ b O + Z × p (1 + p δ b O + ) ! Definition 3.3.2. (1) Let H ⊂ E [ p n ] be the universal cyclic subgroup of order p n , seen as a local systemover ( Rig/X ) proket . We denote by T Iw n the left Iw n -torsor over ( Rig/X ) proket T Iw n = Isom(( T p E, H ) , ( Z p , Z p /p n Z p ⊕ . (2) We let T c I w n ( δ ) be the c I w n ( δ )-extension of T Iw n . Namely, the torsor T c I w n ( δ ) = c I w n ( δ ) × Iw n T Iw n . Remark . The previous torsors can be describe as the quotient sheaves T Iw n = ( X ∞ × Iw n ) / Iw n T c I w n ( δ ) = ( X ∞ × c I w n ( δ )) / Iw n . Let κ be a character in V ( K ) or equal to κ V . We denote by b O + , ( κ ) X the bounded completed structure sheafof ( Rig/X ) proket or ( Rig/X ) proket × V an endowed with the κ -action of T b T δ , we set b O ( κ ) X = b O + , ( κ ) X [ p ]. Definition 3.3.3. (1) The sheaf of geometric δ -analytic modular functions is the continuous H om -sheafin ( Rig/X ) proket or ( Rig/X ) proket × V an O A δ, + κ := O A δ, + κ = H om T b T δ ( b N \ T c I w n ( δ ) , b O + , ( κ ) X )We denote O A δκ = O A δ, + κ [ p ], and O A δ, (+) V = O A δ, (+) κ V .(2) Dually, the sheaf of geometric δ -analytic distributions is the continuous dual O D δ, + κ := O D δκ = H om b O + X ( O A r, + κ , b O + X )We set O D δκ = O D δ, + κ [ p ], and O D δ, (+) V = O D δ, (+) κ V . Remark . It is important to understand where the sheaves O A δκ and O D δκ are defined. Namely, the site( Rig/X ) proket if κ = k ∈ V ( K ), and the site ( Rig/X ) proket × V an if κ = κ V . The use of the big sites guaranteesthat the torsor T c I w n ( δ ) has a representable dense subsheaf, and that the H om functor defining the δ -analyticfunctions and distributions takes the right value.Let B be a base of affinoid perfectoid of ( Rig/X ) proket trivialising T Iw n Proposition 3.3.1.
Let U = Spa( S, S + ) be an object in B . Given a sheaf F over ( Rig/X ) proket or ( Rig/X ) proket ×V an , we denote by F | U its restriction to U proket and U proket × V an respectively. Let { e i } i ∈ I be a O + K -basis of O + ( N N δ ) , then there are canonical isomorphisms O A δ, + κ | X ∞ ∼ = dM b O + X ∞ ( κe i ) O D δ, + κ | X ∞ ∼ = Y i b O + X ∞ ( κe i ) ∨ , where κe i : c I w n ( δ ) → b O + is the function κe i ( ntn ) = κ ( t ) e i ( n ) . UAL EICHLER-SHIMURA MAPS FOR THE MODULAR CURVE 27
Proof.
The torsor T c I w n ( r ) restricted to X ∞ is trivial and isomorphic to X ∞ × c I w n ( r ). By the Iwahori decom-position of c I w n ( r ) we get that O A δ, + κ | X ∞ ∼ = H om T b T r ( T b T r × N b N r , b O + , ( κ ) X )= H om ( N b N r , b O + X ) . But N b N r is a disjoint union of completed sheaves b O + X . By Proposition 2.2.4 the last sheaf is equal to O + ( N N r ) b ⊗ K b O + X . The proposition follows. (cid:3) Definition 3.3.4.
We define the following filtrations in geometric δ -analytic functions and distributionsFil s O A δ, + κ := ker( O A δ, + κ → O A δ +1 , + κ /p s O A δ +1 , + κ )Fil s O D r, + κ := ker( O D δ, + κ → O D δ − , + κ /p s O D δ − , + κ ) . We define O b A δ, + κ := lim ←− i ( O A δ, + κ / Fil i ) and O D c,δ, + κ = H om b O X ( O b A δ, + κ , b O + X ). Remark . Proposition 3.3.1 also implies O b A δ, + κ | X ∞ ∼ = Q i b O + X ∞ ( κe i ) and O D c,δ, + κ = cL i b O + X ∞ ( κe i ) ∨ . More-over, we have factorizations O b A δκ → O A δ +1 κ → O b A δκ and O D δκ → O D c,δ − , + κ → O D δ − κ . The weak dual of O A δ, + κ (resp. of O D c,δ, + κ ) is O D δκ (resp. O b A δ, + κ ). Conversely, the strong dual of O D δ, + κ (resp.of O b A δ, + κ ) is O A δ, + κ (resp. O D c,δ, + κ ). Corollary 3.3.1. (1)
The sheaves O A δκ , O D δκ , O b A δκ and O D c,δκ are completed uniform sheaves as in Defini-tion 2.2.9. (2) O A δκ and O D c,δ are locally pro-Kummer-´etale on X ON Banach sheaves of b O X -modules, with O A δ, + κ and O D c,δ, + κ open an Bounded ON Banach b O + X -sheaves respectively. (3) There are topological isomorphisms A δ, + k b ⊗ O K b O + X ∼ −→ O A δ, + k A δ, + V b ⊗ b O + V b O + X ×V ∼ −→ O A δ, + V . The same holds for O b A δ, + κ , O D δ, + κ and O D c,δ, + κ .Proof. This follows from the descriptions of the previous proposition, and the fact that O b A δ, + κ and O D δ, + κ areendowed with the product topology (which is the same as the weak topology), so the completed tensor productis defined with respect to the filtrations Fil i O A δ, + κ and Fil O D δ, + κ respectively. (cid:3) Theorem 3.3.1.
The following holds (1)
Let k ∈ V ( C p ) . The complexes R Γ proket ( X C p , O A δk ) and R Γ proket ( X C p , O D δk ) are represented by perfect C p -Banach complexes. In particular, the complexes hocolim δ →∞ R Γ proket ( X C p , O b A δk ) = hocolim δ →∞ R Γ proket ( X C p , O A δk )Rlim δ →∞ R Γ proket ( X C p , O D δk ) = Rlim δ →∞ R Γ proket ( X C p , O D c,δk ) = are represented by a perfect complex of compact type and a perfect complex of nuclear Fr´echet spacesover C p respectively. (2) The complexes R Γ proket ( X C p , O A δ V ) and R Γ proket ( X C p , O D δ V ) are represented by perfect complexes of Banach O V b ⊗ K C p -modules over V an . In particular, the complexes hocolim δ →∞ R Γ( X C p , O b A δ V ) = hocolim δ →∞ R Γ( X C p , O A δ V )Rlim δ →∞ R Γ( X C p , O D δ V ) = Rlim δ →∞ R Γ( X C p , O D c,δ V ) are represented by a perfect complex of compact type and a perfect complex of nuclear Fr´echet O V b ⊗ K C p -modules respectively. Proof.
In the following we let R + to be O K or O + ( V ) depending on κ . By Proposition 2.2.5, part (2) followsfrom the argument down below. Let κ denote k ∈ V ( K ) or κ un , and let F be one of the sheaves O A δ, + κ or O D c,δ, + κ . Let ν an : X proket → X an be the morphism of sites, then R Γ proket ( X C p , F ) = R Γ an ( X C p , Rν an, ∗ F ) . The perfectoid cover X ∞ → X is Galois of group Iw n , then by a Hochschild-Serre spectral sequence we havethat Rν ∗ F = R Hom Iw n ( Z p , F ) is the continuous group cohomology of F . Let Iw n = Iw n ∩ Γ ( p ), it fits in anexact sequence 0 → Iw n → Iw n → T ( F p ) → . By a theorem of Lazard [Laz65, Theo. V.3.2.7], Z p has a finite projective resolution by finite free Z p [[Iw n ]]-algebras as a trivial Iw n -module, let P • be such a resolution. Then Hom Iw n (Iw n , P • ) = Hom Z p ( Z p [Iw n / Iw n ] , P • )is a projective resolution of Z p [Iw n / Iw n ] by finite free Z p [[Iw n ]]-modules. But Z p [Iw n / Iw n ] ∼ = Z p ⊕ I as Iw n -module, where I is the augmentantion submodule (the quotient Iw n / Iw n = T ( F p ) has order ( p − ). Thus, P • also induces a bounded resolution of Z p by finite projective Z p [[Iw n ]]-modules. Then, we have that R Γ proket ( X C p , F ) = R Γ an ( X C p , Hom Iw n ( P • , F )) . Let { U i } be a finite over of X by affinoid spaces such that U i, ∞ := X ∞ × X U i is affinoid perfectoid (such a coverexists). The module F | U ∞ ,i has almost zero higher pro-Kummer-´etale cohomology being a complete direct sumof the complete structure sheaf b O + X . By a Leray spectral sequence using the ˇCech covering { U i } , we obtain that R Γ proket ( X C p , F ) = ˇ C ( { U i, ∞ } , Hom Iw n ( P • , F )) , where the right hand side denote the total complex of the double ˇCech complex ˇ C • ( { U i, ∞ } , Hom Iw n ( P • , F )).The terms of the ˇCech complex are complete direct sums of modules of the form ( R + b ⊗ O K b O + ( U i )) b ⊗ O K O C p ,which is almost a complete direct sum of R + b ⊗ O K O C p -modules. The theorem follows. (cid:3) Definition 3.3.5.
With the notations of Theorem 3.3.1 we define the complexes of overconvergent modularsymbols as R Γ( A κ ) := hocolim δ R Γ proket ( X C p , O A δκ ) R Γ( D κ ) := Rlim δ R Γ proket ( X C p , O D c,δκ )For any i ∈ Z , we denote R i Γ( A κ ) (resp. R i Γ( D κ )) the i -th cohomology of R Γ( A κ ) (resp. R Γ( D κ )).In the following we consider almost mathematics with respect to O C p and its maximal ideal. Proposition 3.3.2. (1)
Let k ∈ V ( K ) . The natural maps R Γ proket ( X C p , b A δ, + k ) b ⊗ O K O C p → R Γ proket ( X C p , O b A δ, + k ) R Γ proket ( X C p , D δ, + k ) b ⊗ O K O C p → R Γ proket ( X C p , O D δ, + k ) are almost quasi-isomorphisms of cohomology complexes of O C p -modules, where the completed tensor ofthe left hand side is taking with respect to the filtration of Definition 3.3.4. (2) Let p : ( Rig/X C p ) proket × V an → V an be the projection. Then the natural maps Rp , ∗ ( b A δ, + V ) b ⊗ O K ( O C p b ⊗ O K O + V ) → Rp , ∗ ( O b A δ, + V ) Rp ∗ , ( D δ, + V ) b ⊗ O K ( O C p b ⊗ O K O + V ) → R , ∗ ( O D δ, + V ) are almost quasi-isomorphisms of cohomology complexes of O C p b ⊗ O K O + V -sheaves. The completed tensorproduct is derived and taken with respect to the filtrations as above.Proof. Let F denote one of the previous sheaves. By Proposition 2.2.5 it is enough to show that for all i thereis an isomorphism of almost O C p -modules R Γ proket ( X C p , F / Fil i ) ⊗ O K O C p = R Γ proket ( X C p , F / Fil i ⊗ b O + ) . In the case of (1), the quotient F / Fil i is a finite Iw n -module and the result follows from the primitive comparisonTheorem. In the second case F / Fil i , is a direct limit of finite O K [Iw n ]-modules of finite lenght, as X C p is qcqsthe equality follows from the primitive comparison theorem. (cid:3) Remark . Let us compare the previous definitions of overconvergent modular symbols with those con-structed in [AIS15]. Let κ = ( κ , κ ) be the universal character over V . As it was mentioned in Remark 3.2.2,we have an equality A δ, + κ = A δ, + κ − κ ⊗ (det) κ as Iw n -representations. The main difference between the twoapproaches is the choice of an affinoid open subset instead of a wide open disk. Fix a covering { U i } of X byopen affinoid subsets such that U i, ∞ = U i × X X ∞ is affinoid perfectoid, the description of R Γ proket ( X C p , F )as a complex of perfect Banach complexes is functorial on V and on the sheaf F . In particular, if ˜ V is a UAL EICHLER-SHIMURA MAPS FOR THE MODULAR CURVE 29 wide open polydisk of W containing V , the complex R Γ( X C p , O D δ, +˜ V ) is still represented by the ˇCech complexˇ C ( { U i, ∞ } , Hom Iw n ( P • , O D δ, + )). Note that the quotients D δ, + e V / Fil i are finite Iw n -modules The second item ofProposition 3.3.2 implies that there are not higher derived limits and that H i ( X C p , D δ, + e V ) b ⊗ O K O C p = ae H i ( X C p , O D δ, + e V ) . where the completion is taken with respect to the filtrarion Fil i . On the other hand, by Mittag-Leffler and thecomparison between the ´etale and Betti cohomology for the modular curve, one finds that R Γ proket ( X, D δ, + e V )is quasi-isomorphic to R Γ( X C , D δ, + e V ) = R Hom Γ( N ) ∩ Γ( p n ) ( Z , D δ, + e V ). The last is represented by a finite complexwhose terms are finite direct sums of D δ, + e V (cf. Section 1 of [AS08]). Then we have a quasi-isomorphism ofcomplexes of O ( V ) b ⊗ K C p -modules R Γ proket ( X C p , O D δ,κ V ) b ⊗ O + ( e V ) O ( V ) ≃ ( R Hom Γ( N ) ∩ Γ( p n ) ( Z , D δ, + e V ) b ⊗ O + ( e V ) O ( V )) b ⊗ K C p . The choice of the weak topology for the distributions D δκ implies that the previous complexes are not of Banachsheaves. Nevertheless, their finite slope theory remains intact as the U p -operator will send D δ V to D c,δ − V whichis an ON Banach O ( V )-module. All the previous discussion can be made with b A δκ without any change.3.4. Overconvergent modular forms.
Let π HT : X ∞ → F ℓ be the Hodge-Tate period map. Let W = { , w } be the Weyl group of GL . Let ǫ ≥ n be a rational number and w ∈ W . Recall from Definition 3.1.2 that C w,n ( ǫ ) ⊂ F ℓ denotes the open ǫ -analytic Bruhat cell containing w .Let X ord ∞ denote the ordinary locus of X ∞ . By Lemma III.3.14 of [Sch15], X ord ∞ = π − HT ( F ℓ ( Q p )). Theordinary locus is a Iw n -invariant open subset which descents to an open X ord ⊂ X . It admits a canonicalsubgroup C cann of order p n . Let H n ⊂ E [ p n ] be the universal cyclic subgroup over X . Define X ord and X ordw asthe locus of X ord where H n ∩ C cann = H n or H n ∩ C can = 0 respectively. Lemma 3.4.1.
The open subspace π − ( C w,n ( ǫ )) is an open affinoid perfectoid of X ∞ . Moreover, it is Iw n -invariant and descends to an open affinoid subspace X w ( ǫ ) containing X ordw .Proof. By Theorem III.3.18 of [Sch15] the open subspace π − ( C w,n ( ǫ )) is affinoid perfectoid. It is the pre-imageof some affinoid subspace of finite level, being Iw n -invariant we can take it as an open affinoid subspace of X . (cid:3) Proposition 3.4.1. (1)
Let m ≥ n . There exists ǫ ≥ n such that X w ( ǫ ) admits a canonical subgroup C canm of order p m . (2) The family { X w ( ǫ ) } ǫ ≥ n is a basis of strict affinoid neighbourhoods of the w -ordinary locus X ordw .Proof. By [AIP15, Theorem 3.1.1] it suffices to prove (2). By Lemmas III 3.6 and III 3.8 of [Sch15], π − ( F ℓ ( Q p ))is the closure of the ordinary locus of X ord ∞ . As {C w,n ( ǫ ) } ǫ ≥ n is a basis of Iw n -stable strict neighbourhoods of w ,then { X w ( ǫ ) } ǫ ≥ n is a basis of strict neighbourhoods of X w ( ǫ ) ∩ X ord . It is enough to show that X w ( ǫ ) ∩ X ord is the w -ordinary locus; we can check this at rank 1 ordinary points.Let x = Spa( C, O C ) → X ord be a geometric ordinary point. Let E/C be the elliptic curve at x and H n ⊂ E [ p n ] the cyclic subgroup. If the relative position of C cann with respect to H n is w , by chosing a basis( e , e ) of E [ p n ] such that h e i = H n , we have by definition C n = h e i A with A ∈ B ( Z /p n Z ) w B ( Z /p n Z ). Thisshows that any lift ˜ x → X ∞ of x satisfies π HT (˜ x ) ∈ B\B w Iw n , and x ∈ X w ( ǫ ) ∩ X ord for all ǫ ≥ n . The converseis clear by the definition of π HT . (cid:3) Consider the Hodge exact sequence0 → ω − E ⊗ O X b O X (1) dlog ∨ −−−→ T p E ∨ ⊗ b Z p b O X dlog −−−→ ω E ⊗ O X b O X → . A priori it is only defined over X proket , but it actually extends to a map over ( Rig/X proket ). In fact,
Lemma 3.4.2.
The map dlog : T p E ∨ ⊗ b Z p b O X → ω E ⊗ O X b O X extends naturally to a continuous map of locallyfinite free b O X -sheaves over ( Rig/X ) proket .Proof. Let V ∈ ( Rig/X ) ket and V ∞ = V × X X ∞ . Write f : V → X . Then f ∗ dlog : T p f ∗ E ∨ → f ∗ ω E ⊗ O V O V ∞ is a Iw n -equivariant map of finite free b O V ∞ -modules. It gives rise to a continuous map of b O V -sheaves over V proket . (cid:3) Let m ≥ n + 1 and assume that ǫ ≥ n is big enough such that X w ( ǫ ) admit a canonical subgroup of order p m . Definition 3.4.1.
We define the sheaves over (
Rig/X n ( ǫ )) proket b ω + E = dlog( T p E ∨ ⊗ b O + X ) and ω + E = b ω + E ∩ ω E . Proposition 3.4.2.
The sheaf ω + E is an invertible O + X w ( ǫ ) ,et -module for the ´etale topology of X w ( ǫ ) . Moreover,there exists < δ < such that the dlog map induces an isomorphism dlog : ( C canm ) ∨ ⊗ O + X /p m − δ ∼ = b ω + E /p m − δ ∼ = ω + E /p m − δ , and b ω + E = ω + E ⊗ O + X b O + X .Proof. As ω + E is dense in b ω + E , it suffices to show that ( C canm ) ⊗ O X /p m − δ ∼ = ω + E /p m − δ for some δ ∈ ]0 , C canm is trivialized by an ´etale covering of X w ( ǫ ), we see that ω + E is a locally free O + X -module for the ´etaletopology. The statement follows from Proposition 3.2.1 of [AIP15]. (cid:3) We deduce an integral Hodge-Tate exact sequence over (
Rig/X w ( ǫ )) proket (3.4) 0 → b ω + , − E (1) dlog ∨ −−−→ T p E ∨ ⊗ b Z p b O + X dlog −−−→ b ω + E → . Let m − δ > r ≥ n be a rational number. By Proposition 3.4.2, the Hodge-Tate period map induces anisomorphism dlog : C ∨ m ⊗ O + X /p r ∼ −→ ω + E /p r . Definition 3.4.2. (1) We define the ´etale Z × p (1 + p r O + X )-torsor ω + , × E ( r ) by ω + , × E ( r ) := { v ∈ ω + E basis | v ∈ dlog( C ∨ m ) mod p r } . (2) Dually, we denote by ω + , − , × E ( r ) the ´etale Z × p (1 + p r O + X )-torsor ω + , − , × E ( r ) := { v ∈ ω + , − E basis | dlog ∨ ( v ) ∈ C ′ m /p r mod p r } . (3) We set T ω E ( r ) := ω + , × E ( r ) × ω + , − , × E ( r ) the left ´etale T T r -torsor endowed with the action ( t , t ) · ( v , v ) =( t v , t v ). We denote b T ω E ( r ) the completion of T ω E ( r ) in the pro-Kummer ´etale site, i.e., the T b T r -extension of T ω E ( r ) . Remark . The torsor T ω E ( r ) is trivial after a finite ´etale extension of X n ( ǫ ), namely Isom( Z /p m Z , C canm ).By Corollary 2.2.1 it is represented by an affinoid space of finite type over X n ( ǫ ). Moreover, we see T ω E ( r ) as a T T r -torsor over ( Rig/X w ( ǫ )) et via Definition 2.2.4 (2). Definition 3.4.3. (1) The sheaf of (bounded) overconvergent modular forms of weight k ∈ V ( K ) is definedas the ´etale sheaf over ( Rig/X w ( ǫ )) et ω k, + E := H om T T r ( T ω E ( r ) , O + , ( w ( k )) X ) , we write ω kE := ω k, + E [ p ].(2) The sheaf of (bounded) p -adic families of overconvergent modular forms of weight κ V is defined as the´etale sheaf over ( Rig/X w ( ǫ )) et × V an defined by ω κ V , + E := H om T T r ( T ω E ( r ) , O + , ( w ( κ V )) X ) , we write ω κ V E := ω κ V , + E [ p ]. Lemma 3.4.3. (1)
Let m − δ > r ′ ≥ r ≥ n . Let κ be a r -analytic character. The natural inclusion ofgroups T T r ′ → T r gives rise to a continuous map of torsors T ω E ( r ′ ) → T T ω E ( r ) such that the naturalmap ω κ, + ,rE → ω κ, + ,r ′ E constructed from r and r ′ is an isomorphism. (2) Let X n +1 denote the modular curve of level Γ ( N ) ∩ Γ ( p n +1 ) . Let p n +1 : X n +1 → X n be the naturalmap. Let m − δ > r ≥ n + 1 . Then p ∗ n +1 ( ω + , × E ( r )) = ω + , × E ( r ) and there is an isomorphism ofoverconvergent modular torsors ω κ, + E,n +1 = p ∗ n +1 ω κ, + E,n . (3) Let ǫ ′ ≥ ǫ . Then ω κ, + E ( ǫ ) | X w ( ǫ ′ ) = ω κ, + E ( ǫ ′ ) .Proof. The lemma follows easily from the definitions as ω κE is an invertible R b ⊗ O X -module, with R = K or O ( V ) depending on κ . (cid:3) Let c : Z × p (1 + p r D ) → G m be a r -analytic character of Z × p . Then the composition(det) c : I w n ( r ) det −−→ Z × p (1 + p r D ) c −→ G m is well defined, and gives rise to a character of the analytic Iwahori subgroup. Lemma 3.4.4.
Let ˜ κ = κ ⊗ (det) c . Then there exists a natural isomorphism of O + X -modules ω κ, + E ∼ = ω ˜ κ, + E . Inparticular, if κ = ( κ , κ ) then the isomorphism class of ω κ, + E only depends on κ − κ . UAL EICHLER-SHIMURA MAPS FOR THE MODULAR CURVE 31
Proof.
We have a natural paring ω + , × E ( r ) × ω + , − , × E ( r ) → Z × p (1 + p r O + X ). We defineΦ : ω κ, + E → ω ˜ κ, + E f (Φ( f ) : ( v , v )
7→ h v , v i c f ( v , v ))for ( v , v ) ∈ T ω E ( r ) . One computesΦ( f )( tv , tv ) = det(diag( t , t )) c κ ( t , t ) h v , v i c f ( v , v ) = ˜ κ ( t , t )Φ( f )( v , v )which shows that Φ( f ) ∈ ω ˜ κ, + E . (cid:3) Remark . When k = ( k , k ) ∈ T ( Z p ) is a classical weight, ω kE = ω k − k E is the usual modular sheaf ofweight k − k , cf. Section 2.1.We define X w ( > ǫ ) := S ǫ ′ >ǫ X w ( ǫ ), it is a stein space such that X w ( ǫ + 1) ⊂ X w ( > ǫ ) ⊂ X w ( ǫ ). Proposition 3.4.3. (1)
Let k ∈ V ( K ) . The complexes R Γ( X w ( ǫ ) , ω kE ) and R Γ X w ( >ǫ +1) ( X w ( ǫ ) , ω kE ) areperfect complexes of K -Banach spaces. (2) Let κ = κ V . The complexes R Γ( X w ( ǫ ) , ω κE ) and R Γ X w ( >ǫ +1) ( X w ( ǫ ) , ( ω κE ) are represented by perfectBanach complexes over V an .Proof. As the space X w ( ǫ ) is an affinoid, and ω κE is an inverible sheaf over X w ( ǫ ), we know that H i ( X w ( ǫ ) , ω κE ) =0 for i > H ( X w ( ǫ ) , ω κE ) is a Banach R -module. This proves the statement for the cohomology withoutsupports.On the other hand, by definition we have a distinguished triangle(3.5) R Γ X w ( >ǫ +1) ( X w ( ǫ ) , ω κE ) → R Γ( X w ( ǫ ) , ω κE ) → R Γ( X w ( ǫ ) \ X w ( > ǫ + 1) , ω κE ) + −→ . Both X w ( ǫ ) and X w ( ǫ ) \ X w ( > ǫ + 1) are strict increasing unions of affinoid spaces, this implies that R Γ X w ( >ǫ +1) ( X w ( ǫ ) , ω κE )is a perfect complex of R -Banach modules. The distinguished triangle (3.5) implies that R Γ X w ( >ǫ +1) ( X w ( ǫ ) , ω κE )is concentrated in degree 1. (cid:3) Lemma 3.4.5. (1)
The restriction map R Γ( X w ( ǫ ) , ω κE ) Res −−→ R Γ( X w ( ǫ + 1) , ω κE ) is compact. (2) The correstriction map R Γ X w ( >ǫ +2) ( X w ( ǫ ) , ω κE ) Cor −−→ Γ X w ( >ǫ +1) ( X w ( ǫ ) , ω κE ) is compact.Proof. This follows from Lemma 2.3.5, the fact that ω κE is an invertible sheaf over X w ( ǫ ), and that the inclusion X w ( ǫ + 1) → X w ( ǫ ) is strict. (cid:3) Definition 3.4.4.
The complexes of overconvergent modular forms of weight κ are the ind-compact (resp.proj-compact) limits of perfect Banach complexes R Γ w ( X, κ ) := hocolim ǫ R Γ( X w ( ǫ ) , ω κE ) , R Γ w,c ( X, κ ) := Rlim ǫ R Γ X w ( >ǫ +1) ( X w ( ǫ ) , ω κE ) . We also write R Γ w,ǫ ( X, κ ) = R Γ( X w ( ǫ ) , ω κE ) and R Γ w,c,ǫ ( X, κ ) = R Γ X w ( >ǫ +1) ( X w ( ǫ ) , ω κE ) . The dlog maps.Lemma 3.5.1. (1) On X ( ǫ ) the dlog map (3.4) induces a continuous morphism of T b T r -equivariant sheaves dlog : ( b N ∩ c I w n ( r )) \ T c I w n ( r ) → b T ω E ( r ) , where T b T r acts by left multiplication on both sheaves. (2) On X w ( ǫ ) the dlog ∨ map (3.4) induces a continous morphism of T b T r -equivariant sheaves, dlog ∨ : b T ω E ( r ) → ( b N ∩ c I w n ( r )) \ T c I w n ( r ) , where T b T r acts on b T ω E ( r ) after twisting by w , and by left multiplication on ( b N ∩ c I w n ( r )) \ T c I w n ( r ) .Proof. Let H n ⊂ E [ p n ] denote the universal cyclic subgroup of order p n , and C m denote the canonical subgroupof E of order p m over X w ( ǫ ). We identify E with its dual E ∨ via the principal polarization. In particular, wealso denote by C m the canonical subgroup of E ∨ [ p n ]. There are modular interpretations of the torsors b T ω E ( r ) = { ( v , v ) ∈ ω + , × E × ω + , − , × E | v ∈ dlog( C ∨ m ) and dlog ∨ ( v ) ∈ C m mod p r } T c I w n ( r ) = { ( e , e ) base of T p E ∨ ⊗ b O + X ( − | e i ∈ T p E ∨ mod p r , h e i = ( E [ p n ] /H n ) ∨ mod p n } (1). Over X ( ǫ ) we have H n = C n . Then dlog( e ) ∈ dlog( C ∨ m ) mod p r is a base of ω + E . Let ˜ e ∈ ω + , − E bethe only element such that h e , e i E = h dlog( e ) , ˜ e i . Thus (dlog( e ) , ˜ e ) ∈ b T ω E ( r ) and we definedlog : ( b N ∩ c I w n ( r )) \ T c I w n ( r ) → b T ω E ( r ) ( e , e ) (dlog( e ) , ˜ e )If v ∈ ω + , − E is a basis then ˜ e = h e ,e i E h dlog( e ) ,v i v , in particular the dlog map of torsors is a continuous mapof sheaves. Moreover, if n = (cid:18) c (cid:19) ∈ b N ∩ c I w n ( r ) we have n (cid:18) e e (cid:19) = (cid:18) e e + ce (cid:19) . Then h e , e i E = h e , e + ce i E and dlog( e , e ) = dlog( e , e + ce ). This implies that dlog factorsthrough the quotient ( b N ∩ c I w n ( r )) \ T c I w n ( r ) as wanted. The T b T r -equivariance is clear from the definition.(2). Over X w ( ǫ ) we have H n ∩ C n = 0. In particular the compositions C n → E ∨ [ p n ] → H ∨ n (3.6) ( E [ p n ] /H n ) ∨ → E ∨ [ p n ] → C ∨ n (3.7) are isomorphisms. Let ( v , v ) ∈ b T ω E ( r ) , then dlog ∨ ( v ) ∈ E ∨ [ p m ] mod p r . By equations (3.6) and (3.7)we can find locally pro-Kummer-´etale an element ˜ v ∈ T p E ∨ ⊗ b O + X such that dlog(˜ v ) = v , ˜ v ∈ E ∨ [ p m ]mod p r and ˜ v ∈ ( E [ p n ] /H n ) ∨ mod p n is a generator. Then (dlog ∨ ( v ) , ˜ v ) ∈ T c I w n ( r ) . Moreover, twodifferent choices of ˜ v differ by an element in ( p n Z p + p r b O + X ) dlog ∨ ( v ). In other words, we have a welldefined map of sheaves dlog ∨ : b T ω E ( r ) → ( b N ∩ c I w n ( r )) \ T c I w n ( r ) ( v , v ) (dlog ∨ ( v ) , ˜ v ) . To check that dlog ∨ is continuous we can work locally pro-Kummer-´etale. Thus, we fix an element( v , v ) ∈ b T ω E ( r ) and ˜ v ∈ T p E ⊗ b O + X as above. Then the map T b T r → T c I w n ( r ) ( t , t ) ( t dlog ∨ ( v ) , t ˜ v )provides a continuous section of dlog ∨ from b T ω E ( r ) to T c I w n ( r ) . This implies that dlog ∨ is continuousand T b T r -equivariant as stated. (cid:3) Definition 3.5.1.
Consider the continuous maps of topological sheaves(3.8) dlog : b N \ T c I w n ( r ) → b T ω E ( r ) over X ( ǫ ) , dlog ∨ : b T ω E ( r ) → b N \ T c I w n ( r ) over X w ( ǫ ) . The maps dlog κ, ∨A : b ω w ( κ ) , + E → O A r, + κ and dlog κ A : O A r, + κ → b ω κ, + E are defined by taking H om T b T r ( − , ( b O X ) + , ( κ ) ) of (3.8) respectively. Dually, the mapsdlog κ D : O D r, + κ → b ω − w ( κ ) , + E and dlog κ, ∨D : b ω − κ, + E → O D r, + κ are defined by taking b O + X -duals. Lemma 3.5.2.
The maps dlog κ and dlog κ, ∨ of Definition 3.5.1 are independent of r and ǫ .Proof. The independence of ǫ is clear by Lemma 3.4.3 (3). Recall that b N ∩ c I w n ( r ) = N n c N r We have commutativesquares of equivariant maps of torsors N n b N r \ T c I w n ( r ) b T ω E ( r ) b T ω E ( r ) N n b N r \ T c I w n ( r ) N n b N r +1 \ T c I w n ( r +1) b T ω E ( r +1) b T ω E ( r +1) N n b N r +1 \ T c I w n ( r +1) . dlog dlog ∨ dlog dlog ∨ UAL EICHLER-SHIMURA MAPS FOR THE MODULAR CURVE 33
The construction of ω κ, + E is independent of r by Lemma 3.4.3 (1). We obtain commutative diagrams b ω w ( κ ) , + E O A r, + κ O A r, + κ b ω κ, + E O A r +1 , + κ O A r +1 , + κ dlog κ, ∨A dlog κ A and O D r, + κ b ω − w ( κ ) , + E b ω − κ, + E O D r, + κ O D r +1 , + κ O D r +1 , + κ . dlog κ D dlog κ, ∨D The lemma follows. (cid:3)
Corollary 3.5.1.
The map dlog κ (resp. dlog κ ) is surjective and strict (resp. injective and strict) in thepro-Kummer-´etale site after inverting p .Proof. It suffices to work over X w ( ǫ ) × X X ∞ . Then dlog : b ω w ( κ ) E → O A δκ is constructed from the B -equivariantinjection κ ⊗ C p → A δκ ⊗ K C p around 1 ∈ F ℓ via Construction 2.1. Similarly, dlog ∨ : O A δκ → b ω κE correspondsto the B -equivariant surjection A δκ ⊗ K C p → ( κ ⊗ C p around w ∈ F ℓ . (cid:3) Next we prove the compatibility of dlog κ and dlog κ, ∨ with the classical maps. Let κ = ( κ , κ ) be adominant weight of T . Recall the conventions for V κ and ω κE made in Section 2.1. We denote b V κ = V κ ⊗ b Q p b O X and b V + κ = V κ ⊗ b Z p b O + X . By Proposition 3.2.3 we have natural maps V + κ → A r, + κ and D r, + κ → V + − w ( κ ) . Theyextend to maps of sheaves b V + κ → O A r, + κ and O D r, + κ → b V + − w ( κ ) . The maps dlog : T p E ∨ b ⊗ b O X → b ω E anddlog ∨ : b ω − E → T p E ∨ b ⊗ b O X induce mapsdlog κ : b V κ → ω κE ⊗ O X b O X and dlog κ, ∨ : ω w ( κ ) E ⊗ O X b O X → b V κ . Proposition 3.5.1.
Let κ = ( k , k ) be a dominant character of T . The following diagrams are commutative O A rκ b ω κE b ω w ( κ ) E O A rκ b V κ ω κE ⊗ O X b O X ω w ( κ ) E ⊗ O X b O X b V κ b ω − κE O D rκ O D rκ b ω − w ( κ ) E ω − κE ⊗ O X b O X b V − w ( κ ) b V − w ( κ ) ω − w ( κ ) E ⊗ O X b O X dlog κ A dlog κ, ∨A dlog κ ≀ ≀ dlog κ, ∨ dlog κ, ∨D ≀ dlog κ D ≀ dlog − w κ ) , ∨ dlog − w κ ) Proof.
It is enough to prove that the first two diagrams are commutative. Let f ∈ V κ , by the proof of Lemma3.5.1, dlog κ A ( f ) : b T ω E ( r ) → b O X is the function whose value at ( v , v ) ∈ ω + , × E ( r ) × ω + , − , × E ( r ) is(3.9) dlog k A ( f )( v , v ) = f (dlog ∨ ( v ) , ˜ v ) , where ˜ v ∈ T p E ∨ ⊗ b Z p b O + X is a lift of v ∈ ω + E satisfying some congruences. By definition of ω κE as global functionsof the T an -torsor ω × E × ω − , × E , an element f ∈ ω κE is the same as a function f : ω × E × ω − , × E → O X such that f ( t v , t v ) = κ ( t , t ) f ( v , v ). Furthermore, dlog κ, ∨ is given by the same formula (3.9) where ˜ v is any lift of v in T p E ∨ ⊗ b O X . This proves that the first diagram is commutative.Now let f ∈ ω w ( κ ) E , then dlog κ, ∨A ( f ) : b N \ T c I w n ( r ) → b O X is the function whose value at a basis ( e , e ) ∈ T c I w n ( r ) is(3.10) dlog κ, ∨A ( f )( e , e ) = f (dlog( e ) , ˜ e ) , where ˜ e ∈ b ω + , − E is the unique element such that h e , ˜ e i = h e , e i . By the same argument as before, theformula (3.10) coincides with the map dlog κ, ∨ for f ∈ ω w ( κ ) E ∨ . (cid:3) Proposition 3.5.2.
The Tate twists of the dlog maps b ω w ( κ ) E ( κ ) dlog ∨A −−−−→ O A rκ , O A rκ dlog A −−−−→ b ω κE ( κ ) O D rκ dlog D −−−−→ b ω − w ( κ ) E ( − κ ) , b ω − κE ( − κ ) dlog ∨D −−−−→ O D rκ , are Galois equivariant.Proof. It is enough to show the proposition for O A rκ . We prove it for dlog ∨ κ , the case of dlog κ being similar. Let f : b T ω E ( r ) → b O X C p be a function in b ω w ( κ ) E . Recall that an element ( e , e ) ∈ T c I w n ( r ) is a basis of T p E ∨ ⊗ b O + ( − ∨A , the function dlog ∨A ( f ) : T c I w n ( r ) → b O X C p is givenby dlog ∨A ( f )( e , e ) = f (dlog( e ) , e e ) . Let γ ∈ G K and note that dlog( e ) ∈ b ω E ( −
1) and ˜ e ∈ b ω − E . Then γ ∗ dlog ∨ A ( f )( e , e ) = γ dlog ∨ A ( f )( γ − e , γ − e ) = γf ( γ − (dlog( e ) , ^ γ − e )) . Let v ∈ ω + E ( r ) be a generator locally ´etale over X , let v ∨ ∈ ω − E be its dual. Then dlog( e ) = cv and e e = h e ,e i c v ∨ . Therefore, γ − dlog( e ) = χ cyc ( γ ) γ − ( c ) v and ^ γ − e = h γ − e ,γ − e i γ − ( c ) v ∨ = γ − h e ,e i γ − ( c ) v ∨ . Thisimplies that γ ∗ dlog ∨A ( f )( e , e ) = χ cyc ( γ ) κ dlog ∨A ( γ ∗ f )( e , e ) , the proposition follows. (cid:3) Hecke Operators
The U p operator. Let C denote the finite flat correspondance over X = X ( N, p n )(4.1) CX X, p p parametrizing ( E, ψ N , H n , H ′ ), where ( E, H n , ψ N ) ∈ X , and H ′ ⊂ E [ p ] is a cyclic subgroup of order p suchthat H n ∩ H ′ = 0. We define p ( E, ψ N , H n , H ′ ) = ( E, ψ N , H n ) and p ( E, ψ N , H n , H ′ ) = ( E/H ′ , ψ N , H n ). Let π : p ∗ E → p ∗ E be the universal isogeny and π ∨ : p ∗ E → p ∗ E its dual.Let k = ( k , k ) be an algebraic weight, recall the conventions of Section 2 saying that ω kE = ω k − k E . Let usgive the definition of Hecke operators for the coherent cohomology. Definition 4.1.1.
Let π ∗ : p ∗ ω E → p ∗ ω E and π ∨∗ : p ∗ ω − E → p ∗ ω − E be the pullback and pushforward maps ofdifferentials. We define the Hecke operator U naive,kp acting on R Γ( X, ω kE ) as the composition R Γ( X, ω kE ) R Γ( C, p ∗ ω kE ) ( π ∗ ) ⊗− k ⊗ ( π ∨∗ ) ⊗− k −−−−−−−−−−−−−−→ R Γ( C, p ∗ ω kE ) R Γ( X, ω kE ) . p ∗ Tr p We define the U t,naivep,k operator shifting the roles of p and p , and composing with the map ( π ∗ ) ⊗ k ⊗ ( π ∨ , ∗ ) ⊗ k . Remark . The U naivep,k above is equal to the operator p − k U naivep,k − k of [BP20].Before defining Hecke operators for overconvergent cohomologies, we need to understand the behaviour ofthe correspondance (4.1) at the level of spaces. Given a subspace U ⊂ X we denote U p ( U ) = p ( p − ( U )) and U tp ( U ) = p ( p − ( U )). Lemma 4.1.1.
Let X w ( ǫ ) be the ǫ -overconvergent neighbourhood of X ordw , cf. 3.4. The following holds (i). U tp ( X ( ǫ )) ⊂ X ( ǫ + 1) and U p ( X ( ǫ )) ⊃ X ( ǫ − , (ii). U tp ( X w ( ǫ )) ⊃ X w ( ǫ − and U p ( X w ( ǫ )) ⊂ X w ( ǫ + 1) .Proof. Let X ∞ be the perfectoid modular curve, let C ∞ = X ∞ × X,p C . The perfectoid curve C ∞ parametrizes( E, ψ N , ( e , e ) , H ′ ) where ( E, ψ N , ( e , e )) ∈ X ∞ , and H ′ ⊂ E [ p n ] is a cyclic subgroup of oder p such that h e i ∩ H ′ = 0 mod p . Write C ∞ = F a ∈ F p C ∞ ,a with C ∞ ,a the locus such that H ′ = h e + ae i . We obtain adiagram F a C ∞ ,a X ∞ X ∞ p p UAL EICHLER-SHIMURA MAPS FOR THE MODULAR CURVE 35 with p ( E, ψ N , ( e , e ) , H ′ ) = ( E, ψ N , ( e , e )) and p ( E, ψ N , ( e , e ) , h e − ae i ) = ( E/H ′ , ψ N , ( π ( e ) , ˜ e )),where ˜ e = p ( π ( e ) + aπ ( e )) for 0 ≤ a < p lifting a . Let U p,a := (cid:18) − a p (cid:19) . Composing with the Hodge-Tateperiod map π HT : X ∞ → F ℓ we obtain a commutative diagram C ∞ ,a F ℓ F ℓ . π HT ◦ p π HT ◦ p U p,a By Lemma 3.1.2 we obtain(4.2) ( π HT ◦ p )(( π HT ◦ p ) − ( C w ,n ( ǫ )) Iw n ⊂ C w ,n ( ǫ + 1)( π HT ◦ p )(( π HT ◦ p ) − ( C ,n ( ǫ )) Iw n ⊃ C ,n ( ǫ − π HT ◦ p )(( π HT ◦ p ) − ( C w ,n ( ǫ )) Iw n ⊃ C w ,n ( ǫ − π HT ◦ p )(( π HT ◦ p ) − ( C ,n ( ǫ )) Iw n ⊂ C ,n ( ǫ + 1) . There is a natural map of correspondances X ∞ C ∞ X ∞ X C X p p p pp p . Moreover, C ∞ = C × p ,X X ∞ . Since X w ( ǫ ) = p ( π − ( C w,n ( ǫ ))), we deduce the lemma from (4.2). (cid:3) Corollary 4.1.1.
Let X ord ⊂ U be an open quasi-compact overconvergent neighbourhood of the -ordinarylocus of X . Suppose that X ordw ⊂ X \ U is an overconvergent neighbourhood. Then there exists n >> such that ( U tp ) n ( U ) ⊂ U is a strict open immersion. Analogous statements hold taking X ordw or U p .Proof. The affinoids X w ( ǫ ) form a basis of oveconvergent neighbourhoods of X ordw . Then, there exists ǫ > n such that X w ( ǫ ) ⊂ X \ U . Let δ > n be such that X ( δ ) ⊂ U . It is enough to show that there exists M >> X \ ( U t,Mp ( X w ( ǫ ))) ⊂ X ( δ + 1). We can work over the perfectoid modular curve and the perfectoidfinite flat correspondance of the previous lemma. The question is reduced to show that there exists M >> F ℓ \ (cid:18) C w ,n ( ǫ ) (cid:18) p M
00 1 (cid:19)(cid:19) ⊂ C ,n ( δ + 1) . Let F ℓ → P be the isomorphism given by mapping a matrix (cid:18) a bc d (cid:19) to ( c : d ). Then C w ,n ( ǫ ) contains thelocus { ( x : y ) | | y/x | ≤ | p | ǫ } . Then { ( x : y ) | | y/x | ≤ | p | ǫ } (cid:18) p M
00 1 (cid:19) = { ( x : y ) | | y/x | ≤ | p | ǫ − M } and its complement is the locus { ( x : y ) | | x/y | < | p | M − ǫ } , taking M − ǫ > δ + 1 we are done. (cid:3) Proposition 4.1.1. (1)
The map π : p ∗ E → p ∗ E induces a T b T r -equivariant map p ∗ ( b N \ T c I w n ( r ) ) π p −→ p ∗ ( b N \ T c I w n ( r ) ) . (2) The map π : p ∗ E → p ∗ E induces T b T r -equivariant maps p ∗ ( T ω E ( r ) ) π ∗ × p π ∨∗ −−−−−−→ p ∗ ( T ω E ( r ) ) over p − ( X ( ǫ )) p ∗ ( T ω E ( r ) ) p π ∗ × π ∨∗ −−−−−−→ p ∗ ( T ω E ( r ) ) over p − ( X w ( ǫ )) . In other words, a map π p : p ∗ ( T ω E ( r ) ) → p ∗ ( T ω E ( r ) ) given by π p := w ( ̟ − )( π ∗ × π ∨∗ ) over X w ( ǫ ) . (3) We have commutative diagrams p ∗ ( N \ T c I w n ( r ) ) p ∗ ( N \ T c I w n ( r ) ) p ∗ ( b T ω E ( r ) ) p ∗ ( b T ω E ( r ) ) p ∗ ( b T ω E ( r ) ) p ∗ ( b T ω E ( r ) ) p ∗ ( N \ T c I w n ( r ) ) p ∗ ( N \ T c I w n ( r ) ) π p dlog dlog π p dlog ∨ dlog ∨ π p π p Proof.
Recall that all the sheaves are considered over the big pro-Kummer ´etale site. Then p ∗ ( b N \ T c I w n ( r ) ) = b N \ p ∗ ( T c I w n ( r ) ), and p ∗ ( T c I w n ( r ) ) is the c I w n ( r )-torsor constructed from the data ( E, H n ) over C . Similarly, p ∗ ( b N \ T c I w n ( r ) ) = b N \ p ∗ ( T c I w n ( r ) ), and p ∗ ( T c I w n ( r ) ) is the c I w n ( r )-torsor constructed from the data ( E/H ′ , H n )over C .(1) The morphism π : E → E/H induces a map of Hodge-Tate exact sequences(4.3) 0 b ω + , − E/H T p ( E/H ) ∨ b ⊗ b O + C b ω + E/H b ω + , − E T p E ∨ b ⊗ b O + C b ω + E π ∨∗ π ∨ π ∗ We we have a map of GL ( b O C )-torsors(4.4) Isom( T p ( E/H ) b ⊗ b O C , b O C ) Isom( T p E b ⊗ b O C , b O C ) p ∗ (Isom( T p E b ⊗ b O X , b O X )) p ∗ (Isom( T p E b ⊗ b O X , b O X )) π p By definition p ∗ i ( T c I w n ( r ) ) is an open subsheaf of p ∗ i (Isom( T p E b ⊗ b O X , b O X )). In the proof of Lemma 4.1.1, we sawthat locally over C ∞ ,a the map π p is identified with right multiplication by U p,a = (cid:18) − a p (cid:19) , 0 ≤ a < p . Weobtain a commutative diagram over C ∞ ,a p ∗ ( T c I w n ( r ) ) ∼ = c I w n ( r ) GL ( b O C ) ∼ = p ∗ (Isom( T p E b ⊗ b O X , b O X )) p ∗ ( T c I w n ( r ) ) ⊂ b N Λ c I w n ( r ) GL ( b O C ) ∼ = p ∗ (Isom( T p E b ⊗ b O X , b O X )) . R Up,a R Up,a
But b N \ c I w n ( r ) ∼ = b N Λ \ b N Λ c I w n ( r ). Then, the map (4.4) quotients through b N Λ \ p ∗ (Isom( T p E b ⊗ b O X , b O X )) b N Λ \ p ∗ (Isom( T p E b ⊗ b O X , b O X )) p ∗ ( b N \ T c I w n ( r ) ) p ∗ ( b N \ T c I w n ( r ) ) π p π p as desired.(2). By Lemma 4.1.1 we know that E and E/H ′ admit canonical subgroups C m and C ′ m of order p m > p r +1 over p − ( X ( ǫ )) (resp. p − ( X w ( ǫ ))).On p − ( X ( ǫ )) we have that H n = C n is the canonical subgroup, and H is anticanonical.. Then π ∗ : ω + E/H ′ → ω + E is an isomorphism and π ∨∗ : ω + , − E/H → ω + , − E is exactly divisible by p . It is clear that π ∗ v ∈ ω + , × E ( r ) for v ∈ ω + , × E/H ′ ,r . Let v ∈ ω + , − , × E/H ( r ), we know h v , v i = p h π ∗ v , π ∨∗ ( v ) i . This implies that p π ∨∗ ( v ) ∈ ω + , − , × E ( r )and that π p : p ∗ ( T ω E ( r ) ) → p ∗ ( T ω E ( r ) ) is well defined.On p − ( X w ( ǫ )) we have H ′ n ∩ C ′ n = 0 and E [ p ] /H ∼ = H ′ . Therefore H ∼ = ( E/H )[ p ] /H ′ ∼ = C ′ n is the canonicalsubgroup of E [ p ]. Then π ∨∗ : ω + , − E/H → ω + , − E is an isomorphism and π ∗ : ω + E/H → ω + E is exactly divisible by p . It is clear that π ∨∗ ( v ) ∈ ω + , − , × E ( r ) for v ∈ ω + , − , × E/H ( r ). By the same argument as above, one checks that π p : p ∗ ( T ω E ( r ) ) → p ∗ ( T ω E ( r ) ) is well defined.(3). We can work locally pro-Kummer-´etale. Let 0 ≤ a < p , let ( e ′ , e ′ ) ∈ p ∗ T c I w n ( r ) | C ∞ ,a , let f ′ , f ′ ∈ T p ( E/H ) be its dual basis and f , f ∈ T p E the basis such that( f ′ , f ′ ) = ( π ( f ) , p ( π ( f ) + aπ ( f ))) . Let ( e , e ) ∈ p ∗ T c I w n ( r ) be the dual basis of ( f , f ). By definition π ∨ ( e ′ , e ′ ) = ( e ′ ◦ π, e ′ ◦ π ) = ( e − ae , pe )and(4.5) π ∨ ( e , e ) ≡ ( π ∨ ( e ′ ) , p π ∨ ( e ′ )) ∈ b N \ p ∗ T c I w n ( r ) . Let us first check commutativity for w = 1. We have π p (dlog( e ′ ) , ˜ e ′ ) = ( π ∗ (dlog( e ′ )) , p π ∨∗ (˜ e ′ )) = (dlog( π ∨ ( e )) , p π ∨∗ (˜ e ′ )) , UAL EICHLER-SHIMURA MAPS FOR THE MODULAR CURVE 37 where ˜ e ′ ∈ b ω − E/H is the unique element such that h dlog( e ′ ) , ˜ e ′ i = h e ′ , e ′ i . On the other hand, by 4.5 we getthat dlog( π p ( e ′ , e ′ )) = (dlog( π ∨ ( e ′ )) , ^ ( 1 p π ∨ ( e ′ ))) , where ^ ( p π ∨ ( e ′ )) ∈ b ω − E is the unique element such that h dlog( π ∨ ( e ′ )) , ^ ( 1 p π ∨ ( e ′ )) i = h π ∨ ( e ′ ) , p π ∨ ( e ′ ) i = h e , e i But dlog( π ∨ ( e ′ )) = π ∗ (dlog( e ′ )), and we have the equality h e ′ , e ′ i = h e ′ , π ( e )) i = h π ∨ ( e ′ ) , e i = h e − ae , e i = h e , e i . This implies p π ∨∗ (˜ e ′ ) = ^ p π ∨ ( e ′ ) proving the commutativity of the diagram.Let us prove the case w = w . Let ( v , v ) ∈ b T ω E/H ( r ) . By definitiondlog( v , v ) = (dlog ∨ ( v ) , ˜ v ) mod b N . with ˜ v ∈ T p ( E/H ) ⊗ b O + X some lifting of v by dlog. Moreover, we can take ˜ v such that π ∨ (˜ v ) is divisible by p in T p E ⊗ b O + X . Therefore π p (dlog ∨ ( v ) , ˜ v ) = ( π ∨ (dlog ∨ ( v )) , p π ∨ (˜ v )) . But π ∨ (dlog ∨ ( v )) = dlog ∨ ( π ∨∗ ( v )) anddlog( 1 p π ∨ (˜ v )) = 1 p π ∗ (dlog(˜ v )) = 1 p π ∗ ( v )which proves the commutativity in this case. (cid:3) Corollary 4.1.2.
The π p maps of torsors give rise to commutative diagrams p ∗ O b A r, + κ p ∗ O b A r, + κ p ∗ b ω − κ, + E p ∗ b ω − κ, + E over p − ( X w ( ǫ )) p ∗ b ω κ, + E p ∗ b ω κ, + E p ∗ O b D r, + κ p ∗ O b D r, + κ p ∗ b ω w ( κ ) , + E p ∗ b ω w ( κ ) , + E p ∗ O b D r, + κ p ∗ O b D r, + κ over p − ( X ( ǫ )) p ∗ O b A r, + κ p ∗ O b A r, + κ p ∗ b ω − w ( κ ) , + E p ∗ b ω − w ( κ ) , + Eπ ∨ p dlog κ A dlog κ A π p dlog κ, ∨D dlog κ, ∨D π ∨ p π p π ∨ p dlog κ, ∨A dlog κ, ∨A π p dlog κ D dlog κ D π ∨ p π p Let Tr i : p i, ∗ O C → O X denote the trace of p i : C → X . It is clear that Tr i maps O + C to O + X . Completing p -adically we obtain a trace Tr i : b O + C → O + X . We define the trace Tr i : p i, ∗ p ∗ i O A r, + κ → O A r, + κ (resp. for O D r, + κ and O b A r, + κ ) to be the composition p i, ∗ p ∗ i O A r, + κ = p i, ∗ ( p ∗ i ( A r, + κ ) b ⊗ b O + C ) = ( p i, ∗ p ∗ i A r, + κ ) b ⊗ ( p i, ∗ b O + C ) µ ⊗ Tr i −−−−→ A r, + κ b ⊗ b O + X = O A r, + κ where µ : p i, ∗ p ∗ i → Definition 4.1.2 (Hecke operators) . (1) We define the Hecke operator U p on modular symbols as thecomposition R Γ proket ( X C p , O D rκ ) p ∗ −→ R Γ proket ( C C p , p ∗ O D rκ ) π p −→ R Γ proket ( C C p , p ∗ O D rκ ) →→ R Γ proket ( X C p p , ∗ p ∗ O D rκ ) Tr −−→ R Γ proket ( X C p , O D rκ ) . Dually, we define the action of U tp on R Γ( X C p ,proket , O b A rκ ) shifting the roles of p and p , and composingwith π ∨ p . (2) We define the action of U p on R Γ ,ǫ ( X, κ ) by the composition R Γ ,ǫ ( X, κ ) p ∗ −→ R Γ( p − ( X ( ǫ )) , p ∗ ω κE ) π p −→ R Γ( p − ( X ( ǫ )) , p ∗ ω κE ) → Res −−→ R Γ( p − ( X ( ǫ − , p ∗ ω κE ) Tr −−→ R Γ ,ǫ − ( X, κ ) Res −−→ R Γ ,ǫ ( X, κ ) . Dually, we define the action of U tp on R Γ w ,ǫ ( X, κ ) shifting the roles of p and p , and composing with π ∨ p .(3) We define the action of U p on R Γ w ,c,ǫ ( X, κ ) by the composition R Γ w ,c,ǫ ( X, κ ) p ∗ −→ R Γ p − ( X w ( >ǫ +1)) ( p − ( X w ( ǫ )) , p ∗ ω κE ) π p −→ R Γ p − ( X w ( >ǫ +1)) ( p − ( X w ( ǫ )) , p ∗ ω κE ) Cor −−→ R Γ p − ( X w ( >ǫ +2)) ( p − ( X w ( ǫ )) , p ∗ ω κE ) Tr −−→ R Γ w ,c,ǫ +1 ( X, κ ) Cor −−→ R Γ w ,c,ǫ ( X, κ ) . Dually, we define the action of U tp on R Γ ,c,ǫ ( X, κ ) by shifting the roles of p and p , and composingwith π ∨ p . Remark . It follows from the definition that the action of the Hecke operator U p on R Γ( X, O D rκ ) = R Γ proket ( X C p , D rκ ) b ⊗ K C p coincides with the action of the double coset U p = [Iw n (cid:18) p
00 1 (cid:19) Iw n ]. Remark . Let k = ( k , k ) be an algebraic weight. We have that U p = p k U naivep over ω kE /X ( ǫ ) and U p = p k U naivep over ω kE /X w ( ǫ ). Dually, we have U tp = p k U t,naivep over ω kE /X ( ǫ ) and U tp = p k U t,naivep over ω kE /X w ( ǫ ). In other words, U p = k ( w (diag( p, U naivep and U tp = k ( w (diag(1 , p ))) U t,naivep . Proposition 4.1.2.
The map π p : p ∗ ( b N \ T c I w n ( r ) ) → p ∗ ( b N \ T c I w n ( r ) ) factors throught p ∗ ( b N \ T c I w n ( r ) ) p ∗ ( b N \ T c I w n ( r ) ) p ∗ ( b N \ ( b T r T c I w n ( r +1) )) . π p Furthermore, the operators π p : p ∗ ( O b D r, + κ ) → p ∗ ( O b D r, + κ ) and π ∨ p : p ∗ ( O b A r, + κ ) → p ∗ ( O b A r, + κ ) factors through p ∗ ( O b D r, + κ ) π p −→ p ∗ ( O b D r +1 , + κ ) → p ∗ ( O b D r, + κ ) and p ∗ ( O b A r, + ) → p ∗ ( O b A r +1 , + κ ) π tp −→ p ∗ ( O b A r, + κ ) .Proof. By the proof of Proposition 4.1.1 (1), locally pro-Kummer ´etale on C the map π p looks like right multi-plication by the matrix U p,a = (cid:18) − a p (cid:19) on b N Λ \ ( b N Λ c I w n ( r )). This gives the desired factorisation at the levelof T b T r -equivariant sheaves. By taking H om T b T r and noticing that H om T b T r ( b N \ ( b T r T c I w n ( r +1) ) , R K b ⊗ O K b O + X ) ∼ = O A r +1 , + κ , the factorization at the the level of sheaves of modules follows. (cid:3) Corollary 4.1.3. (1)
The Hecke operators of the systems { H proket ( X C p , O A rκ ) } and { H proket ( X C p , O D rκ ) } act equivariantly. (2) The Hecke operators of overconvergent modular symbols and overconvergent coherent cohomologies arecompact operators.Proof.
Part (1) follows from the previous proposition and the definition of the Hecke operators via finite flatcorrespondances. Part (2) follows by the same result, Lemma 3.4.5 and the remark after Proposition 3.3.2. (cid:3)
Definition 4.1.3. (1) We define the U p -operator on the modular symbols R Γ( D κ ) (resp. the U tp operatoron R Γ( A κ )) as the inverse limit of the U p operators of H proket ( X C p , D rκ ) (resp. the colimit of the U tp operators of H proket ( X C p , A rκ )).(2) We define the U p and U tp operators on the perfect complex of compact type R Γ w ( X, κ ) (resp. the perfectFrechet complex R Γ w,c ( X, κ )) as the direct limit of the U p and U tp operators of R Γ w,ǫ ( X, κ ) (resp. theinverse limit of the U p and U tp operators of R Γ w,c,ǫ ( X, κ )).We obtain the following proposition regarding the finite slope
UAL EICHLER-SHIMURA MAPS FOR THE MODULAR CURVE 39
Proposition 4.1.3.
We have isomorphisms of finite slope sheaves R Γ( A κ ) fs = lim −→ r H proket ( X, O A rκ ) fs , R Γ( D κ ) fs = lim ←− r H proket ( X, O D rκ ) fs H w ( X, κ ) fs = lim −→ ǫ H w,ǫ ( X, κ ) fs , H w,c ( X, κ ) fs = lim ←− ǫ H w,c,ǫ ( X, κ ) fs , where all the transition maps of the (co)limits are isomorphisms. Slope estimates of the U p -operator. Modular symbols.
Let k = ( k , k ) be a dominant weight in V ( K ), and let K rk = ker( D rk → V − w ( k ) ) and CK rk := coker( V rk → A rk ). Lemma 4.2.1 ( [AS08, Theo. 3.11.1]) . Let k = ( k , k ) ∈ V ( K ) be an algebraic weight. The action of ̟ = (cid:18) p (cid:19) on CK rk (resp. of ̟ t = (cid:18) p
00 1 (cid:19) on K rk ) satisfies || ̟ || CK rk ≤ | p | k − k +1 , (cid:0) resp. || ̟ t || K rk ≤ | p | k − k +1 . (cid:1) Proof.
Let L ( k ) = { f : BB → A K | f ( bx ) = k ( b ) f ( x ) for all b ∈ B } be the space of k -homogeneous functionsover the big cell of GL . Then V k ⊂ L ( k ) is the subspace of functions which extend to GL . Moreover, L ( k ) isthe admissible dual of the Verma module V ( k ). Thus the maximal weight of L ( k ) /V k is ( k − , k + 1). Thespace L ( k ) is a dense subspace of A rk . Moreover, the normalized action of ̟ on f ∈ L ( k ) is( ̟ · f )( x ) = k − ( ̟ ) f ( x̟ ) . By looking at weight vectors one finds || ̟ || L ( k ) /V k ≤ | p | k − k +1 , which gives the same bounds on CK rk . By taking duals, since the center of GL acts trivially on D rk , one obtainsthe bound for K rk . (cid:3) With the same lines of Theorem 6.4.1 in loc. cit applied to b A rκ one deduces the next result Theorem 4.2.1.
Let k = ( k , k ) be a dominant weight in V ( K ) . The maps D rk → V − w ( k ) and V k → b A rk induce isomorphisms of the ( < k − k + 1) -slope part for the action of the U p operators H proket ( X C p , O D rk ) U p This follows by Lemma 4.2.1, the definition of the Hecke operators and Remark 3.3.4, see [AS08]. (cid:3) Remark . It is not longer true that the U p -operators are compact using just the complex constructed inTheorem 3.3.1, one has to pass through the quasi-isomorphism with Betti cohomology. The reason is that astrict map of perfectoid affinoids Spa( A, A + ) ⊂ Spa( B, B + ) does not give a compact map B → A (which holdstrue for adic spaces locally of finite type). Indeed, let K be any perfectoid field and take B = K h T /p ∞ i and A = K h ( Tp ) /p ∞ i . However, the theory of locally analytic vectors for the modular curve developed by Lue Panin [Pan20] provides some insights of whats should be the solution: the cohomology hocolim δ R Hom Iw n ( Z p , O A δκ )is the κ -isotipic component of the sheaf O laX of locally analytic vectors of b O X , in particular concentrated in degreezero. This last can be computed using small ˇCech covers, and if U ⊂ U ′ is a strict immersion of (small enough)affinoids of X , then O δ − an ( U ′ ) → O δ +1 − an ( U ) is compact, cf. Theorem 4.3.9 of loc. cit. Overconvergent modular forms. We will need the following slope bounds for overconvergent coherentcohomology. Lemma 4.2.2 ( [BP20, Lemma 5.3] ) . Let k = ( k , k ) ∈ be an algebraic weight. Then U naivep has slopes ≥ − k on R Γ ,ǫ ( X, k ) and slopes ≥ − k on R Γ w ,c,ǫ ( X, k ) . Dually, the U t,naivep operator has slopes ≥ − k on R Γ ,c,ǫ ( X, k ) and slopes ≥ − k on R Γ w ,ǫ ( X, k ) . Define normalizations of U p and U tp acting on overconvergent cohomologies U goodp = p U p over R Γ ( X, κ ) ,U p over R Γ w ,c ( X, κ ) p − min { − k , − k } over R Γ( X, ω k ) , and U t,goodp = p U tp over R Γ w ( X, κ ) ,U tp over R Γ ,c ( X, κ ) p − min { − k , − k } U t,naivep over R Γ( X, ω k )One obtains the classicality theorem of coherent overconvergent cohomologies. Theorem 4.2.2. Let k = ( k , k ) be an algebraic weight. (1) The U goodp -operator has slopes ≥ on H ( X, k ) and H w ,c ( X, k ) . (2) The U t,goodp operator has slopes ≥ on H w ( X, k ) and H ,c ( X, k ) .Futhermore, we have isomorphisms of small slope cohomologies H ( X, k ) U goodp Classical Eichler-Shimura decomposition. Let X = X ( N, p n ) C p and π : E → X be the universalgeneralized elliptic curve over X . Let k = ( k , k ) ∈ X ∗ ( T ) be a dominant weight. Consider the k -th fold tensorproduct of the dlog maps(5.1) b ω w ( k ) E ( k ) dlog ∨ −−−→ V k ⊗ b O X , V k ⊗ b O X dlog −−−→ b ω kE ( k ) . Let ν : X proket → X ket be the natural map of sites. We have the projection formula R ν ∗ b O X (1) = Ω X (log). Tak-ing pro-Kummer-´etale cohomology in 5.1, and composing with the connection maps we obtain G K -equivariantmaps(5.2) H ( X an , ω w ( k ) E )( k ) ES ∨ −−−→ H ( X ket , V k ) ⊗ K C p ES −−→ H ( X an , ω kE ⊗ Ω X (log))( k − . We have the following theorem of Faltings Theorem 5.1.1. The composition ES ◦ ES ∨ of (5.2) is zero. Moreover, ES ∨ is injective and ES has a naturalsection. In particular, we have an Eichler-Shimura decomposition H ket ( X, V k ) ⊗ K C p (1) ∼ = H ( X an , ω w ( k ) E )( k + 1) ⊕ H ( X an , ω kE ⊗ Ω X (log))( k ) . Next, we present a different proof of this fact using a pro´etale version of the dual BGG decompositionof [FC90]. Let Y ⊂ X be the open modular curve and let H dR = H dR ( E/Y ) denote the relative de Rhamcohomology of E/Y . The relative comparison Theorem [Sch13a, Theo. 8.8] says that there is a naturalisomorphism T p E ( − ⊗ b Z p O B dR ∼ = H dR ⊗ O X O B dR . Let H dR also denote the natural extension of the de Rham cohomology to X , cf. [FC90]. Lue Pan showsin [Pan20, § 4] that we still have a canonical isomorphism in the pro-Kummer-´etale site T p E (1) ⊗ b Z p O B dR, log ∼ = H dR ⊗ O X O B dR, log . Furthermore, he proves the following Proposition 5.1.1. The ω E -twist of the Hodge-Tate exact sequence is isomorphic to the Faltings extension → b O X (1) → gr O B + dR, log → b Ω E (log) → via the opposite of the Kodaira-Spenser isomorphism − KS : ω E ∼ −→ Ω E (log) .Proof. Let ξ ∈ B + dR be a local generator of θ : B + dR → b O X . Consider the B + dR -lattices in H dR ⊗ O X O B dR, log M = ( T p E ( − ⊗ O B + dR, log ) ∇ =0 M = ( H dR ⊗ O X O B + dR, log ) ∇ =0 . The Hodge filtration of H dR being concentrated in degrees 0 and 1 implies that ξ M ⊂ M ⊂ M . By Proposition7.9 of [Sch13a] we have that M / M = gr ( H dR ) ⊗ b O X ∼ = b ω E and M /ξ M ∼ = gr ( H dR ) ⊗ b O X ∼ = b ω − E . Therefore,0 → M /ξ M → M /ξ M → M / M → → ξ M /ξ M → M /ξ M → M /ξ M → UAL EICHLER-SHIMURA MAPS FOR THE MODULAR CURVE 41 recover the Hodge-Tate and the Hodge exact sequences of T p E ⊗ b O X ( − 1) and H dR ⊗ O X b O X respectively. Wehave a morphism of complexes0 M M ⊗ B + dR O B + dR, log M ⊗ B + dR O B + dR, log ⊗ Ω X (log) 00 M M ⊗ B + dR O B + dR, log M ⊗ B + dR O B + dR, log ⊗ Ω X (log) 0 ∇∇ Let e θ : O B + dR, log → b O X be the augmentation map. Taking the first graded piece of the first exact sequenceabove, and its pullback by the image of the quotient of the second one, we obtain a short exact sequence(5.3) 0 → ξ M /ξ M → M ⊗ ker e θ + ξ M ⊗ O B + dR, log M ⊗ (ker e θ ) + ξ M ⊗ ker e θ ∇ −→ M ⊗ O B + dR, log M ⊗ ker e θ + ξ M ⊗ O B + dR, log ⊗ Ω X (log) → M /ξ M ⊗ Ω X (log) ∼ = b ω − E ⊗ Ω X (log). The middle term equalsgr ( H dR ⊗ O B + dR, log ) = ξ M ξ M ⊕ M ξ M ⊗ gr ( O B + dR, log )= gr ( H dR ) ⊗ b O X ⊕ gr ( H dR ) ⊗ gr ( O B ∗ dR, log ) ∼ = b ω E ⊕ b ω − E ⊗ gr ( O B + dR, log ) . The restriction of ∇ to b ω E coincides with the Kodaira-Spencer map by construction, and its restriction to b ω − ⊗ gr ( O B + dR, log ) coincides with the ω − E -twist of ∇ : gr ( O B + dR, log ) ∼ −→ b Ω X (log). Tensoring with ω E we get ashort exact sequence 0 → T p E ⊗ b ω E → b ω E ⊕ gr ( O B + dR, log ) ( KS, ∇ ) −−−−−→ b Ω X (log) → T p E ⊗ b ω E → gr ( O B + dR, log ) is an isomorphism. Indeed, the terms involved are locallypro-Kummer-´etale finite free b O X -modules, and KS : b ω E → b Ω X (log) is an isomorphism. Moreover, we have anisomorphism of extensions0 b O X (1) T p E ⊗ b ω E b ω E b O X (1) gr ( O B + dR, log ) b Ω X (log) 0 Φ − KS ∇ as wanted. (cid:3) Recall from Section 2 the period sheaf O C log ( • ) = gr • ( O B dR, log ). Let let θ : B + dR → b O X and ξ ∈ ker θ a localgenerator. The 0-th graded piece of O C log ( • ) can be described as O C log := O C log (0) = lim −→ n ξ − n gr n O B + dR, log . More explicitely, let U → X be a log affinoid perfectoid such that O B + dR, log | U ∼ = B + dR [[ T ]]. Then O C log ∼ = b O U (cid:20) Tξ (cid:21) and ξ − n gr n O B + dR, log ⊂ O C log is the b O U -submodule of polynomials in T /ξ of degree ≤ n . By Example 2.2.1this holds locally analytic in the perfectoid modular curve X ∞ . Let g , b denote the Lie algebra of GL and B respectively, we use similar notation for the Lie algebras of the others algebraic groups. Let p : X ∞ → X bethe projection and π HT : X ∞ → F ℓ the Hode-Tate period map. Lemma 5.1.1. Let L (0) = { f : BB → A K | f ( bg ) = f ( g ) } be the space of algebraic left B -invariant functionsin the big cell of F ℓ , endowed with the right regular action of ( g , B ) . Let L (0) be the GL -equivariant quasi-coherent sheaf over F ℓ attached to L (0) . There is a natural Galois-equivariant isomorphism ( p ∗ π ∗ HT ( L (0))) / Iw n ∼ = O C log . Proof. Recall the conventions made in Section 2.1. For a character k ∈ X ( T ), let L ( k ) = L (0) ⊗ k = { f : BB → A K | f ( bg ) = k ( b ) f ( g ) } . The dual of the standard representation of GL has maximal weight k = (1 , St ∨ ⊂ L ( k ) correspond to the polynomials of the degree ≤ L ( k ) ∼ = O ( N ).Let v ∈ St be the dominant weight vector such that v ( bb ) = k ( b ). The pullback of St via π HT is naturallyisomorphic to T p E . We have a natural isomorphism of locally algebraic B -modules L (0) = lim −→ n Sym n ( St ⊗ k − ) where the transition maps are provided by x x ⊗ v . We obtain a GL -equivariant isomorphism of quasi-coherent sheaves over F ℓ L (0) = lim −→ n Sym n ( S t ⊗ L (0 , − π HT and using the previous proposition, we get Iw n -equivariant isomorphisms of sheavesover X ∞ π ∗ HT L (0) = lim −→ n Sym n ( T p E ⊗ b ω − E ( − −→ n Sym n ( ξ − gr O B + dR, log )= lim −→ n ξ − n gr n O B + dR, log = O C log (0) . Taking the invariants of the projection to X we get the result. (cid:3) The quotient b \ g is a right ( g , B ) for which N acts trivially. Then we can consider ( b \ g ) ∨ as a left B -modulevia its projection to T . Proposition 5.1.2. The dual of the standard resolution → K → L (0) → ( b \ g ) ∨ ⊗ L (0) → pullbacks via π ∗ HT ( − ) and ( p ∗ ( − )) / Iw n to the de Rham complex → b O X → O C log (0) ∇ −→ O C log ( − ⊗ Ω X (log) → . Proof. Proposition 2.3.9 of [CS17] implies that the pullback of a T -representation to X via the Hodge-Tateperiod map, or the standard principal principal bundle are the same. Let α = (1 , − ∈ T . Section 4 of [FC90,Ch. IV] implies that the isomorphism w ( α ) K ∼ = ( b \ g ) ∨ : 1 X ∨ := (cid:18) (cid:19) ∨ pullbacks to the Kodaira-Spencer isomorphism ω αE ∼ = ω E KS −−→ Ω X (log) over X .The B -module L (0) has an increasing filtration Fil n L (0) given by the polynomials of degree ≤ n . Thedifferential d : L (0) → ( b \ g ) ∨ ⊗ L (0) ∼ = L (0) ⊗ ( b \ g ) ∨ is by definition d ( f ) = X ∨ ⊗ ddn ( f ) = − ddn ( f ) ⊗ X ∨ , where g ∈ BB = NTN is written as g = ntn . Thus, it suffices to show that d : Fil L (0) /K ∼ −→ ( b \ g ) ∨ maps f : bn n to − X ∨ which is clear. (cid:3) Theorem 5.1.2. Let k = ( k , k ) be an algebraic weight, α = (1 , − and w · k = w ( k ) − α the dot action.Let L ( k ) := k ⊗ L (0) . The dual BGG exact sequence of ( g , B ) -modules (5.4) 0 → V k → L ( k ) → L ( w · ( k )) → pullbacks via π ∗ HT ( − ) and ( p ∗ ( − )) /Iw n to a short exact sequence (5.5) 0 → b V k → b ω w ( k ) E ⊗ O C log ( k ) → b ω k + αE ⊗ O C log ( k − → . Moreover, (5.5) is a quasi-isomorphic direct summand of the de Rham complex → V k ⊗ b O X → V k ⊗ O C log (0) ∇ −→ V k ⊗ O C log ( − ⊗ Ω X (log) → . Proof. Since L ( k ) = k ⊗ L (0) as B -modules, we have that π ∗ HT ( L ( k )) = ω w ( k ) E ⊗ O C log (0), similar for w · ( k ). Let Z ( g ) be the center of the enveloping algebra of g , let H, X, Y ∈ sl ⊂ g be the standard basis and Z = (cid:18) (cid:19) .Then Z ( g ) = K [ C, Z ] is a polynomial algebra in C and Z , where C = H + 1 + 2 XY + 2 Y X is the Casimiroperator.By definition, the sequence (5.4) is equal to the common eigenspace C = ( k − k + 1) and Z = k + k ofthe standard resolution(5.6) 0 → V k → L (0) ⊗ K V k → (( b \ g ) ∨ ⊗ L (0)) ⊗ K V k → . Moreover, the inclusion of (5.4) in (5.6) is a homotopy equivalence. By Proposition 5.1.2, the short exactsequence (5.4) pullbacks to (5.5) when seen as a direct summand of the standard resolution. The theoremfollows. (cid:3) UAL EICHLER-SHIMURA MAPS FOR THE MODULAR CURVE 43 Corollary 5.1.1. Let ν : X proket → X ket be the projection of sites. Then (1) Rν ∗ b V k = ω w ( k ) E ( k )[0] ⊕ ω k + αE ( k − − . (2) We have the Eichler-Shimura decomposition H ( Y et , V k ) ⊗ C p = H ( X an , ω w ( k ) E )( k ) ⊕ H ( X an , ω k + αE )( k − . Proof. By the previous theorem, (5.5) is a resolution of b V k . Let ν : X proket → X ket be the projection of sites,we have the projection formula [Sch13a, Prop. 6.16] Rν ∗ O C log ( i ) = O X ( i ) . Then Rν ∗ b V k = ω w ( k ) E ( k ) → ω kE ( k − H et above and the primitive comparison Theorem of [DLLZ19]. (cid:3) Example 5.1.1. The standard representation St has maximal weight (1 , X is T p E . Let k ≥ H ( Y et , Sym k ( T p E )) ⊗ Q p C p (1) = H ( X an , ω − kE )( k + 1) ⊕ H ( X an , ω k +2 E ) . We recall Faltings proof of compatibility of Poincar´e and Serre duality and the Eichler-Shimura decomposi-tion.Let j ket : Y et → X ket be the open immersion of modular curves, let ι : D → X be the cusp divisor. Given a F p -local system on Y et , we define R Γ ket,c ( Y, L ) = R Γ ket ( X, j ! L ), cf. Section 5 of [DLLZ19]. For j ≥ H jket,i ( Y, L ) as the image of H jket,c ( Y, L ) in H jket ( X, j ∗ L ) = H jet ( Y, L ), they are called the Kummer-´etaleinterior cohomology groups. Lemma 5.1.2. Let j et : Y et → X et be the natural morphism of sites. (1) The is a natural isomorphism R Γ et,c ( Y, L ) ∼ −→ R Γ ket,c ( Y, L ) . In particular, the interior ´etale cohomology H jet,i ( Y, L ) is equal to the interior Kummer-´etale cohomology. (2) We have a distinguished triangle R Γ et,c ( Y, L ) → R Γ ket ( X, j ket, ∗ L ) → R Γ ket ( D, ι ∗ j ket, ∗ L ) + −→ Proof. Let ǫ ket : X ket → X et be the natural map of sites. There is an isomorphism j et, ! ∼ −→ ǫ ket, ∗ ◦ j ket, ! . Then,we have a map j et, ! L → Rǫ ket, ∗ j ket, ! L . But Lemma 4.4.27 of [DLLZ19] implies that the right hand side complex has no higher cohomology, this showsthat the previous arrow is an equality. Taking ´etale cohomology we obtain (1). For (2) notice that there is ashort exact sequence in X ket → j ket, ! L → j ket, ∗ L → ι ∗ j ket, ∗ L → . (cid:3) The previous lemma extends straightforwardly to b Q p -local systems. Let k = ( k , k ) be a dominant weight.We can compute explicitely the cohomology of R Γ ket ( D, ι ∗ V k ) Lemma 5.1.3. Let ξ ∈ D be a cusp endowed with the induced log structure from X . Let γ ∈ GL ( Z ) be anelement such that γ − Bγ ∩ Iw n is the stabilizer of ξ in Iw n , and let N γ denote its unipotent radical. Then thereis an isomorphism R Γ proket ( ξ, V k | ξ ) = R Γ cont ( N γ , V k ) . Moreover, H i ( N γ , V k ) = if i > ω w ( k ) E | ξ ( k ) if i = 0 ω kE | ξ ( k − if i = 1 . Proof. First note that a cusp belongs to the ordinary locus of X . To compute the pro-Kummer-´etale cohomologyof V k at the log point ξ , it suffices to find a log perfectoid point e ξ over ξ trivializing V k . If e ξ is Galois of groupΠ over ξ ,, then a Hochschild-Serre spectral sequence shows that R Γ proket ( ξ, V k | ξ ) = R Γ cont (Π , V k | ξ ) . Consider the fiber X ∞ ,ξ of ξ in the perfectoid modular curve, any point ξ ∈ X ∞ ,ξ satisfies π HT ( ξ ) ∈ F ℓ ( Q p ).In particular, there exists a section ξ → X ∞ ,ξ /N γ by taking an adequate basis of ω E | ξ via the canonical group.Using this section, the fiber of ξ in over X ∞ → ( X ∞ /N γ ) gives the log perfectoid point ˜ ξ . In particular R Γ proket ( ξ, V ψ | ξ ) = R Γ cont ( N γ , V ψ ) . On the other hand, the pro-finite group N γ is isomorphic to Z p (1) as we are taking p -th power roots of anelement in M X | ξ . Without loss of generality, suppose that N = N γ = N ∩ Iw n . Let n α ∈ N be such that n α − N . Then H i ( N, V ψ ) = ker( V ψ n ψ − −−−−→ V ψ ) if i = 0 , coker( V ψ n ψ − −−−−→ V ψ ) if i = 1 , i > . The lemma follows from the B -filtration of V k , cf Section 2.1. (cid:3) Proposition 5.1.3 ( [Fal87, Lemma 4.1]) . We have a commutative diagram H ket ( X, V k ) ⊗ Q p C p H ( X, ω kE ⊗ Ω X (log))( k − Q ξ ∈ D H proket ( ξ, V k | ξ ) Q ξ ∈ D ω kE | ξ ( k − ES res ∼ In particular, the Eichler-Shimura isomorphism induces a Galois-equivariant decomposition H et,i ( Y, V k ) ⊗ Q p C p (1) = H an ( X, ω w ( k ) E )( k + 1) ⊕ H an ( X, ω kE ⊗ Ω E )( k ) . Proof. We already know that the maps of the commutative diagram are Galois-equivariant, we forget the actionof G K in the following.It is enough to prove the commutativity for one ξ ∈ D . Without loss of generality we can assume thatthe monodromy of ξ is N ⊂ Iw n . Since the cohomology of ι ∗ V k depends only on a neighbourhood of ξ , themap H ket ( X, V k ) ⊗ Q p C p → H proket ( ξ, V k | ξ ) factors through H ( U, R ν ∗ ( b V k )) = H ( U, ω ψE ⊗ Ω X (log)) for someaffinoid neighbourhood U of ξ no containing other cusp. We use the Faltings exact sequence(5.7) 0 → b V k (1) → V k ⊗ Q p O C log (1) → V k ⊗ Q p O C log (0) ⊗ O X Ω X (log) → ξ . By Theorem 5.1.2, it is enough to check this at the level of representations. Consider the dualBGG resolution of V k → V k → L ( w ( k )) → L ( k ) ⊗ ( b \ g ) ∨ . Let S denote the variable of N , and let dS ∈ ( b \ g ) ∨ be a generator. Take β ∈ k C p . Then β ⊗ dS ∈ L ( k ) ⊗ ( b \ g ) ∨ is invariant under the action of N . Therefore, it defines a class in H ket ( ξ, V k ) given as follows: ,Let n α = (cid:18) (cid:19) ∈ N . Identify L ( w ( k )) = w ( k ) ⊗ L (0) = w ( k ) ⊗ C p [ T ]. Then βk − k + 1 w ( k ) ⊗ T k − k +1 is a lift of β ⊗ dT , and( n α − βk − k + 1 w ( k ) ⊗ T k − k +1 = βk − k + 1 w ( k )(( T + 1) k − k +1 − T k − k +1 ) ≡ β ⊗ w ( k ) T k − k mod ( n α − V k = β ⊗ k ∈ V k / ( n α − V k = k C p , which ends the proof. (cid:3) Corollary 5.1.2. Let α = (1 , − ∈ X ∗ ( T ) . The interior cohomology has an Eichler-Shimura decomposition H et,i ( Y, V k ) ⊗ K C p (1) ∼ = H an ( X, ω w ( k ) E )( k + 1) ⊕ H an ( X, ω kE ⊗ Ω X )( k ) . To prove the compatibility of Poincar´e and Serre duality it suffices to show the compatibility of Poincar´e andSerre duality traces for algebraic curves in the pro-´etale framework. We recall the Lemma 3.24 of [Sch13b] UAL EICHLER-SHIMURA MAPS FOR THE MODULAR CURVE 45 Lemma 5.1.4. Let X be a proper smooth adic space over Spa( C p , O C p ) . Consider the exact sequence in X proet → b Z p (1) → lim ←− p G m → G m → . The boundary map G m,et → R ν ∗ Z p (1) induces a commutative diagram of sheaves over X et G m,et R ν ∗ b Z p (1)Ω X,et R ν ∗ b O X (1) dlog F E where the lower horizontal map is given by the Faltings’s extension. Proposition 5.1.4. Let X be a proper smooth curve over Spec( C p ) , and let X be its analytification over Spa( C p , O C p ) . The following holds (1) H proet ( X , b O X (1)) = H et ( R ν ∗ b O X (1)) . (2) Let Tr P : H et ( X, Z p (1)) → Z p denote the Poincar´e duality trace. Then the composition H ( X, Ω X ) F E −−→ H proet ( X , b O X (1)) = H et ( X, Z p (1)) ⊗ Z p C p Tr P −−→ C p is the Serre duality Trace.Proof. Consider the spectral sequence H pet ( X , R q ν ∗ b O X (1)) ⇒ H p + qproet ( X , b O X (1)) . We know that R p ν ∗ b O X (1) is a coherent sheaf for all p, then H pet ( X , R p ν ∗ b O X (1)) = 0 for q > 1. The degeneracyof the spectral sequence implies part (1).(2) Recall that the isomorphism H et ( X, Z p (1)) ⊗ Z p C p = H proet ( X , b O X (1)) is induced by the inclusion b Z p (1) ֒ → b O X (1). By Lemma 5.1.4 we have a commutative diagram H ( X, G m ) H et ( X, Z p (1)) H ( X, Ω X,et ) H et ( X , b O X (1)) C p dlog Tr P F E Tr P Since the image of dlog is dense, it suffices to show that for α ∈ H ( X, G m ), the composition Tr P ◦ F E (dlog( α ))is the Serre duality trace. Let L /X be a line bundle, we can see L as a subsheaf of the rational functions K X of X . Let { U i } i be a covering of X by affines trivializing L , and L | U i = O U i e i ⊂ K X a trivialization. Wewrite as usual U ij = U i ∩ U j . Let c ij = e i e − j ∈ O × U ij , then dlog( L ) = ( dc ij c ij ) ij ∈ H ( X, Ω X,et ). We have thatTr P L = X x ∈ X v x ( L )where for x ∈ U i , v x ( L ) = v x ( e i ) is the valuation at x . On the other hand, the Serre trace of dlog( L ) is X x ∈ X res x (dlog( L ))where res x (dlog( L )) is the residue at x of dlog e i for x ∈ U i . But we know thatres x (dlog e i ) = v x ( e i ) , which finish the proof. (cid:3) Corollary 5.1.3. The Poincar´e pairing H et,i ( Y, V k )(1) × H et,i ( Y, V − w ( k ) ) ∪ −→ H et,i ( Y, Q p (1)) Tr P −−→ Q p (1) is compatible with the Serre pairings H ( X, ω − kE ) × H ( ω kE ⊗ Ω X ) ∪ −→ H ( X, Ω X ) Tr S −−→ C p (resp. ω w ( k ) E and ω − w ( k ) E ), and the Eichler-Shimura decomposition.Proof. By definition of interior cohomology the corollary reduces to prove that the Poincar´e and Serre dualitytraces are compatible on X which is exactly Proposition 5.1.4. (cid:3) Overconvergent Eichler-Shimura decomposition. Let V ⊂ W be an open affinoid subspace of theweight space. Let Y = Y ( N, p n ) C p be the modular curve of level Γ ( N ) ∩ Γ ( p n ) and X = X ( N, p n ) its naturalcompactification. We take r ≥ n adapted to V as in Proposition 3.2.1. In Sections 3.3 and 3.4 we have definedcomplexes of perfect Banach spaces over V . We proved that the finite slope of their cohomology groups glue tocoherent sheaves over A W \{ } , whose support is quasi-finite over W . Namely H proket ( X, O D W ) fs , H proket ( X, O A W ) fs , H w ( X, κ ) fs and H w,c ( X, κ ) fs . Then, in Section 3.5 we have constructed dlog maps between the sheaves of overconvergent modular forms andsheaves of principal series and their duals. As a byproduct we have the following lemma Lemma 5.2.1. Let κ = ( κ , κ ) be κ V or k ∈ V ( K ) . Let α = (1 , − ∈ X ∗ ( T ) . The dlog κ and dlog κ, ∨ mapsgive rise Galois and U goodp -equivariant applications of cohomology groups H proket ( X, O D rκ ) ES D −−−→ H ,ǫ ( X, − w ( κ ) + α )( − κ − over X ( ǫ ) H w ,c,ǫ ( X, − κ )( − κ ) ES ∨D −−−→ H proket ( X, O D rκ ) over X w ( ǫ ) . Dually, we have Galois and U t,goodp -equivariant maps H ,c,ǫ ( X, w ( κ ))( κ ) ES ∨A −−−→ H proket ( X, O A rκ ) over X ( ǫ ) H proket ( X, O A rκ ) ES A −−−→ H w ,ǫ ( X, κ + α )( κ − over X w ( ǫ ) Proof. Let ν an : X proket → X an be the natural projection. The dlog map induces morphisms of complexes Rν an, ∗ b ω w ( κ ) E Rν an, ∗ dlog ∨ −−−−−−−−→ Rν an, ∗ O A rκ over X ( ǫ ) Rν an, ∗ O A rκ Rν an, ∗ dlog −−−−−−−→ Rν an, ∗ c ω Eκ over X w ( ǫ )Tensoring the Poincar´e complex (2.6) of O C log with ω κE one gets a short exact sequence0 → b ω κE → ω κE ⊗ O X O C log (0) → ω κE ⊗ O X O C log ( − ⊗ Ω X (log) → . As X w ( ǫ ) is affinoid, not considering the Galois action we have that Rν an, ∗ b ω κE = ω κE [0] ⊕ ω κ + αE [ − . Therefore, taking H -hypercohomology we obtain maps ES ∨A : H ,c,ǫ ( X, w ( κ )) → H X ( >ǫ +1) ( X ( ǫ ) , Rν an, ∗ O A rκ ) Cor −−→ H proket ( X, O A rκ ) over X ( ǫ ) ES A : H proket ( X, O A rκ ) Res −−→ H ( X w ( ǫ ) , Rν an, ∗ O A rκ ) → H w ,ǫ ( X, κ + α ) over X w ( ǫ ) . The maps for O D rκ are defined in a similar fashion. The Galois and Hecke equivariance follows from Proposition3.5.2 and Corollary 4.1.2 respectively. (cid:3) Let us state the main theorem of this paper. We will focus in the case of O D , the statements for O A arecompletely analogous. Theorem 5.2.1. Keep the previous notation. (1) The composition of the following maps is zero (5.8) H w ,c,ǫ ( X, − κ un V )( − κ un ) ES ∨D −−−→ H proket ( X, O D r V ) ES D −−−→ H ,ǫ ( X, − w ( κ un V ) + α )( − κ un − . Moreover, the finite slope part glue to maps of sheaves over A W \{ } H w ,c ( X, − κ un ) fs ( − κ un ) ES ∨D −−−→→ H proket ( X, O D W ) fs ES D −−−→ H ( X, − w ( κ un ) + α ) fs ( − κ un − . (2) Let ν an : X proket → X an be the projection of sites. Let k = ( k , k ) ∈ V ( K ) be a dominant weight. Wehave commutative diagrams UAL EICHLER-SHIMURA MAPS FOR THE MODULAR CURVE 47 H w ,c,ǫ ( X, − κ un V )( − κ un ) H w ,c,ǫ ( X, − k )( − k ) H an ( X, ω − kE )( − k ) H proket ( X, O D r V ) H proket ( X, O D rk ) H ket ( X, V − w ( k ) ) ⊗ C p H ,ǫ ( X, − w ( κ un V ) + α )( − κ un − H ,ǫ ( X, − w ( k ) + α )( − k − H an ( X, ω − w ( k )+ αE )( − k − H X w ( >ǫ +1) ( X, Rν an, ∗ O D r V ) H ( X ( ǫ ) , Rν an, ∗ O D r V ) ES ∨D ES D ES ∨ ES Cor ResCor Res Similarly for O A shifting and w . (3) The maps of (2) are Galois and Hecke equivariant with respect to the U goodp operators. In particular,their finite slope part fits in a similar commutative diagram. (4) Locally around k ∈ V , the ( < k − k + 1) -slope part of (5.8) is a short exact sequence of finite free O ( V ) -modules.Proof. Part (1) follows immediately from parts (2) and (3) as we have a distinguished triangle R Γ X \ X ( ǫ ) ( X, Rν an, ∗ O D rκ ) → R Γ( X, Rν an, ∗ O D rκ ) → R Γ( X ( ǫ ) , Rν an, ∗ O D rκ ) + −→ and X w ( > ǫ + 1) ⊂ X \ X ( ǫ ). Indeed, the formation of the cohomologies in(5.8), and the maps ES , ES ∨ ,commute with open embeddings V ′ ⊂ V as this holds true for the dlog maps. In particular, their finite slopecan be glued to a sequence of coherent sheaves over A W \{ } by Corollary 4.1.3 and Proposition 2.3.2. Parts (2)and (3) and the classicality Theorems 4.2.1 and 4.2.2 imply part (4) as proved in [AIS15]; one has a sequence oflocally free O ( V )-modules which is short exact at the point k ∈ V . Taking the connected component containing k one may assume O ( V ) a domain, then locally Zariski in Supp k ∈ Spec( O ( V )) we have the desired short exactsequence. The commutativity of part (2) is a direct consequence of the definition of the classical Eichler-Shimuramaps, the Lemma 5.2.1, and Proposition 3.5.1.It is left to prove Galois and Hecke equivariance of the diagram in (2). Hecke equivariance follows from thedefinition of the U p -operators for overconvergent cohomologies (Definition 4.1.2), Corollary 4.1.3, and the factthat the U naivep -operator is U goodp for ω αE ∼ = Ω E (log). Finally, Galois equivariance follows from Proposition 3.5.2and the construction of Lemma 5.2.1. (cid:3) Compatibility with Poincar´e and Serre duality. Let V ⊂ W be an open affinoid, X = X ( N, p n ) C p , Y ⊂ X the affine modular curve and D = X \ Y . Let F be a sheaf over X proket . We define R Γ proket,c ( Y, F ) asthe cone R Γ proket,c ( Y, F ) → R Γ proket ( X, F ) → R Γ proket ( D, F | D ) + −→ If F is a F p -local system over X this notion coincides with the usual cohomology with compact supports.Similarly, if p : X proket × V an → V an is the projection and F is a sheaf over X proket × V an , we define Rp , ∗ ,c ( Y, F ) in a similar way. Lemma 5.3.1. Let κ be κ V or k ∈ V ( K ) . For w ∈ W denote D w = D ∩ X ordw . (1) Let ξ ∈ D , then there is a natural map H proket ( ξ, O D rκ ) → ω − w ( κ ) E | ξ . such that the following diagram is commutative H proket ( X, O D rκ ) H ( X ( ǫ ) , ω − w ( κ ) E ⊗ Ω X (log)) H proket ( ξ, O D rκ ) ω − w ( κ ) E | ξES D res (2) Let ξ ∈ D w , then there is a natural map H proket ( ξ, O A rκ ) → ω κE | ξ . such that the following diagram is commutative H proket ( X, O A rκ ) H ( X w ( ǫ ) , ω κE ⊗ Ω X (log)) H proket ( ξ, O A rκ ) ω κE | ξ . ES A res8 JUAN ESTEBAN RODR´IGUEZ CAMARGO Proof. Let γ ∈ GL ( Z p ) be such that γ − Bγ ∩ Iw n is the stabilizer of ξ in I w n , and let N γ denote its unipotentradical. By the proof of Lemma 5.1.3 we know that the pro-Kummer-´etale cohomology of O D rκ and O D rκ at ξ is just the continuous group cohomology with respect to N γ . Furthermore, after twisting by an element of Iw n we may assume that γ = w ∈ W . We fall in two different cases N = (cid:26)(cid:18) x (cid:19) : x ∈ Z p (cid:27) and N n = (cid:26)(cid:18) p n x (cid:19) : x ∈ Z p (cid:27) . Let T denote the variable of N . An element ϕ ∈ O A rκ | ξ can be written as ϕ = κ ⊗ h ( T ) , where h ( T ) is an analytic function of the affinoid space N N r . Let n α and n α be the generators of N and N given by n α = (cid:18) (cid:19) , n α = (cid:18) (cid:19) . Then( n α − ϕ = κ ⊗ ( h ( T + 1) − h ( T ))( p n n α − ϕ = κ ⊗ ((1 + p n T ) κ − κ h ( T / (1 + p n T )) − h ( T ))This implies that κ ⊗ ∈ ker( n α − T = 0 gives rise a map O A rκ / ( p n n α − O A rκ → ( κ ⊗ C p . In other words, we have H ( N, O D rκ ) → ( − w ( κ ) ⊗ C p = ω − w ( κ ) E | ξ and H ( N n , O A rκ ) → ( κ ⊗ C p = ω κE | ξ . Which makes the diagrams of (1) and (2) commutative by the definition of the dlog maps. (cid:3) Remark . When κ = κ V the object R Γ proket ( D, O A rκ ) is represented by a perfect complex of Banachmodules over V . Indeed, for ξ ∈ D let N γ ⊂ Iw n be the unipotent monodromy of ξ . Then R Γ proket ( D, O A rκ ) = R Hom N γ ( Z p , O A rκ | ˜ ξ ), where ˜ ξ ∈ X ∞ is the fiber of ξ . But N γ ∼ = Z p (1) and Z p is represented by 0 → N γ n γ − −−−→ N γ → n γ ∈ N γ a topological generator. A similar property holds for O D rκ . It is also clear thatthese complexes admit finite slope decompositions with respect to the U p -operators and that the natural map R Γ proket ( X, O A rκ ) → R Γ proket ( D, O A rκ | D ) is Hecke-equivariant.We let H pproket,i ( Y, O A rκ ) denote the image of H pproket,c ( Y, O A rκ ) in H pproket ( X, O A rκ ). Let α = (1 , − ∈ X ∗ ( T ). Similar for O D rκ . For w ∈ W we define H ,cuspw,ǫ ( X, κ + α ) to be the kernel of the residue map H w,ǫ ( X, κ + α ) res −−→ Q ξ ∈ D ω κE | ξ , equivalently we have H ,cuspw,ǫ ( X, κ + α ) = H ( X w ( ǫ ) , ω κE ⊗ Ω X ). Corollary 5.3.1. With the hypothesis of Theorem 5.2.1, we have Galois and Hecke equivariant maps H ,c,ǫ ( X, w ( κ un ))( − κ ) H proket,i ( Y, O A r V ) H ,cuspw ,ǫ ( X, κ un + α )( κ − 1) 00 H w ,c,ǫ ( X, − κ un )( − κ ) H proket,i ( Y, O D rκ ) H ,cusp ,ǫ ( X, − w ( κ un ) + α )( − κ − 1) 0 . ES ∨A ES A ES ∨D ES D The finite slope of the previous arrows glue to morphisms of coherent sheaves over A W \{ } . Moreover, locallyaround k ∈ V the ( < k − k + 1) -slope of the previous diagrams are exact.Proof. This is a consequence of Theorem 5.2.1 and Lemma 5.3.1. (cid:3) The pairings. We have natural (Σ − n , Σ + n )-equivariant pairings D r V × A r V → O V ( V ). These maps inducethe Yoneda’s cup product of the interior cohomologies(5.9) ∪ : H proket,i ( Y, O D r V (1)) × H proket,i ( Y, O A r V ) → H proket,i ( Y, b O W ( V ) ⊗ b O Y (1)) . Lemma 5.3.2. Let p : X proket × V an → V a n be the projection. The pairings (5.9) glue to pairings of sheavesover V ∪ : H p proket,i ( Y, O D r V (1)) × H Γ proket,i ( Y, O A r V ) → R p , ∗ ,i ( p − ( O V )(1)) ⊗ K C p . Moreover, the Poincar´e trace Tr P : H et ( Y, Z p (1)) ∼ −→ Z p gives rise to an isomorphism Tr P : R p , ∗ ,i ( O V ) ∼ −→ O V ,and a Poincar´e pairing h− , −i P : H proket,i ( Y, O D r V (1)) × H proket,i ( Y, O A r V ) → O V ⊗ K C p . Proof. This is clear from functoriality of the objects involved and Proposition 2.2.5. (cid:3) On the other hand, in [AIP15] the authors define overconvergent Serre pairings in families h− , −i S : H w,c,ǫ ( X, − κ un ) × H ,cuspw,ǫ ( X, κ un + α ) → O ( V ) ⊗ K C p . compatible with the classical Serre pairings. They are constructed by taking the Yoneda’s product H ,cuspw,ǫ ( X, κ + α ) × H w,c,ǫ ( X, − κ ) ∪ −→ H X w ( >ǫ +1) ( X w ( ǫ ) , Ω X b ⊗ O ( V )) , UAL EICHLER-SHIMURA MAPS FOR THE MODULAR CURVE 49 and composing with the Serre trace map of X Tr S : H X w ( >ǫ +1) ( X w ( ǫ ) , Ω X b ⊗ O ( V )) Cor −−→ H ( X, Ω X b ⊗ O ( V )) → O ( V ) b ⊗ K C p . Sheafifing we obtain a Serre pairing with values in O V b ⊗ K C p . Theorem 5.3.1. Let V ⊂ W be an open affinoid subspace, and let k ∈ V be a dominant weight of X ∗ ( T ) . (1) The Poincar´e and Serre pairings of overconvergent cohomologies are compatible with the U p -operators.Moreover, they are compatible with the Eichler-Shimura maps of Corollary 5.3.1. (2) Let h < k − k + 1 . Locally around k ∈ V we have perfect pairings of finite free O V -modules h− , −i P : H proket,i ( Y, O D r V (1)) ≤ h × H proket,i ( Y, O A r V ) ≤ h → O V ⊗ K C p . and h− , −i S : H w,c,ǫ ( X, − κ un ) ≤ h × H ,cuspw,ǫ ( X, κ un + α ) ≤ h → O V ⊗ K C p compatible with the Eichler-Shimura exact sequence.Proof. The Hecke operators are compatible since they are defined by finite flat correspodances, and U tp is definedas the dual of U p , cf. Corollary 4.1.2. Then, we have a commutative diagram of Yoneda’s product H X ( >ǫ +1) ( X ( ǫ ) , Rν an, ∗ b ω w ( κ ) E ) × H proket ( X ( ǫ ) , b ω − w ( κ ) E ) H X ( >ǫ +1) ( X ( ǫ ) , Rν an, ∗ b O X ) H proket,i ( X, O A rκ ) × H proket,i ( X, O D rκ ) H proket ( X, b O X ) H proket ( X w ( ǫ ) , b ω κE ) × H X w ( >ǫ +1) ( X w ( ǫ ) , b ω − κE ) H X w ( >ǫ +1) ( X w ( ǫ ) , Rν an, ∗ b O ) CorCor On the other hand, we also have compatible pairings provided by the Faltings extension H w,c,ǫ ( X, − κ un ) × H w,ǫ ( X, κ un + α ) H ( X, Ω X (log)) H X w ( >ǫ +1) ( X w ( ǫ ) , Rν an, ∗ b ω − κ un ) × H proket ( X w ( ǫ ) , b ω κ un ) H proket ( X, b O X ) Cor ◦∪ F E Cor ◦∪ The compatibility of Poincar´e and Serre traces gives (1). Part (2) follows the same lines of the proof of Theorem5.2.1 using the fact that the pairings are perfect for classical objects. (cid:3) References [AIP15] Fabrizio Andreatta, Adrian Iovita, and Vincent Pilloni. p -adic families of Siegel modular cuspforms. Ann. of Math. 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