aa r X i v : . [ m a t h . N T ] J un THE HALF-INTEGRAL WEIGHT EIGENCURVE by Nick Ramsey
Abstract . —
In this paper we define Banach spaces of overconvergent half-integralweight p -adic modular forms and Banach modules of families of overconvergent half-integral weight p -adic modular forms over admissible open subsets of weight space.Both spaces are equipped with a continuous Hecke action for which U p is moreovercompact. The modules of families of forms are used to construct an eigencurve pa-rameterizing all finite-slope systems of eigenvalues of Hecke operators acting on thesespaces. We also prove an analog of Coleman’s theorem stating that overconvergenteigenforms of suitably low slope are classical. Contents
1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23. Some modular functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84. The spaces of forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135. Hecke operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176. Classical weights and classical forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277. The half-integral weight eigencurve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Appendix A. Properties of the stack X ( M p, p ) over Z ( p ) by Brian Conrad. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
1. Introduction
In [ ], the author set up a geometric theory of modular forms of weight k/ k , complete with geometrically defined Hecke operators. This This research is supported in part by NSF Grant DMS-0503264.
NICK RAMSEY approach naturally led to a theory of overconvergent p -adic modular forms of suchweights equipped with a Hecke action for which U p is compact.In this paper we define overconvergent half-integral weight p -adic modular forms ofgeneral p -adic weights, as well as rigid-analytic families thereof over admissible opensubsets of weight space. We use the latter spaces and Buzzard’s eigenvariety ma-chine ([ ]) to construct a half-integral weight eigencurve parameterizing all systemsof eigenvalues of Hecke operators occurring on spaces of half-integral weight overcon-vergent eigenforms of finite slope. In contrast to the integral weight situation, thisspace does not parameterize actual forms because a half-integral weight form that isan eigenform for all Hecke operators is not always characterized by its weight andcollection of Hecke eigenvalues. We also prove an analog of Coleman’s result thatoverconvergent eigenforms of suitably low slope are classical.This paper lays the foundation for a forthcoming paper of the author in whichwe construct a map from our half-integral weight eigencurve to its integral weightcounterpart (at least after passage to the underlying reduced spaces) that rigid-analytically interpolates the classical Shimura lifting introduced in [ ]. Acknowledgments
The author extends his thanks to Brian Conrad for writing the appendix, as well asfor numerous helpful discussions and suggestions about the technical issues in Section2. The author would also like to thank the referee for several suggestions on themanuscript and for directing him to some good references to help deal with the case p = 2.
2. Preliminaries2.1. General Notation. —
Fix a prime number p . The symbol K will alwaysdenote a complete and discretely-valued field extension of Q p . For such K we denotethe ring of integers by O K and the maximal ideal therein by m K . The absolute valueon K will always be normalized by | p | = 1 /p . For positive integers N and n , X ( N ) and X ( N, n ) willdenote the usual moduli stacks of generalized elliptic curves with level structure. Theformer classifies generalized elliptic curves with a point P of order N while the latterclassifies generalized elliptic curves with a pair ( P, C ) consisting of a point P of order N and a cyclic subgroup C of order n meeting the subgroup generated by P trivially(plus a certain ampleness condition for non-smooth curves). This level structure willalways be taken to be the Drinfeld-style level structure found in [ ], [ ], and theappendix to this paper, and in all cases the base ring will be a Z ( p ) -algebra.Throughout this paper we will make extensive use of certain admissible opens inrigid spaces associated to some of these modular curves. Traditionally these openswere defined using the Eisenstein series E p − , but this requires that we pose unfavor-able restrictions on p and N . Fortunately, more recent papers of Buzzard ([ ]) and HE HALF-INTEGRAL WEIGHT EIGENCURVE Goren-Kassaei ([ ]) define these opens and explore their properties in greater gen-erality using alternative techniques. These authors define a “measure of singularity” v ( E ) ∈ Q ≥ associated to an elliptic curve over a complete extension of Q p . In case v ( E ) ≤ p/ ( p + 1), one may associate to E a canonical subgroup H p ( E ) of order p in an appropriately functorial manner. Moreover, one understands v ( E/C ) for finitecyclic subgroups C ⊆ E as well as the canonical subgroup of E/C when it exists.Inductively applying this with C = H p ( E ), one can define (upon further restricting v ( E )) canonical subgroups H p m ( E ) of higher p -power order. For details regardingthese constructions and facts, we refer the reader to Section 3 of [ ] and Section 4 of[ ].We will denote the Tate elliptic curve over Z (( q )) by Tate( q ) (see [ ]). Our no-tational conventions concerning the Tate curve differ from those often found in theliterature as follows. In the presence of, for example, level N structure, previous au-thors (e.g., [ ]) have preferred to consider the curve Tate( q N ) over the base Z (( q )).Points of order N on this curve are used to characterize the behavior of a modularform at the cusps, and are all defined over the fixed ring Z (( q ))[ ζ N ] (where ζ N is someprimitive N th root of 1). We prefer to fix the curve Tate( q ) and instead consider extensions of the base. Thus, in the presence of level N structure, we introduce theformal variable q N , and define q = q NN . Then the curve Tate( q ) is defined over thesub-ring Z (( q )) of Z (( q N )) and all of its N -torsion is defined over the ring Z (( q N ))[ ζ N ].To be precise, the N -torsion is given by ζ iN q jN , ≤ i, j ≤ N − . Cusps will always be referred to by specifying a level structure on the Tate curve.Suppose that N ≥ X ( N ) Q p and let K/ Q p be a finite extension (which will generally be fixed in applications). If r ∈ [0 , ∩ Q ,then the region in the rigid space X ( N ) an K whose points correspond to pairs ( E, P )with v ( E ) ≤ r is an admissible affinoid open. We denote by X ( N ) an ≥ p − r the connectedcomponent of this region that contains the cusp associated to the datum (Tate( q ) , ζ N )for some (equivalently, any) choice of primitive N th root of unity ζ N . Similarly, X ( N, n ) an ≥ p − r will denote the connected component of the region defined by v ( E ) ≤ r in X ( N, n ) an K containing the cusp associated to (Tate( q ) , ζ N , h q n i ) for any such ζ N .For smaller N one defines these spaces by first adding prime-to- p level structureto rigidify the moduli problem and proceeding as above, and then taking invariants.Similarly, the space X ( N ) an ≥ p − r is defined as the quotient of X ( N ) an ≥ p − r by the actionof the diamond operators. The reader may wish to consult Section 6 of [ ] for a moredetailed discussion of these quotients. If X is an admissible formal scheme over O K (in the sense of [ ]),we will denote its (Raynaud) generic fiber by X rig and its special fiber by X . In case X = Spf( A ) is a formal affine we have X rig = Sp( A ⊗ O K K ) and X = Spec( A /π A )where π ∈ O K is a uniformizer. We recall for later use that the natural specializationmap sp : X rig −→ X is surjective on the level of closed points (see Proposition 3.5 of [ ]). NICK RAMSEY
Assume that X is reduced and let L be an invertible sheaf on X (that is to say, asheaf of modules on this ringed space that is Zariski-locally free of rank one). For apoint x ∈ X rig ( L ) let b x : Spf( O L ) −→ X denote the unique extension of x to the formal model. Then the canonical identifica-tion H (Sp( L ) , x ∗ L rig ) = H (Spf( O L ) , b x ∗ L ) ⊗ O L L furnishes a norm | · | x on this one-dimensional vector space by declaring the formalsections on the right to be the unit ball. Now for any admissible open U ⊆ X rig andany f ∈ H ( U , L rig ) we define k f k U = sup x ∈ U | x ∗ f | x . Note that, in case, L = O X , this is simply the usual supremum norm on functions.There is no reason for k f k U to be finite in general, but in case U is affinoid thenthis is indeed finite and endows H ( U , L rig ) with the structure of a Banach space over K as we now demonstrate. Lemma 2.1 . —
Let X be a reduced quasi-compact admissible formal scheme over O K , let L be an invertible sheaf on X , and let U be an admissible affinoid open in X rig . Then H ( U , L rig ) is a K -Banach space with respect to k·k U .Proof . — By Raynaud’s theorem there is quasi-compact admissible formal blowup π : X ′ −→ X and an admissible formal open U in X ′ with generic fiber U . For x ∈ U let b x ′ denote the unique extension to an O L -valued point of U and let b x denote itsimage in X (which is the same b x as above by uniqueness). Then we have H (Spf( O L ) , b x ′∗ π ∗ L ) = H (Spf( O L ) , b x ∗ L )as lattices in H (Sp( L ) , L rig ). It follows that | f | x = | π ∗ f | x and we may compute k f k U using the models X ′ and π ∗ L , and hence we may as well assume that U is thegeneric fiber of an admissible formal open U in X . Furthermore, we may just wellreplace X by U and assume that U is the generic fiber of X itself.Cover X by a finite collection of admissible formal affine opens U i trivializing L and pick a trivializing section ℓ i of L on U i . Let U i = ( U i ) rig , so that the U i forman admissible cover of U by admissible affinoid opens. Then, for any section f ∈ H ( U , L rig ), we may write f | U i = a i ℓ i for a unique a i ∈ O ( U i ), and one easily checksthat k f k U = max i k a i k sup . The desired assertion now follows easily from the analogous assertion about the supre-mum norm on a reduced affinoid.The following lemma and its corollary establish a sort of maximum modulus prin-ciple for these norms.
HE HALF-INTEGRAL WEIGHT EIGENCURVE Lemma 2.2 . —
Let X = Spf( A ) be a reduced admissible affine formal scheme over O K and let U ⊆ X be a Zariski-dense open subset of the special fiber. Suppose thatthe generic fiber X = Sp( A ⊗ O K K ) is equidimensional. Then, for any a ∈ A ⊗ O K K ,the supremum norm of a over X is achieved on sp − ( U ) .Proof . — Let us first prove the lemma in the case that A is normal. First note that if k a k sup = 0, then the result is obvious. Otherwise, since the supremum norm is power-multiplicative we may assume that k a k sup is a norm from K and scale to reduce tothe case k a k sup = 1. By Theorem 7.4.1 of [ ] it follows that a ∈ A (this is wherenormality is used). If the reduction a ∈ A = A /π A vanishes at every closed pointof U , then it vanishes everywhere by density, so a n = 0 in A for some n , which is tosay that π | a n in A . But this is impossible because by power-multiplicativity we have k a n k sup = 1 for all n ≥
1. Thus a must be non-vanishing at some point of U . Bythe surjectivity of the specialization map we can find a point x reducing to this point.Clearly then | a ( x ) | = 1, which establishes the normal case.Suppose that X is equidimensional of dimension d . We claim that it follows thatthe special fiber X must be equidimensional of dimension d as well. Indeed, insideeach irreducible component of this special fiber we can find a nonempty Zariksi-opensubset V that does not meet any of the other irreducible components. The genericfiber V rig is an admissible open in X and therefore has dimension d . It follows that V has dimension d , and the claim follows.Let f : e X −→ X be the normalization map (meaning Spf applied to the nor-malization map on algebras) and note that this map is finite by general excellenceconsiderations. By Theorem 2.1.3 of [ ] the generic fiber of this map coincides withthe normalization of X . Thus e X rig is also equidimensional of dimension d and theargument above shows that e X is equidimensional of dimension d as well. Now since f is finite it follows that f carries generic points to generic points. In particular wesee that f − ( U ) is Zariski-dense in e X . Thus by the normal case proven above thereexists x ∈ e X rig reducing to f − ( U ) at which a (thought of as an element of e A ⊗ O K K )attains its supremum norm. But then f ( x ) is a point in X reducing to U with thesame property, since the supremum norm of a is the same thought of on X or on e X (since e X −→ X is surjective). Remark 2.3 . — Note that the proof in the normal case did not use the equidimen-sionality hypothesis. This hypothesis may not be required in the general case, butthe above proof breaks down without it since it is not clear how to control the specialfiber under normalization in general, especially if X is non-reduced (as is often thecase for us). Corollary 2.4 . —
Let X be a reduced quasi-compact admissible formal scheme over O K , let U ⊆ X be a Zariski-dense open, and let L be an invertible sheaf on X .Assume that X rig is equidimensional. Then, for any f ∈ H ( X rig , L rig ) we have k f k X rig = sup x ∈ sp − ( U ) | x ∗ f | x = max x ∈ sp − ( U ) | x ∗ f | x . NICK RAMSEY
Proof . — Cover X be a finite collection of admissible formal affine opens trivializing L and apply Lemma 2.2 on each such affine separately.The invertible sheaves whose sections we will be taking norms of in this paperwill all be of the form O X ( D ) for some divisor D on X = X ( N ) K or X ( N, n ) K supported on the cusps. In the end, the main consequence of the previous Corollary(namely, Lemma 2.5) will be that these norms are equal to the supremum norm of therestriction of the section in question to the complement of the residue disks aroundthe cusps (where it is simply an analytic function). We feel that it is worthwhileto give more natural definitions using the above norm machinery in the cases thatit applies to (those where we have nice moduli schemes to work with) in the hopesthat the techniques used and the above Corollary will be useful in other similarsituations. The reader who is content with this equivalent “ad hoc” definition (thatis, the supremum norm on the complement of the residue disks around the cusps) canskip to Section 2.4 and ignore the appendix all together.In order to endow spaces of sections of a line bundle as in the previous paragraphwith norms using the techniques above, we need formal models of the spaces X andsheaves O ( D ). For technical reasons (involving regularity of certain moduli stacks)we are forced to work over Z p in going about this. The formal models over O K willthen be obtained by extension of scalars. The general procedure for obtaining formalmodels over Z p goes as follows. Let X denote one the stacks X ( N ) or X ( N, n )over Z p and assume that the generic fiber X Q p is a scheme. Let D be a divisor on X Q p that is supported on the cusps. If the closure D of D in X lies in the maximalopen subscheme X sch of X and this subscheme is moreover regular along D , then thisclosure is Cartier and we may associate to it the invertible sheaf O ( D ) on X sch . Let( X sch ) b and O ( D ) b denote the formal completions of these objects along the specialfiber.In case X = X ( N ) or X ( N, n ) with p ∤ n , assume that N has a divisor thatis prime to p and at least 5. Then X sch = X by Theorem 4.2.1 of [ ], and X ismoreover regular (at least over Z ( p ) ) by Theorem 4.1.1 of [ ]. That passage to Z p preserves regularity follows by excellence considerations from the fact that Z ( p ) −→ Z p is geometrically regular. Strictly speaking the results of [ ] do not apply to X ( N, n )as stated, but since p ∤ n the proofs of these results are still valid over Z ( p ) , as isobserved in the appendix. Since X is proper over Z p , we have b X rig = X an Q p (theanalytification of the algebraic generic fiber of X ) and hence we have a formal model( b X, O ( D ) b ) of ( X an Q p , O ( D )).Now suppose that X = X ( M p, p ) for an integer M ≥ p . Let D beany divisor supported on the cusps in the connected component X ( M p, p ) an ≥ of theordinary locus. By Theorem A.11 of the appendix, the closure D of D in X lies in X sch and is Cartier. Thus we obtain a formal model (( X sch ) b , O ( D ) b ) of (( X sch ) b rig , O ( D )).Observe that, by Lemma A.9 and the comments that follow it, X sch is simply thecomplement of a finite collection of cusps on the characteristic p fiber (namely, the HE HALF-INTEGRAL WEIGHT EIGENCURVE ones with nontrivial automorphisms). It follows that the open immersion(1) ( X sch ) b rig ֒ → ( X sch Q p ) an ∼ = X an Q p identifies the Raynaud generic fiber on the left with the complement of the residuedisks around the cusps in the analytification on the right that reduce to the missingpoints in characteristic p . Thus (1) is an isomorphism when restricted to any con-nected component of the locus defined by v ( E ) ≤ r that contains no such cusps. Inparticular, it is an isomorphism when restricted to X ( M p, p ) an ≥ p − r by Theorem A.11of the appendix.Given a complete discretely-valued extension K/ Q p , we may extend scalars on theformal models of O ( D ) we have obtained to arrive at norms on the following spaces. – sections of O ( D ) over any admissible open U in X = X ( N ) an K (resp. X ( N, n ) an K with p ∤ n ), where D is (the scalar extension of) a divisor on X ( N ) Q p (resp. X ( N, n ) Q p ) and N is divisible by an integer that is prime to p and at least 5 – sections of O ( D ) over any admissible open U in X = X ( M p, p ) an ≥ p − r , where D is (the scalar extension of) a divisor supported on the cusps in X ( M p, p ) an Q p and M is an integer that is prime to p and at least 5 Lemma 2.5 . —
Let X , D , and U be as in either of the two cases above and assumethat U contains every component of the ordinary locus that it meets. Let U ′ denote thecomplement of the residue disks around the cusps in U . Then for any f ∈ H ( U , O ( D )) we have k f k U = k f | U ′ k sup . Proof . — We will treat the case of X = X ( N ) an K ; the other cases are proven inexactly the same manner. First note that, since points in U ′ reduce to points outsideof the support of D , the claim is equivalent to the assertion that k f k U = k f | U ′ k U ′ . That is, the norm on U ′ that we have defined using formal models happens to beequal to the supremum norm on U ′ .Note that the supersingular loci of U and U ′ coincide, so the contributions tothe above norms over this locus are equal, and it suffices to check the assertionupon restriction to the ordinary locus. By assumption, the ordinary locus in U isa finite union of connected components of the ordinary locus in X ( N ) an K . Eachsuch component corresponds via reduction to an irreducible component of the specialfiber. Let X denote the admissible formal open in X ( N ) b given by the union of thecomponents so obtained with the supersingular points removed. Then X rig is preciselythe ordinary locus in U , and the result now follows from Corollary 2.4 with U equalto the complement of the cusps in X . Remark 2.6 . — There remain some curves on which we will need to have normsfor sections of O ( D ) but to which the norm machinery as set up here does not apply.Namely, for p = 2 we have the curves X (4 p m ) an K and X (4 p m , p ) an K , while for p = 2 wehave X (2 m +1 N ) an K and X (2 m +1 N, an K , where m ≥ N ∈ { , } . The previouslemma suggests an ad hoc workaround to this problem. In case we are working with NICK RAMSEY sections of O ( D ) for a cuspidal divisor on one of these curves, we simply define thenorm to be the supremum norm of the restriction of our section to the complement ofthe residue disks about the cusps. A more natural definition would likely result fromconsiderations of “formal stacks,” but this norm would surely turn out to be equal toours by an analog of Lemma 2.5. Throughout most of this paper W will denote p -adic weightspace (everywhere except for the beginning of Section 7 where it is allowed to be ageneral reduced rigid space for the purpose of reviewing a general construction). Thatis, W is a rigid space over Q p whose points with values in an extension K/ Q p are W ( K ) = Hom cont ( Z × p , K × ) . Define q = p if p = 2 and q = 4 if p = 2. Let τ : Z × p −→ ( Z / q Z ) × −→ Q × p denote reduction composed with the Teichmuller lifting, and let h x i = x/τ ( x ) ∈ q Z p . For a weight κ we have κ ( x ) = κ ( h x i ) κ ( τ ( x )) = κ ( h x i ) τ ( x ) i for a unique integer i with 0 ≤ i < ϕ ( q ) (where ϕ denotes Euler’s function). Moreover,this breaks up the space W as the admissible disjoint union of ϕ ( q ) admissible opens W i , each of which is isomorphic to a one-dimensional open ball.For each positive integer n , let W n denote the admissible open subspace of W whose points are those κ with | κ (1 + q ) p n − − | ≤ | q | . Then W in := W i ∩ W n is an affinoid disk in W i and the { W in } n form a nested admissiblecover of W i .To each integer λ we may associate the weight x x λ . This weight, which byabuse of notation we simply refer to as λ , lies in W i for the unique i ≡ λ (mod ϕ ( q )).Also, if λ is an integer and ψ : ( Z / q p n − Z ) × −→ C × p is a character, then x x λ ψ ( x )is a point in W (with values in Q p ( µ p n − )) which lies in W n , as standard estimatesfor | ζ − | for roots of unity ζ demonstrate.
3. Some modular functions
Our definition of the spaces of half-integral weight modular forms will follow thegeneral approach of [ ] (in the integral weight p -adic situation) and [ ] (in the half-integral weight situation). The motivating idea behind this approach is to reduce toweight zero by dividing by a well-understood form of the same weight. For example,if f is a half-integral weight p -adic modular form of weight k/ θ is the usual Jacobitheta function of weight 1 /
2, and E λ is the weight λ = ( k − / f / ( E λ θ ) should certainly be a meromorphic modular functionof weight zero. As we have no working notion of “half-integral weight p -adic modularform” we simply use the weight zero forms so obtained as the definition of this notion.One must of course work out issues such as exactly what kind of poles are introduced, HE HALF-INTEGRAL WEIGHT EIGENCURVE how dividing by θE λ affects the nebentypus character, and how to translate theclassical Hecke action into an action on these new forms. The precise definition willbe given in the next section.We remark that this was carried out by the author in [ ] by dividing by θ k insteadof θE λ . That approach had the disadvantage of limiting us to classical weights k/ p -adic weights (and indeed,for families of modular forms) since E λ interpolates nicely in the variable λ .This technique of division to reduce to weight zero in order to define modularforms forces us to modify the usual construction of the Hecke operators using theHecke correspondences on the curve X ( N ) by multiplying by certain functions onthe source spaces of these correspondences. Our first task is to define these functionsand to establish their overconvergence properties. Since we are dividing by E λ θ toreduce to weight zero, we will require, for each prime number ℓ , a modular functionwhose q -expansion (at the appropriate cusp, on the appropriate space, which dependson whether or not ℓ = p ) is E λ ( q ℓ ) θ ( q ℓ ) E λ ( q ) θ ( q ) . Factoring this into its Eisenstein part and theta part we split the problem into twoproblems, the first of which is nearly done in the integral-weight literature (see [ ],[ ]),and the second of which is done in an earlier paper of the author ([ ]). We brieflyreview both here. See the aforementioned references for additional details. Note thatall analytic spaces in this section are taken over Q p .Let c denote the cusp on X (4) Q corresponding to the point ζ q of order 4 on theTate curve. Define a Q -divisor Σ N on the curve X (4 N ) Q byΣ := 14 π ∗ [ c ]where π : X (4 N ) Q −→ X (4) Q is the obvious degeneracy map. This divisor is set up to look like the divisor of zerosof the pullback of the Jacobi theta function θ to X (4 N ) Q and will later be used tocontrol poles introduced in dividing by E λ θ .In [ ] we defined a rational function Θ ℓ on X (4 , ℓ ) Q with divisordiv(Θ ℓ ) = π ∗ Σ − π ∗ Σ such that Θ ℓ (Tate( q ) , ζ , h q ℓ i ) = P n ∈ Z q n ℓ P n ∈ Z q n = θ ( q ℓ ) θ ( q ) . Here π and π are the maps comprising the ℓ Hecke correspondence on X (4) andare defined in Section 5.1. Strictly speaking, we had assumed ℓ = 2 in the argumentsin [ ], but if one is only interested in the result above, then one can easily checkthat the arguments work for ℓ = 2 verbatim.Let us now turn to the Eisenstein part of the above functions. For further detailsand proofs of the assertions in this paragraph, we refer the reader to Sections 6 and NICK RAMSEY ]. Let E ( q ) := 1 + 2 ζ p ( κ ) X n X d | n , p ∤ d κ ( d ) d − q n ∈ O ( W )[[ q ]]be the q -expansion of the p -deprived Eisenstein family over W . Note that thereare no problems with zeros of ζ p since we are restricting our attention to W . For aparticular choice of κ ∈ W , we denote by E κ ( q ) the expansion obtained by evaluatingall of the coefficients at κ . In particular, for a positive integer λ ≥ ϕ ( q ), E λ ( q ) is the q -expansion of the usual p -deprived classical Eisenstein series of weight λ and level p .Let ℓ be a prime number. If ℓ = p , then there exists a rigid analytic function E ℓ on X ( pℓ ) an ≥ × W whose q -expansion at (Tate( q ) , µ pℓ ) is E ( q ) /E ( q ℓ ). If ℓ = p , thenthe same holds with X ( pℓ ) an ≥ replaced by X ( p ) an ≥ and µ pℓ replaced by µ p . In [ ] itis shown that there exists a sequence of rational numbers1 p + 1 > r ≥ r ≥ · · · ≥ r n ≥ · · · > r i < p − i / q (1 + p ) such that, when restricted to X ( pℓ ) an ≥ × W n (respec-tively, X ( p ) an ≥ × W n if ℓ = p ), E ℓ analytically continues to an invertible functionon X ( pℓ ) an ≥ p − rn × W n (respectively, X ( p ) an ≥ p − rn × W n if ℓ = p ). Fix such a sequenceonce and for all. Let us first extend these results to square level. Lemma 3.1 . —
Let ℓ = p be a prime number. There exists an invertible function E ℓ on X ( pℓ ) an ≥ × W whose q -expansion at (Tate( q ) , µ pℓ ) is E ( q ) /E ( q ℓ ) . More-over, the function E ℓ , when restricted to W n , analytically continues to an invertiblefunction on X ( pℓ ) an ≥ p − rn × W n .There exists an invertible function E p on X ( p ) an ≥ × W whose q -expansion at (Tate( q ) , µ p ) is E ( q ) /E ( q p ) . Moreover, the function E p , when restricted to W n ,analytically continues to an invertible function on X ( p ) an ≥ p − rn/p × W n .Proof . — Let ℓ be a prime different from p . There are two natural maps X ( pℓ ) an Q p −→ X ( pℓ ) an Q p , namely those given on noncuspidal points by( E, C ) d ℓ, ( E, ℓC )( E, C ) d ℓ, ( E/pℓC, C/pℓC )Both of these restrict to maps d ℓ, , d ℓ, : X ( pℓ ) an ≥ p − rn −→ X ( pℓ ) an ≥ p − rn . We define E ℓ to be the invertible function(2) E ℓ := d ∗ ℓ, E ℓ · d ∗ ℓ, E ℓ ∈ O ( X ( pℓ ) an ≥ p − rn × W n ) × . HE HALF-INTEGRAL WEIGHT EIGENCURVE The q -expansion of E ℓ at (Tate( q ) , µ pℓ ) is E ℓ ( d ℓ, (Tate( q ) , µ pℓ )) E ℓ ( d ℓ, (Tate( q ) , µ pℓ ))= E ℓ (Tate( q ) , µ pℓ ) E ℓ (Tate( q ) /µ ℓ , µ pℓ /µ ℓ )= E ℓ (Tate( q ) , µ pℓ ) E ℓ (Tate( q ℓ ) , µ pℓ )= E ( q ) E ( q ℓ ) E ( q ℓ ) E ( q ℓ ) = E ( q ) E ( q ℓ )One must take additional care if ℓ = p . Then there is a well-defined map d : X ( p ) an ≥ p − rn/p −→ X ( p ) an ≥ p − rn ( E, C ) ( E/C, H p /C )where H p is the canonical subgroup of E of order p . This follows form the fact that X ( p ) an ≥ p − rn/p consists of pairs ( E, C ) with C equal to the canonical subgroup of E of order p , and standard facts about quotienting by such subgroups (see for exampleTheorem 3.3 of [ ]). We define an invertible function by E p := E p · d ∗ E p ∈ O ( X ( p ) an ≥ p − rn/p × W n ) × where we have implicitly restricted E p to X ( p ) an ≥ p − rn/p × W n ⊆ X ( p ) an ≥ p − rn × W n . The q -expansion of E p at (Tate( q ) , µ p ) is E p (Tate( q ) , µ p ) E p ( d (Tate( q ) , µ p )) = E p (Tate( q ) , µ p ) E p (Tate( q ) /µ p , µ p /µ p )= E p (Tate( q ) , µ p ) E p (Tate( q p ) , µ p )= E ( q ) E ( q p ) E ( q p ) E ( q p ) = E ( q ) E ( q p )Let π : X ( p, ℓ ) an Q p −→ (cid:26) X ( pℓ ) an Q p ℓ = pX ( p ) an Q p ℓ = p denote the map given on noncuspidal points by( E, P, C ) (cid:26) ( E/C, ( h P i + E [ ℓ ]) /C ) ℓ = p ( E/C, h P i /C ) ℓ = p Note that we have(3) π (Tate( q ) , ζ p , h q ℓ i ) = (cid:26) (Tate( q ℓ ) , µ pℓ ) ℓ = p (Tate( q p ) , µ p ) ℓ = p This observation suggests that perhaps the components X ( p, ℓ ) an ≥ p − r should be re-lated to (via π ) the components X ( pℓ ) an ≥ p − r . NICK RAMSEY
Lemma 3.2 . — If ℓ = p , then the map π restricts to π : X ( p, ℓ ) an ≥ p − r −→ X ( pℓ ) an ≥ p − r for all r < p/ (1 + p ) .In case ℓ = p , the map π restricts to X ( p, p ) an ≥ p − p r −→ X ( p ) an ≥ p − r for all r < /p (1 + p ) .Proof . — First suppose ℓ = p . Let U denote the entirety of the locus in X ( pℓ ) an Q p defined by v ( E ) ≤ r . First note that, since quotienting by a subgroup of order primeto p does not change its measure of singularity, the map π restricts to a map X ( p, ℓ ) an ≥ p − r −→ U . The inverse images of the two connected components of U under this map are disjointadmissible opens that admissibly cover a connected space, and π − ( X ( pℓ ) an ≥ p − r ) isnonempty by (3), so this must be all of X ( p, ℓ ) an ≥ p − r , and the result follows.Now suppose that ℓ = p . Let U denote the entirety of the locus in X ( p ) an Q p definedby v ( E ) ≤ r . Once we verify that π restricts to X ( p, p ) an ≥ p − p r −→ U , the argument may proceed exactly as above. We claim, moreover, that if ( E, P, C )is a point in X ( p, p ) an ≥ p − p r , then v ( E/C ) = v ( E ) /p . This would follow if we knewthat C met the canonical subgroup of E trivially (again by standard facts aboutquotienting by canonical and non-canonical subgroups of order p , as in Section 3 of[ ]), so it suffices to prove that h P i is the canonical subgroup of E .The natural map X ( p, p ) −→ X ( p )( E, P, C ) ( E, h P i )restricts to X ( p, p ) an ≥ p − r −→ X ( p ) an ≥ p − r by the same connectivity argument used in the ℓ = p case (since this map clearlydoesn’t change v ( E )). But the locus X ( p ) an ≥ p − r is well-known to consist of pairs( E, C ) with C equal to the canonical subgroup of E .We may pull back the Eisenstein family of Lemma 3.1 for ℓ = p through the map π to arrive at an invertible function on X ( p, ℓ ) an ≥ p − rn × W n . By the previous lemma, wemay also pull back the family for ℓ = p through π to arrive at an invertible function on X ( p, p ) an ≥ p − prn × W n . For any ℓ , it follows from (3) that the function π ∗ E ℓ satisfies π ∗ E ℓ (Tate( q ) , ζ p , h q ℓ i ) = E ( q ℓ ) E (( q ℓ ) ℓ ) = E ( q ℓ ) E ( q ) . To arrive at the functions that we need, we simply multiply π ∗ E ℓ and Θ ℓ (whichis constant in the weight). Of course, to do so we must first pull these functions back HE HALF-INTEGRAL WEIGHT EIGENCURVE so that they lie on a common curve. The natural (“smallest”) curve to use dependson whether or not p = 2, since 2 already lies in the Γ part of the level of Θ ℓ . Thefollowing proposition summarizes the properties of the resulting functions. Proposition 3.3 . —
Let p be and ℓ be primes. There exists an element H ℓ of (cid:26) H ( X (4 p, ℓ ) an ≥ × W , O ( π ∗ Σ p − π ∗ Σ p )) p = 2 H ( X (4 , ℓ ) an ≥ × W , O ( π ∗ Σ − π ∗ Σ )) p = 2 whose q -expansion at (cid:26) (Tate( q ) , µ p , h q ℓ i )) p = 2(Tate( q ) , µ , h q ℓ i )) p = 2 is equal to E ( q ℓ ) θ ( q ℓ ) E ( q ) θ ( q ) . Moreover, there exists a sequence of rational numbers r n such that
11 + p > r ≥ r ≥ · · · > with r i < p − i / q (1 + p ) such that H ℓ , when restricted to W n , analytically continuesto the region X (4 p, ℓ ) an ≥ p − rn × W n p = 2 , ℓ = pX (4 p, p ) an ≥ p − prn × W n p = 2 , ℓ = pX (4 , ℓ ) an ≥ − rn × W n p = 2 , ℓ = 2 X (4 , an ≥ − rn × W n p = ℓ = 2Finally, we wish to extend H ℓ and E ( q ) to all of W . To do this, we simply pullback through the natural map W −→ W (4) κ κ ◦ hi When restricted to W i , this map is simply the isomorphism κ κ/τ i . Remark 3.4 . — We have chosen in the end to use Γ -structure on the curves onwhich the H ℓ lie both to rigidify the associated moduli problems over Q p as wellas because these are the curves that will actually turn up in the sequel. We note,however, that the H ℓ are invariant under all diamond automorphisms.
4. The spaces of forms
In this section we define spaces of overconvergent p -adic modular forms as wellas families thereof over admissible open subsets of W . Again, the motivating ideabehind these definitions is that we have reduced to weight 0 via division by the well-understood forms E λ θ . By “well-understood” we essentially mean two things here.The first is that we understand their zeros once we eliminate part of the supersingular NICK RAMSEY locus (and thereby remove the zeros of the Eisenstein part). The second is that, bythe previous section, we know that there are modular functions with q -expansions E λ ( q ℓ ) θ ( q ℓ ) E λ ( q ) θ ( q )that interpolate rigid-analytically in λ , a fact that we will need to define Hecke oper-ators on families in the next section.Before defining the spaces of forms, we need to make a couple of remarks aboutdiamond automorphisms. For a positive integer N and an element d ∈ ( Z /N Z ) × , let h d i denote the usual diamond automorphism of X ( N ) given on (noncuspidal) pointsby ( E, P ) ( E, dP ). Now suppose we are given a factorization N = N N intorelatively prime factors, so the natural reduction map( Z /N Z ) × ∼ −→ ( Z /N Z ) × × ( Z /N Z ) × is an isomorphism. For a ∈ ( Z /N Z ) × and b ∈ ( Z /N Z ) × we let ( a, b ) ∈ ( Z /N Z ) × denote the inverse image of the pair ( a, b ) under the this map. For a ∈ ( Z /N Z ) × ,we define h a i N := h ( a, i , and we refer to these automorphisms as the diamondautomorphisms at N . The diamond automorphisms at N are defined similarly, andwe have a factorization h d i = h d i N ◦ h d i N . Finally, we observe that the diamond operators on X (4 N ) an K preserve the sub-spaces X (4 N ) an ≥ p − r and the divisor Σ N in the sense that h d i − ( X (4 N ) an ≥ p − r ) = X (4 N ) an ≥ p − r and h d i ∗ Σ N = Σ N , respectively. Convention 4.1 . —
By the symbol O (Σ) for a Q -divisor Σ we shall always mean O ( ⌊ Σ ⌋ ) , where ⌊ Σ ⌋ is the divisor obtained by taking the floor of each coefficient oc-curring in Σ . First we define the spaces of forms of fixed weight. Let N be a positive integer andsuppose that either p ∤ N or that p = 2 and p ∤ N . Definition 4.2 . — Let κ ∈ W i ( K ) and pick n such that κ ∈ W in . Then, for anyrational number r with 0 ≤ r ≤ r n , we define the space of p -adic half-integral weightmodular forms of weight κ , tame level 4 N (or rather N if p = 2) , and growthcondition p − r over K to be f M κ (4 N, K, p − r ) := ( H ( X (4 N p ) an ≥ p − r , O (Σ Np )) τ i × { κ } p = 2 H ( X (4 N ) an ≥ − r , O (Σ N )) ( − / · ) i τ i × { κ } p = 2where () τ i denotes the τ i eigenspace for the action of the diamond automorphisms at p , and similarly for ( − / · ) i τ i if p = 2. Remarks 4.3 . — – For p = 2, we have chosen to remove p from the level and only indicate the tamelevel in the notation because, as we will see, these spaces contain forms of all p -power level. However, for p = 2, we have left the 4 in as a reminder that theforms have at least a 4 in the level, as well as for some uniformity in notation. HE HALF-INTEGRAL WEIGHT EIGENCURVE – Note that this space has been “tagged” with the weight κ because the actualspace has only a rather trivial dependence on κ ( κ serves only to restrict theadmissible K and r and to determine i ). The point is that, as we will see, theHecke action on this space is very sensitive to κ . The tag will generally beignored in what follows as the weight will be clear from the context. – This space is endowed with a norm which is defined as in Subsection 2.3 and isa Banach space over K with respect to this norm. – We call the forms belonging to spaces with r > overconvergent . The space ofall overconvergent forms (of this weight and level) is the inductive limit f M † κ (4 N, K ) = lim r → f M κ (4 N, K, p − r ) . – In case κ is the character associated to an integer λ ≥
0, the space of formsdefined above would classically be thought of having weight λ + 1 /
2. Our choiceof p -adic weight character book-keeping seems to be the most natural one (theShimura lifting has the effect of squaring the weight character, for example). – In case κ is the weight associated to an integer λ ≥
0, then the definition here issomewhat less general than the definition of the space of forms of weight λ + 1 / ]) due to the need to eliminateenough of the supersingular locus to get rid of the Eisenstein zeros. The twodefinitions are (Hecke-equivariantly) isomorphic whenever they are both defined,as we will see in Proposition 6.2. – The tilde is an homage to the metaplectic literature and will be used in forth-coming work on all half-integral weight objects in order to distinguish themfrom their integral weight counterparts.We now turn to the spaces of families of modular forms.
Definition 4.4 . — Let X be a connected affinoid subdomain of W . Then X ⊆ W i for some i since X is connected and moreover X ⊆ W in for some n since X is affinoid.For any rational number r with 0 ≤ r ≤ r n , we define the space of families of half-integral weight modular forms of tame level 4 N and growth condition p − r on X tobe f M X (4 N, K, p − r ) := ( H ( X (4 N p ) an ≥ p − r , O (Σ Np )) τ i b ⊗ K O ( X ) p = 2 H ( X (4 N ) an ≥ − r , O (Σ N )) ( − / · ) i τ i b ⊗ K O ( X ) p = 2 Remarks 4.5 . — – We endow f M X (4 N, K, p − r ) with the completed tensor product norm obtainedfrom the norms we have defined in Section 2.3 and the supremum norm on O ( X ). The space f M X (4 N, K, p − r ) with this norm is a Banach module over theBanach algebra O ( X ). – As in the case of fixed weight, the definition depends rather trivially on X butthe Hecke action will be very sensitive to X . – In general, if X is an affinoid subdomain of W , we define f M X to be the directsum of the spaces corresponding to the connected components of X . Also, just NICK RAMSEY as for particular weights, we can talk about the space of all overconvergentfamilies of forms on X , namely f M † X (4 N, K ) = lim r → f M X (4 N, K, p − r ) . – Using a simple projector argument, one sees easily that we have a canonicalidentification H ( X (4 N p ) an ≥ p − r , O (Σ Np )) τ i b ⊗ K O ( X ) ∼ = ( H ( X (4 N p ) an ≥ p − r , O (Σ Np )) b ⊗ K O ( X )) τ i , and similarly at level 4 N if p = 2, a comment that will prove to be useful in thenext section.For each X as above and each L -valued point κ ∈ X , evaluation at x induces aspecialization map f M X (4 N, K, p − r ) −→ f M κ (4 N, L, p − r ) . In the next section we will define a Hecke action on both of these spaces for whichsuch specialization maps are equivariant and which recover the usual Hecke operatorson the right side above (in the sense that they are given by the usual formulas on q -expansions).Each of the spaces of forms that we have defined has a cuspidal subspace consistingof forms that “vanish at the cusps.” This notion is a little subtle in half-integral weightbecause there are often cusps at which all forms are forced to vanish. To explain thiscomment and motivate the subsequent definition of the space of cusp forms, let usgo back to the motivation behind our definitions of the spaces of forms. If F is aform of half-integral weight in our setting, then F θE (where E is an appropriateEisenstein series) is what we would “classically” like to think of as a half-integralweight form. Indeed, in case F is classical (this notion is defined in Section 6) then F θE can literally be identified with a classical holomorphic modular form of half-integral weight over C . The condition div( F ) ≥ − Σ Np (we are assuming p = 2for the sake of this motivation) in our definition is exactly the condition that F θE be holomorphic at all cusps. Likewise, the condition that this inequality be strictat all cusps is the condition that
F θE be cuspidal. But since div( F ) has integralcoefficients, the non-strict inequality implies the strict inequality at all cusps whereΣ Np has non-integral coefficients.With this in mind, we are led to the following definition of cusp forms. For aninteger M , let C M be the divisor on X (4 M ) an Q p given by the sum of the cusps atwhich Σ M has integral coefficients. To define the cuspidal subspace of any of theabove spaces of forms, we replace the divisor Σ Np (resp. Σ N if p = 2) by the divisorΣ Np − C Np (resp. Σ N − C N if p = 2). We will denote the cuspidal subspaces bythe letter S instead of M . Thus, for example, if κ ∈ W in ( K ) and 0 ≤ r ≤ r n , wedefine e S κ (4 N, K, p − r ) = ( H ( X (4 N p ) an ≥ p − r , O (Σ Np − C Np )) τ i × { κ } p = 2 H ( X (4 N ) an ≥ − r , O (Σ N − C N )) ( − / · ) i τ i × { κ } p = 2 HE HALF-INTEGRAL WEIGHT EIGENCURVE Remarks 4.3 and 4.5 apply equally well to the corresponding spaces of cusp forms.
5. Hecke operators
Before we construct Hecke operators, we need to make some remarks on diamondoperators and nebentypus. Since the p -part of the nebentypus character is encodedas part of the p -adic weight character, we need to separate out the tame part of thediamond action. Fix a weight κ ∈ W i ( K ). In order to define the tame diamond oper-ators in a manner compatible with the classical definitions and that in [ ] we musttwist (at least in the case p = 2) those obtained via pull-back from the automorphism hi N by ( − / · ) i . That is, for F ∈ f M κ (4 N, K, p − r ), we define h d i N,κ F = (cid:18) − d (cid:19) i h d i ∗ N F if p = 2and h d i N,κ F = h d i ∗ N F if p = 2Without this twist in the p = 2 case, the definition would not agree with the classicalone because of the particular nature of the automorphy factor of the form θ used in theidentification of our forms with classical forms. The same formulas define operators hi N,X and hi N,X on the space of families of modular forms over X ⊆ W i . For a moregeneral X ⊆ W , we break into the components in W i for each i and define hi N,X and hi N,X component by component. For a character χ modulo 4 N (resp. modulo N if p = 2), we define the space of forms of tame nebentypus χ to be the χ -eigenspace of f M κ (4 N, K, p − r ) for the operators hi N,κ (resp. hi N,κ if p = 2). The same definitionapplies to families of forms. These subspaces are denoted by appending a χ to thelist of arguments (e.g. f M κ (4 N, K, p − r , χ )).Let X and Y be rigid spaces equipped with a pair of maps π , π : X −→ Y and let D be a Q -divisor on Y such that π ∗ D − π ∗ D has integral coefficients. Let Z ⊆ X be an admissible affinoid open and let H ∈ H ( Z , O ( π ∗ D − π ∗ D )) . Let U , V ⊆ Y be admissible affinoid opens such that π − ( V ) ∩ Z ⊆ π − ( U ) ∩ Z , andsuppose that π : π − ( V ) ∩ Z −→ V is finite and flat. Then there is a well-defined map H ( U , O ( D )) −→ H ( V , O ( D )) NICK RAMSEY given by the composition H ( U , O ( D )) π ∗ / / H ( π − ( U ) ∩ Z , O ( π ∗ D )) res / / H ( π − ( V ) ∩ Z , O ( π ∗ D )) EDBCGF · H @A / / H ( π − ( V ) ∩ Z , O ( π ∗ D )) π ∗ / / H ( V , O ( D ))where π ∗ is the trace map corresponding to the finite and flat map π . Let N be as above, let ℓ be anyprime number, and let π , π : (cid:26) X (4 N p, ℓ ) an K −→ X (4 N p ) an K p = 2 X (4 N, ℓ ) an K −→ X (4 N ) an K p = 2be the maps defined on noncuspidal points of the underlying moduli problem by π : ( E, P, C ) ( E, P ) π : ( E, P, C ) ( E/C, P/C )Suppose that ℓ = p . Then ( π − ( X (4 N p ) an ≥ p − r ) = π − ( X (4 N p ) an ≥ p − r ) p = 2 π − ( X (4 N ) an ≥ − r ) = π − ( X (4 N ) an ≥ − r ) p = 2for any r < p/ (1 + p ) since quotienting an elliptic curve by a subgroup of order primeto p does not change its measure of singularity. Fix a weight κ ∈ W i ( K ) and let H ℓ ( κ ) denote the specialization of H ℓ to κ ∈ W (which, recall, is defined to bethe specialization of H ℓ to κ/τ i ∈ W ). Pick n such that κ ∈ W in and suppose0 ≤ r ≤ r n . Applying the general construction above with p = 2 p = 2 X X (4 N p, ℓ ) an K X (4 N, ℓ ) an K Y X (4 N p ) an K X (4 N ) an K Z X (4 N p, ℓ ) an ≥ p − r X (4 N, ℓ ) an ≥ − r D Σ Np Σ N H H ℓ ( κ ) H ℓ ( κ ) U = V X (4 N p ) an ≥ p − r X (4 N ) an ≥ − r we arrive an endomorphism of the K -vector space (cid:26) H ( X (4 N p ) an ≥ p − r , O (Σ Np )) p = 2 H ( X (4 N ) an ≥ − r , O (Σ N )) p = 2One checks easily that since the diamond operators act trivially on H ℓ (see Remark3.4), this endomorphism commutes with the action of the diamond operators, andtherefore induces an endomorphism of f M κ (4 N, K, p − r ). We define T ℓ (or U ℓ if ℓ | N ) to be the quotient of this endomorphism by ℓ . HE HALF-INTEGRAL WEIGHT EIGENCURVE Now suppose that ℓ = p . Note that ( π − ( X (4 N p ) an ≥ p − p r ) ⊆ π − ( X (4 N p ) an ≥ p − r ) p = 2 π − ( X (4 N ) an ≥ − r ) ⊆ π − ( X (4 N ) an ≥ − r ) p = 2for any r < /p (1 + p ). This follows from repeated application of the observation(made, for example, in [ ], Theorem 3.3 ( v )) that if v ( E ) < p/ (1 + p ) and C is asubgroup of order p other than the canonical subgroup, then v ( E/C ) = v ( E ) /p andthe canonical subgroup of E/C is E [ p ] /C .If κ ∈ W in and r is chosen so that 0 ≤ r ≤ r n , then we may apply the constructionabove with p = 2 p = 2 X X (4 N p, p ) an K X (4 N, an K Y X (4 N p ) an K X (4 N ) an K Z X (4 N p, p ) an ≥ p − pr X (4 N, an ≥ − r D Σ Np Σ N H H p ( κ ) H ( κ ) U X (4 N p ) an ≥ p − r X (4 N ) an ≥ − r V X (4 N p ) an ≥ p − pr X (4 N ) an ≥ − r to arrive at a linear map (cid:26) H ( X (4 N p ) an ≥ p − r , O (Σ Np )) −→ H ( X (4 N p ) an ≥ p − pr , O (Σ Np )) p = 2 H ( X (4 N ) an ≥ − r , O (Σ N )) −→ H ( X (4 N ) an ≥ − r , O (Σ N )) p = 2This map commutes with the diamond operators and restricts to a map f M κ (4 N, K, p − r ) −→ f M κ (4 N, K, p − pr ) . When composed with the natural restriction map(5) f M κ (4 N, K, p − pr ) −→ f M κ (4 N, K, p − r )and divided by p , we arrive at an endomorphism of f M κ (4 N, K, p − r ) which we denoteby U p . Proposition 5.1 . —
The Hecke operators defined above are continuous.Proof . — Each of the spaces arising in the construction is a Banach space over K ,so it suffices to show that each of the constituent maps of which our Hecke operatorsare the composition has finite norm. By Lemma 2.5 we may ignore the residue disksaround the cusps when computing norms, thereby reducing ourselves to the supremumnorm on functions. It follows easily that the pullback, restriction, and trace mapshave norm not exceeding 1 and that multiplication by H has norm not exceeding thesupremum norm of H on the complement of the residue disks around the cusps. Thelatter is finite since this complement is affinoid. Remarks 5.2 . — – In the overconvergent case, i.e. when we have r >
0, the restriction map (5) iscompact (see Proposition A5.2 of [ ]). It follows that U p is compact as it isthe composition of a continuous map with a compact map. NICK RAMSEY – The Hecke operators T ℓ and U ℓ preserve the space of cusp forms, as can beseen by simply constructing them directly on this space in the same manner asabove. The operator U p is compact on a space of overconvergent cusp forms. Let X ⊆ W be a connected admissibleaffinoid open. We wish to define endomorphisms of f M X (4 N, K, p − r ) that interpolatethe endomorphisms T ℓ and U ℓ constructed above for fixed weights κ ∈ X .Suppose that ℓ = p and let p = 2 p = 2 U = V X (4 N p ) an ≥ p − r X (4 N ) an ≥ − r Z X (4 N p, ℓ ) an ≥ p − r X (4 N, ℓ ) an ≥ − r Σ Σ Np Σ N In the interest of keeping notation under control, let us for the remainder of thissection assume the following definitions. M = H ( U , O (Σ)) N = H ( π − ( U ) ∩ Z , O ( π ∗ Σ)) L = H ( π − ( V ) ∩ Z , O ( π ∗ Σ)) P = H ( π − ( V ) ∩ Z , O ( π ∗ Σ − π ∗ Σ)) Q = H ( π − ( V ) ∩ Z , O ( π ∗ Σ))The Hecke operator T ℓ (or U ℓ if ℓ | N ) at a fixed weight was constructed in theprevious section by first taking the composition of the following continuous maps: apullback M → N , a restriction N → L , multiplication by an element of H ∈ P toarrive at an element of Q , and a trace Q → M , and then restricting to an eigenspaceof the diamond operators at p and dividing by ℓ .The module of families of forms on X is an eigenspace of M b ⊗ K O ( X ) (by the finalremark in Remarks 4.5). To define T ℓ (or U ℓ ) we begin as in the fixed weight caseby defining an endomorphism of M b ⊗ K O ( X ) and then observing that it commuteswith the diamond automorphisms and therefore restricts to an operator on families ofmodular forms. To define this endomorphism, we modify the above sequence of mapsby first applying b ⊗ K O ( X ) to all of the spaces and taking the unique continuous O ( X )-linear extension of each map, with the exception of the multiplication step, where weopt instead to multiply by H ℓ | X ∈ P b ⊗ K O ( X ). In so doing we arrive at an O ( X )-linear endomorphism of M b ⊗ K O ( X ) that is easily seen to commute with the diamondautomorphisms, thereby inducing an endomorphism of the module f M X (4 N, K, p − r ). Lemma 5.3 . —
The Hecke operators defined above for families are continuous.Proof . — By definition, each map arising in the construction is continuous exceptperhaps for the multiplication map. The proof of the continuity of this map requiresseveral simple facts about completed tensor products, all of which can be found insection 2.1.7 of [ ]. HE HALF-INTEGRAL WEIGHT EIGENCURVE It follows trivially from Lemma 2.5 that the multiplication map L × P −→ Q is a bounded K -bilinear map and therefore extends uniquely to a bounded K -linearmap L b ⊗ K P −→ Q. Extending scalars to O ( X ) and completing we arrive at a bounded O ( X )-linear map( L b ⊗ K P ) b ⊗ K O ( X ) −→ Q b ⊗ K O ( X ) . There is an isometric isomorphism( L b ⊗ K P ) b ⊗ K O ( X ) ∼ = ( L b ⊗ K O ( X )) b ⊗ O ( X ) ( P b ⊗ K O ( X ))so we conclude that the O ( X )-bilinear multiplication map( L b ⊗ K O ( X )) b ⊗ O ( X ) ( P b ⊗ K O ( X )) −→ Q b ⊗ K O ( X )is bounded. In particular, multiplication by H ∈ P b ⊗ K O ( X ) is a bounded (and hencecontinuous) map · H : L b ⊗ K O ( X ) −→ Q b ⊗ K O ( X )as desired. Remarks 5.4 . — – The construction of a continuous endomorphism U p is entirely analogous andonce again we find that U p is compact in the overconvergent case, that is,whenever r > – The endomorphisms T ℓ and U ℓ can be extended to f M X (4 N, K, p − r ) for gen-eral admissible affinoid opens X in the usual manner working component bycomponent. – All of the the Hecke operators defined on families preserve the cuspidal sub-spaces, as a direct construction on these spaces demonstrates. Again, the oper-ator U p is compact on a module of overconvergent cusp forms. q -expansions. — In this section we will work out the effect of theHecke operators that we have defined on q -expansions. As in [ ], we must adjustthe naive q -expansions obtained by literally evaluating our forms on Tate curves withlevel structure to get at the classical q -expansions. In particular, by the q -expansionof a form F ∈ f M κ (4 N, K, p − r ) at the cusp associated to (Tate( q ) , ζ ) where ζ is aprimitive 4 N p th root of unity if p = 2 and a primitive 4 N th root of unity if p = 2, wemean F (Tate( q ) , ζ ) θ ( q ) E κ ( q )Similarly, for a family F ∈ M X (4 N, K, p − r ) the corresponding q -expansion is F (Tate( q ) , ζ ) θ ( q ) E ( q ) | X and has coefficients in the ring of analytic functions on X . NICK RAMSEY
Proposition 5.5 . —
Let F be an element of f M κ (4 N, K, p − r ) or f M X (4 N, K, p − r ) and let P a n q n be the q -expansion of F at (Tate( q ) , ζ ) . The corresponding q -expansionof U p F is then P a p n q n .Proof . — We prove the theorem for U p acting on f M κ (4 N, K, p − r ). To obtain theresult for families one could either proceed in the same manner or deduce the resultfor families over X from the result for fixed weight by specializing to weights in X .Let F ∈ f M κ (4 N, K, p − r ) and suppose that F (Tate( q ) , ζ ) θ ( q ) E κ ( q ) = X a n q n . The expansion we seek is1 p π ∗ ( π ∗ F · H p ( κ ))(Tate( q ) , ζ ) · θ ( q ) E κ ( q ) . The cyclic subgroups of order p that intersect the subgroup generated by ζ triviallyare exactly those of the form h ζ ip q p i , 0 ≤ i ≤ p −
1. Thus we have π ∗ ( π ∗ F · H p ( κ ))(Tate( q ) , ζ ) = p − X i =0 ( π ∗ F · H p ( κ ))(Tate( q ) , ζ, h ζ ip q p i )= p − X i =0 F (Tate( q ) / h ζ ip q p i , ζ/ h ζ ip q p i ) H p ( κ )(Tate( q ) , ζ, h ζ ip q p i )= p − X i =0 F (Tate( ζ ip q p ) , ζ ) H p ( κ )(Tate( q ) , ζ, h ζ ip q p i )= p − X i =0 P a n ( ζ ip q p ) n θ ( ζ ip q p ) E κ ( ζ ip q p ) θ ( ζ ip q p ) E κ ( ζ ip q p ) θ ( q ) E κ ( q ) = p P a p n q n θ ( q ) E κ ( q )The same analysis also proves the following. Proposition 5.6 . —
Suppose that either ℓ | N . Let F be an element of f M κ (4 N, K, p − r ) or f M X (4 N, K, p − r ) and let P a n q n be the q -expansion of F at (Tate( q ) , ζ ) . Then thecorresponding q -expansion of U ℓ F is then P a ℓ n q n . In order to work out the effect of T ℓ for ℓ ∤ N p on q -expansions, we will needseveral more q -expansions of Θ ℓ and E ℓ . For the former, we refer the reader to [ ].The latter will follow from the following lemma. For x ∈ Z × p , we denote by [ x ] theanalytic function on W defined by [ x ]( κ ) = κ ( x ). Lemma 5.7 . —
For ℓ = p we have E ℓ (Tate( q ) , µ p + h q ℓ i ) = [ h ℓ i ] E ( q ) E ( q ℓ ) and E ℓ (Tate( q ) , µ pℓ ) = E ( q ) E ( q ℓ ) . HE HALF-INTEGRAL WEIGHT EIGENCURVE Proof . — The second equality is how we chose to characterize E ℓ in the first place.We will use it to give an alternative characterization, which we will in turn use toprove the first equality.By definition, E ℓ and the coefficients of E ( q ) are pulled back from their restrictionsto W through the map (4). Clearly [ h ℓ i ] is the pull-back of [ ℓ ] through this map, soit suffices to prove that E ℓ (Tate( q ) , µ p + h q ℓ i ) = [ ℓ ] E ( q ) E ( q ℓ )where the coefficients are now though of as function only on W . Moreover, it sufficesto prove the equality after specialization to integers λ ≥ ϕ ( q ), as suchintegers are Zariski-dense in W . Let E λ ( τ ) denote the classical analytic p -deprivedEisenstein series of weight λ and level p (normalized to have q -expansion E λ ( q )).Then E an ℓ ( λ ) := E λ ( τ ) /E λ ( ℓτ )is a meromorphic function on X ( pℓ ) an C with rational q -expansion coefficients, and byGAGA and the q -expansion principle yields a rational function on the algebraic curve X ( pℓ ) Q p . By comparing q -expansions it is evident that the restriction of this functionto the region X ( pℓ ) an ≥ is equal to the specialization, E ℓ ( λ ), of E ℓ to λ ∈ W .It follows that E ℓ ( λ )(Tate( q ) , µ p + h q ℓ i ) = E an ℓ ( λ )(Tate( q ) , µ p + h q ℓ i ). The rightside can be computed using the usual yoga where one pretends to specialize q to e πiτ and then computes with analytic transformation formulas (see Section 5 of [ ] for arigorous explanation of this yoga). So specializing, we get E an ℓ ( λ )(Tate( q ) , µ p + h q ℓ i )( τ ) = E an ℓ ( λ )( C / h , τ i , h /p i + h τ /ℓ i ) . Choosing a matrix γ = (cid:18) a bc d (cid:19) ∈ SL ( Z )such that p | c and ℓ | d we arrive at an isomorphism( C / h , τ i , h /p i + h τ /ℓ i ) ∼ −→ ( C / h , γτ i , h /pℓ i ) z zcτ + d Thus E an ℓ ( λ )( C / h , τ i , h /p i + h τ /ℓ i ) = E an ℓ ( λ )( C / h , γτ i , h /pℓ i ) = E λ ( γτ ) E λ ( ℓγτ ) . Now ℓγτ = ( aℓ )( τ /ℓ ) + bc ( τ /ℓ ) + d/ℓ , so we have E λ ( γτ ) E λ ( ℓγτ ) = ( cτ + d ) λ E λ ( τ )(( cτ + d ) /ℓ ) λ E λ ( τ /ℓ ) = ℓ λ E λ ( τ ) E λ ( τ /ℓ )and the result follows. NICK RAMSEY
Proposition 5.8 . —
Let F ∈ f M κ (4 N, K, p − r , χ ) with κ ∈ W i and let P a n q n bethe q -expansion of F at (Tate( q ) , ζ ) . Then the corresponding q -expansion of T ℓ F is P b n q n where b n = a ℓ n + κ ( ℓ ) χ ( ℓ ) ℓ − (cid:18) ( − i nℓ (cid:19) a n + κ ( ℓ ) χ ( ℓ ) ℓ − a n/ℓ . Let F ∈ f M X (4 N, K, p − r , χ ) with X a connected affinoid in W i , and let the q -expansion of F be P a n q n as above. Then the corresponding q -expansion of T ℓ F is P b n q n where b n = a ℓ n + [ ℓ ] χ ( ℓ ) ℓ − (cid:18) ( − i nℓ (cid:19) a n + [ ℓ ] χ ( ℓ ) ℓ − a n/ℓ . Proof . — We prove the first assertion. The second assertion may either be provendirectly in the same manner or simply deduced from the first via specialization toindividual weights in X . Let κ ∈ W ( K ), let F ∈ f M κ (4 N, K, p − r , χ ), and let F (Tate( q ) , ζ ) θ ( q ) E κ ( q ) = X a n q n be the q -expansion of F at (Tate( q ) , ζ ). The corresponding q -expansion of T ℓ F is(6) 1 ℓ π ∗ ( π ∗ F · H ℓ ( κ )) · θ ( q ) E κ ( q ) . The cyclic subgroups of Tate( q ) of order ℓ are the subgroups µ ℓ , h ζ iℓ q ℓ i ≤ i ≤ ℓ − , and h ζ jℓ q ℓ i ≤ j ≤ ℓ − . We examine the contribution of each of these types of subgroups to π ∗ ( π ∗ F · H ℓ ( κ ))separately.First, we have F (Tate( q ) /µ ℓ , ζ/µ ℓ ) H ℓ ( κ )(Tate( q ) , ζ, µ ℓ )= F (Tate( q ℓ ) , ζ ℓ )Θ ℓ (Tate( q ) , ζ , µ ℓ ) π ∗ E ℓ ( κ )(Tate( q ) , ζ p , µ ℓ )= F (Tate( q ℓ ) , ζ ℓ )Θ ℓ (Tate( q ) , ζ , µ ℓ ) · E ℓ ( κ )(Tate( q ) /µ ℓ , ( µ p + Tate( q )[ ℓ ]) /µ ℓ )= F (Tate( q ℓ ) , ζ ℓ )Θ ℓ (Tate( q ) , ζ , µ ℓ ) E ℓ ( κ )(Tate( q ℓ ) , µ p + h q i )From the definition (2) and Lemma 5.7 we have E ℓ (Tate( q ℓ ) , µ p + h q i )= E ℓ (Tate( q ℓ ) , µ p + h q ℓ i ) E ℓ (Tate( q ℓ ) / h q ℓ i , ( µ p + h q i ) / h q ℓ i )= E ℓ (Tate( q ℓ ) , µ p + h q ℓ i ) E ℓ (Tate( q ℓ ) , µ p + h q i )= [ h ℓ i ] E ( q ℓ ) E ( q ℓ ) · [ h ℓ i ] E ( q ℓ )( q ) = [ h ℓ i ] E ( q ℓ ) E ( q ) HE HALF-INTEGRAL WEIGHT EIGENCURVE When specialized to κ , this becomes κ ( h ℓ i ) E κ ( q ℓ ) E κ ( q ) . Referring to [ ] we find Θ ℓ (Tate( q ) , ζ , µ ℓ ) = ℓ θ ( q ℓ ) θ ( q ) . Thus the contribution of this first subgroup is χ ( ℓ ) τ ( ℓ ) i P a n q ℓ n θ ( q ℓ ) E κ ( q ℓ ) ℓ θ ( q ℓ ) θ ( q ) κ ( h ℓ i ) E κ ( q ℓ ) E κ ( q ) = ( κ ( h ℓ i ) χ ( ℓ ) τ ( ℓ ) i ) ℓ P a n q ℓ n θ ( q ) E κ ( q )The subgroups h ζ aℓ q ℓ i contribute ℓ − X a =0 F (Tate( q ) / h ζ aℓ q ℓ i , ζ/ h ζ aℓ q ℓ i ) H ℓ ( κ )(Tate( q ) , ζ, h ζ aℓ q ℓ i )= ℓ − X a =0 F (Tate( ζ aℓ q ℓ ) , ζ )Θ ℓ (Tate( q ) , ζ , h ζ aℓ q ℓ i ) · π ∗ E ℓ ( κ )(Tate( q ) , ζ p , h ζ aℓ q ℓ i )= ℓ − X a =0 F (Tate( ζ aℓ q ℓ ) , ζ )Θ ℓ (Tate( q ) , ζ , h ζ aℓ q ℓ i ) · E ℓ ( κ )(Tate( q ) / h ζ aℓ q ℓ i , ( µ p + Tate( q )[ ℓ ]) / h ζ aℓ q ℓ i )= ℓ − X a =0 F (Tate( ζ aℓ q ℓ ) , ζ )Θ ℓ (Tate( q ) , ζ , h ζ aℓ q ℓ i ) · E ℓ ( κ )(Tate( ζ aℓ q ℓ ) , µ pℓ )By (2) we have E ℓ (Tate( ζ aℓ q ℓ ) , µ pℓ ) = E ℓ (Tate( ζ aℓ q ℓ ) , µ pℓ ) E ℓ (Tate( ζ aℓ q ℓ ) /µ ℓ , µ pℓ /µ ℓ )= E ℓ (Tate( ζ aℓ q ℓ ) , µ pℓ ) E ℓ (Tate( ζ aℓ q ℓ ) , µ pℓ )= E ( ζ aℓ q ℓ ) E ( ζ aℓ q ℓ ) E ( ζ aℓ q ℓ ) E ( q ) = E ( ζ aℓ q ℓ ) E ( q )Referring to [ ], we findΘ ℓ (Tate( q ) , ζ , h ζ aℓ q ℓ i ) = θ ( ζ aℓ q ℓ ) θ ( q ) . Thus the total contribution of this collection of subgroups is ℓ − X a =0 P a n ( ζ aℓ q ℓ ) n θ ( ζ aℓ q ℓ ) E κ ( ζ aℓ q ℓ ) θ ( ζ aℓ q ℓ ) θ ( q ) E κ ( ζ aℓ q ℓ ) E κ ( q ) = ℓ P a ℓ n q n θ ( q ) E κ ( q ) . NICK RAMSEY
The subgroups h ζ bℓ q ℓ i contribute ℓ − X b =1 F (Tate( q ) / h ζ bℓ q ℓ i , ζ/ h ζ bℓ q ℓ i ) H ℓ ( κ )(Tate( q ) , ζ, h ζ bℓ q ℓ i )= ℓ − X b =1 F (Tate( ζ bℓ q ) , ζ ℓ )Θ ℓ (Tate( q ) , ζ , h ζ bℓ q ℓ i ) · π ∗ E ℓ ( κ )(Tate( q ) , ζ p , h ζ bℓ q ℓ i )= ℓ − X b =1 F (Tate( ζ bℓ q ) , ζ ℓ )Θ ℓ (Tate( q ) , ζ , h ζ bℓ q ℓ i ) · E ℓ ( κ )(Tate( q ) / h ζ bℓ q ℓ i , ( µ p + Tate( q )[ ℓ ]) / h ζ bℓ q ℓ i )= ℓ − X b =1 F (Tate( ζ bℓ q ) , ζ ℓ )Θ ℓ (Tate( q ) , ζ , h ζ bℓ q ℓ i ) · E ℓ ( κ )(Tate( ζ bℓ q ) , µ p + h q ℓ i )By (2) and Lemma 5.7 we have E ℓ (Tate( ζ bℓ q ) , µ p + h q ℓ i )= E ℓ (Tate( ζ bℓ q ) , µ p + h q i ) E ℓ (Tate( ζ bℓ q ) /µ ℓ , ( µ p + h q ℓ i ) /µ ℓ )= E ℓ (Tate( ζ bℓ q ) , µ pℓ ) E ℓ (Tate( q ℓ ) , µ p + h q i )= E ( ζ bℓ q ) E ( q ℓ ) · [ h ℓ i ] E ( q ℓ ) E ( q ) = [ h ℓ i ] E ( ζ bℓ q ) E ( q )When specialized to κ , this becomes κ ( h ℓ i ) E κ ( ζ bℓ q ) E κ ( q ) . Referring to [ ] we findΘ ℓ (Tate( q ) , ζ , h ζ bℓ q i ) = (cid:18) − ℓ (cid:19) g ℓ ( ζ bℓ ) θ ( ζ bℓ q ) θ ( q )where g ℓ ( ζ ) = ℓ − X m =1 (cid:16) mℓ (cid:17) ζ m HE HALF-INTEGRAL WEIGHT EIGENCURVE is the Gauss sum associated to the ℓ th root of unity ζ . Thus the total contribution ofthis third collection of subgroups is ℓ − X b =1 χ ( ℓ )( − /ℓ ) i τ ( ℓ ) i P a n ( ζ bℓ q ) n θ ( ζ bℓ q ) E κ ( ζ bℓ q ) (cid:18) − ℓ (cid:19) g ℓ ( ζ bℓ ) θ ( ζ bℓ q ) θ ( q ) κ ( h ℓ i ) E κ ( ζ bℓ q ) E κ ( q )= κ ( h ℓ i ) χ ( ℓ ) (cid:18) − ℓ (cid:19) i +1 τ ( ℓ ) i g ℓ ( ζ ℓ ) θ ( q ) E κ ( q ) X n a n ℓ − X b =1 ζ bnℓ (cid:18) bℓ (cid:19)! q n = κ ( h ℓ i ) χ ( ℓ ) (cid:18) − ℓ (cid:19) i +1 τ ( ℓ ) i g ℓ ( ζ ℓ ) θ ( q ) E κ ( q ) X n a n (cid:16) nℓ (cid:17) g ℓ ( ζ ℓ ) q n = κ ( h ℓ i ) χ ( ℓ ) (cid:18) − ℓ (cid:19) i τ ( ℓ ) i ℓ P (cid:0) nℓ (cid:1) a n q n θ ( q ) E κ ( q )Adding all this up and plugging into (6) we see that the q -expansion of T ℓ F is P b n q n where b n = a ℓ n + κ ( h ℓ i ) ℓ − χ ( ℓ ) (cid:18) − ℓ (cid:19) i τ ( ℓ ) i (cid:16) nℓ (cid:17) a n + κ ( h ℓ i ) ℓ − χ ( ℓ ) τ ( ℓ ) i a n/ℓ = a ℓ n + κ ( ℓ ) ℓ − χ ( ℓ ) (cid:18) ( − i nℓ (cid:19) a n + κ ( ℓ ) ℓ − χ ( ℓ ) a n/ℓ .
6. Classical weights and classical forms
In this section we define classical subspaces of our spaces of modular forms andprove the following analog of Coleman’s theorem on overconvergent forms of low slope.Throughout this section k will denote an odd positive integer and we set λ = ( k − / Theorem 6.1 . —
Let m be a positive integer, let ψ : ( Z / q p m − Z ) × −→ K × be acharacter, and define κ ( x ) = x λ ψ ( x ) . If F ∈ f M † κ (4 N, K ) satisfies U p F = αF with v ( α ) < λ − , then F is classical. Our proof follows the approach of Kassaei ([ ]), which is modular in nature andbuilds the classical form by analytic continuation and gluing. The term “analyticcontinuation” has little meaning here since we have only defined our modular formsover restricted regions on the modular curve, owing to the need to avoid Eisensteinzeros. To get around this difficulty, we must invoke the previous formalism of theauthor for p -adic modular forms of classical half-integral weight (see [ ]).Let N be a positive integer. In [ ] we defined the space of modular forms ofweight k/ N over a Z [1 / N ]-algebra R to be the R -module f M ′ k/ (4 N, R ) := H ( X (4 N ) R , O ( k Σ N )) . Note that this space was denoted M k/ (4 N, R ) and k Σ N was denoted Σ N,k in [ ].Roughly speaking, in this space of forms we have divided by θ k to reduce to weight NICK RAMSEY zero instead of E λ θ . Let r ∈ [0 , ∩ Q and define f M ′ k/ (4 N p m , K, p − r ) = H ( X (4 N p m ) an ≥ p − r , O ( k Σ Np m )) . It is an easy matter to check that the construction of the Hecke operators T ℓ and U p in Section 5 (using H = Θ kℓ ) adapts to this space of forms and furnishes us withHecke operators having the expected effect on q -expansions. We will briefly reviewthe construction of U p in this context later in this section.The next proposition relates these spaces of p -adic modular forms to the onesdefined in this paper, and will ensure that the latter spaces (and consequently theeigencurve defined later in this paper) see the classical half-integral weight modularforms of arbitrary p -power level. Note that this identification requires the knowledgeof the action of the diamond operators at p because this data is part of the p -adicweight character. Proposition 6.2 . —
Let m be a positive integer, let ψ : ( Z / q p m − Z ) × −→ K × be acharacter, and define κ ( x ) = x λ ψ ( x ) . Then, for ≤ r ≤ r m , the space f M ′ (4 N p m +1 / q , K, p − r ) hi ∗ q pm − = ψ = ( f M ′ k/ (4 N p m , K, p − r ) hi ∗ pm = ψ p = 2 f M ′ k/ (2 m +1 N, K, p − r ) hi ∗ m +1 = ψ p = 2 is isomorphic to f M κ (4 N, K, p − r ) in a manner compatible with the action of the Heckeoperators and tame diamond operators.Proof . — Let i be such that κ ∈ W i . The complex-analytic modular forms θ k − and E κτ − i are each of weight λ . If p = 2, then the former is invariant under the h d i ∗ q p m − while if p = 2 it has eigencharacter ( − / · ) i . The latter has eigencharacter ψτ − i for this action in both cases. Standard arguments using GAGA and the q -expansion principle show that the ratio θ k − /E κτ − i furnishes an algebraic rationalfunction on X (4 N p m +1 / q ) K . Passing to the p -adic analytification and restrictingto X (4 N p m +1 / q ) an ≥ p − r , we see that this function has divisor ( k − Np m +1 / q , since E κτ − i is invertible in this region for r as in the statement of the proposition (because κ ∈ W m ).Let F ′ ∈ f M ′ k/ (4 N p m +1 / q , K, p − r ) be a form with eigencharacter ψ for hi ∗ q p m − and let F = F ′ · θ k − E κτ − i Then, for d ∈ ( Z / q p m − Z ) × we have h d i ∗ q p m − F = τ ( d ) i ( − / · ) i F . In particular, F isfixed by h d i ∗ p m with d ≡ q ). The construction of the canonical subgroup oforder q p m − (defined because r ≤ r m < p − m / q (1 + p )) ensures that the map(7) X (4 N p m +1 / q ) an ≥ p − r / {h d i q p m − | d ≡ q ) } −→ (cid:26) X (4 N p ) an ≥ p − r p = 2 X (4 N ) an ≥ − r p = 2induced by ( E, P ) ( E, aP ) HE HALF-INTEGRAL WEIGHT EIGENCURVE where the integer a is chosen so that (cid:26) a ≡ p m − (mod p m ) and a ≡ N ) p = 2 a ≡ m − (mod 2 m +1 ) and a ≡ N ) p = 2is an isomorphism. This map pulls the divisor Σ Np (or Σ N if p = 2) back to Σ Np m (resp. Σ m +1 N if p = 2), so we conclude that F descends to a section of O (Σ Np )on X (4 N p ) an ≥ p − r (resp. a section of O (Σ N ) on X (4 N ) an ≥ − r ) and that this sectionsatisfies h d i ∗ p F = τ ( d ) i F (resp. h d i ∗ F = τ ( d ) i ( − /d ) i F ) for all d ∈ ( Z / q Z ) × . Thus wemay regard F as an element of f M κ (4 N, K, p − r ). Conversely, for F ∈ f M κ (4 N, K, p − r ),it is easy to see that F · E κτ − i θ k − ∈ f M ′ k/ (4 N p m +1 / q , K, p − r ) hi q pm − = ψ (where F is implicitly pulled back via the above map (7)) and that this furnishes aninverse to the above map F ′ F . That these maps are equivariant with respectto the Hecke action is a formal manipulation with the setup in Section 5 used todefine the action on both sides. That it is equivariant with respect to tame diamondoperators is trivial, but relies essentially on the “twisted” convention for this actionon f M κ (4 N, K, p − r ) (for p = 2).In general, if U is a connected admissible open in X (4 N p m +1 / q ) an K containing X (4 N p m +1 / q ) an ≥ p − r and F ∈ f M κ (4 N, K, p − r ) (with κ as in the previous propo-sition) we will say that F analytically continues to U if the corresponding form F ′ ∈ f M ′ k/ (4 N p m +1 / q , K, p − r ) analytically continues to an element of(8) H ( U , O ( k Σ Np m +1 / q )) . Note that, in case U is preserved by the diamond operators at p , this analytic con-tinuation automatically lies in the ψ -eigenspace of (8) since G − h d i ∗ q p m − G vanisheson the nonempty admissible open X (4 N p m +1 / q ) an ≥ p − r for all d , and hence must van-ish on all of U . In particular, in case U = X (4 N p m +1 / q ) an K we make the followingdefinition. Definition 6.3 . — Let κ ( x ) = x λ ψ ( x ) be as in Proposition 6.2. An element F ∈ f M κ (4 N, K ) † is called classical if it analytically continues in the sense described aboveto all of X (4 N p m +1 / q ) an K . That is, if it is in the image of the (injective) map H ( X (4 N p m +1 / q ) an K , O ( k Σ Np )) hi pm = ψ −→ f M ′ k/ (4 N p m +1 / q , K, p − r m ) hi pm = ψ ∼ = f M κ (4 N, K, p − r m ) ֒ → f M κ (4 N, K ) † The analytic continuation used to prove Theorem 6.1 will proceed in three steps.All of them involve the construction of the operator U p on f M ′ k/ (4 N p m +1 / q , K, p − r ),which goes as follows. Let π , π : X (4 N p m +1 / q , p ) an K −→ X (4 N p m +1 / q ) an K NICK RAMSEY be the usual pair of maps and let Θ p denote the rational function on X (4 , p ) Q fromSection 3. For any pair of admissible open U and V in X (4 N p m +1 / q ) an K with π − V ⊆ π − U we have the map H ( U , O ( k Σ Np m +1 / q )) −→ H ( V , O ( k Σ Np m +1 / q )) F p π ∗ ( π ∗ F · Θ kp )Note that there is no need to introduce the space Z as in Section 5 since our “twisting”section Θ kp is defined on all of X (4 N p m +1 / q , p ) an K . Also, recall from Section 5 thatif 0 ≤ r < /p (1 + p ) we have π − ( X (4 N p m +1 / q ) an ≥ p − p r ) ⊆ π − ( X (4 N p m +1 / q ) an ≥ p − r )Thus if F ∈ f M ′ k/ (4 N p m +1 / q , K, p − r ) with r < /p (1 + p ) then U p F analyticallycontinues to X (4 N p m +1 / q ) an ≥ p − p r . From this simple observation we get the first andeasiest analytic continuation result. Proposition 6.4 . —
Let r > and let F ∈ f M ′ k/ (4 N p m +1 / q , K, p − r ) . Suppose thatthere exists a polynomial P ( T ) ∈ K [ T ] with P (0) = 0 such that P ( U p ) F analyticallycontinues to X (4 N p m +1 / q ) an ≥ p − / (1+ p ) . Then F analytically continues to this regionas well.Proof . — Write P ( T ) = P ( T ) + a with P (0) = 0 and a = 0. Then F = 1 a (cid:0) P ( U p ) F − P ( U p ) F (cid:1) . If 0 < r < /p (1+ p ), then the right side analytically continues to X (4 N p m +1 / q ) an ≥ p − p r ,and hence so does F . Since r >
0, we may repeat this process until we have ana-lytically continued F to X (4 N p m +1 / q ) an ≥ p − s for some s ≥ /p (1 + p ). Now restrict F to X (4 N p m +1 / q ) an ≥ p − /p p ) and apply the process once more to get the desiredresult.The second analytic continuation step requires that we introduce some admissibleopens in X (4 N p m +1 / q ) an Q p defined by Buzzard in [ ]. The use of the letter W in thispart of the argument is intended to keep the notation parallel to that in [ ] and shouldnot be confused with weight space. If p = 2, we let W ⊆ X (4 N, p ) an Q p denote theadmissible open subspace whose points reduce to the irreducible component on thespecial fiber of X (4 N, p ) in characteristic p that contains the cusp associated to thedatum (Tate( q ) , P, µ p ) for some (equivalently, any) point of order 4 N on Tate( q ). Al-ternatively, W can be characterized as the compliment of the connected component ofthe ordinary locus in X (4 N, p ) an Q p containing the cusp associated to (Tate( q ) , P, h q p i )for some (equivalently, any) choice of P . If p = 2, we let W ⊆ X ( N, an Q p denote theadmissible open subspace whose points reduce to the irreducible component on thespecial fiber of X ( N,
2) in characteristic 2 that contains the cusp associated to the
HE HALF-INTEGRAL WEIGHT EIGENCURVE datum (Tate( q ) , P, µ ) for some (equivalently, any) point of order N on Tate( q ). Al-ternatively, W can be characterized as the compliment of the connected componentof the ordinary locus in X ( N, an Q p containing the cusp associated to (Tate( q ) , P, h q i )for some (equivalently, any) choice of P . In particular W always contains the entiresupersingular locus. The reader concerned about problems with small N in thesedescriptions should focus on the “alternative” versions and the remarks in Section 2.2about adding level structure and taking invariants.In [ ], Buzzard introduces a map v ′ : W −→ Q defined as follows. If x ∈ W isa cusp, then set v ′ ( x ) = 0. Otherwise, x ∈ W corresponds to a triple ( E/L, P, C )with
E/L an elliptic curve, P a point of order 4 N ( N if p = 2) on E , and C ⊂ E acyclic subgroup of order p . If E has bad or ordinary reduction, then set v ′ ( x ) = 0.Otherwise, if 0 < v ( E ) < p/ (1 + p ), then E has a canonical subgroup H of order p ,and we define v ′ ( x ) = (cid:26) v ( E ) H = C − v ( E/C ) H = C Finally, if v ( E ) ≥ p/ (1 + p ) we define v ′ ( x ) = p/ (1 + p ). Note that v ′ does not dependon the point P . For a nonnegative integer n , we let V n denote the region in W definedby the inequality v ′ ≤ − /p n − (1 + p ). Buzzard proves that V n is an admissibleaffinoid open in W for each n , and that W is admissibly covered by the V n .Let f : X (4 N p m +1 / q ) an Q p −→ (cid:26) X (4 N, p ) an Q p p = 2 X ( N, an Q p p = 2denote the map characterized by( E, P ) (cid:26) ( E/ h N pP i , p m P/ h N pP i , h N P/ h N pP ii p = 2( E/ h N P i , m +1 P/ h N P i , h N P/ h N P ii ) p = 2on noncuspidal points. Define W = f − ( W ) and Z n = f − ( V n ) for n ≥
0. Itfollows from the above that W is an admissible open in X (4 N p m +1 / q ) an K and that W is admissibly covered by the admissible opens Z n . The latter are affinoid since f is finite. Lemma 6.5 . —
The inclusion π − ( Z n +2 ) ⊆ π − ( Z n ) holds for all n ≥ .Proof . — Since the maps π and π are finite, the stated inclusion is between affinoidsand can be checked on noncuspidal points. Then the assertion follows immediatelyfrom two applications of Lemma 4.2 (2) of [ ].We can now state and prove the second analytic continuation result. Proposition 6.6 . —
Let r > and let F ∈ f M ′ k/ (4 N p m +1 / q , K, p − r ) . Suppose thatthere exists a polynomial P ( T ) ∈ K [ T ] with P (0) = 0 such that P ( U p ) F extends to W . Then F extend to this region as well.Proof . — Note that X (4 N p m +1 / q ) an ≥ p − / (1+ p ) = Z ⊆ W NICK RAMSEY so that by Proposition 6.4, F extends to Z . Now we proceed inductively to extend F to each Z n . Let P ( T ) = P ( T ) + a with P (0) = 0 and a = 0. Then F = 1 a ( P ( U p ) F − P ( U p ) F ) . Suppose F extends to Z n for some n ≥
0. By hypothesis P ( U p ) F extends to allof W , and by the construction of U p and Lemma 6.5, P ( U p ) F extends to Z n +2 ,and hence so does F . Thus by induction F extends to Z n for all n , and since W isadmissibly covered by the Z n , F extends to W .If p = 2 and m = 1 (that is, if there is only one p in the level), then this is the endof the second analytic continuation step. In all other cases, Buzzard’s techniques in[ ] allow us to analytically continue to more connected components of the ordinarylocus. Define m = ord p ( q p m − ) = (cid:26) m p = 2 m + 1 p = 2Following Buzzard, for 0 ≤ r ≤ m let U r denote the admissible open in X (4 N p m +1 / q ) an K whose non-cuspidal points parameterize pairs ( E, P ) that areeither supersingular or satisfy H p m − r ( E ) = (cid:26) H p m − r ( E ) = h N p r P i p = 2 H m +1 − r ( E ) = h N r P i p = 2We have W = U ⊆ U ⊆ · · · ⊆ U m = X (4 N p m +1 / q ) an K The last goal of the second step is to analytically continue eigenforms to U m − . Lemma 6.7 . —
For ≤ r ≤ m − we have π − ( U r +1 ) ⊆ π − ( U r ) .Proof . — As usual, it suffices to check this on non-cuspidal points. Moreover, itsuffices to check it on ordinary points, since the entire supersingular locus is containedin each U r . For brevity we will assume p = 2. The case p = 2 is proven in exactlythe same manner. Let ( E, P, C ) ∈ π − ( U r +1 ) be such a point. Then H p m − r − ( E ) = h N p r +1 P i and since r + 1 < m , we conclude that H p m − r − ( E ) ∩ C = 0. NowProposition 3.5 of [ ] implies that H p r ( E/C ) is indeed generated by the image of4
N p r P in E/C , so (
E, P, C ) ∈ π − ( U r ). Proposition 6.8 . —
Let r > and let F ∈ f M ′ k/ (4 N p m +1 / q , K, p − r ) . Suppose thatthere exists a polynomial P ( T ) ∈ K [ T ] with P (0) = 0 such that P ( U p ) F extends to U m − . Then F extend to this region as well.Proof . — Since U = W , Proposition 6.6 ensures that F analytically continues to U . Now we proceed inductively to extend F to each U r , 0 ≤ r ≤ m −
1. Let P ( T ) = P ( T ) + a with P (0) = 0 and a = 0. Then F = 1 a ( P ( U p ) F − P ( U p ) F ) . Suppose F extends to U r for some 0 ≤ r ≤ m −
2. By hypothesis P ( U p ) F extendsto all of U m − , and by the construction of U p and Lemma 6.7, P ( U p ) F extends to HE HALF-INTEGRAL WEIGHT EIGENCURVE U r +1 , and hence so does F . Proceeding inductively, we see that F can be extendedall the way to U m − .The third and most difficult analytic continuation step is to continue to the restof the curve X (4 N p m +1 / q ) an K . If p = 2, we let V denote the admissible open in X (4 N, p ) an K whose points reduce to the irreducible component on the special fiber incharacteristic p that contains the cusp associated to (Tate( q ) , P, h q p i ) for some (equiv-alently, any) choice of P . On the other hand, if p = 2, we let V denote the admissibleopen in X ( N, an K whose points reduce to the irreducible component on the specialfiber in characteristic 2 that contains the cusp associated to (Tate( q ) , P, h q i ) for some(equivalently, any) choice of P . Let V denote the preimage of V under the finite map g : X (4 N p m +1 / q ) an Q p −→ (cid:26) X (4 N, p ) an Q p p = 2 X ( N, an Q p p = 2( E, P ) (cid:26) ( E, p m P, h N p m − P i ) p = 2( E, m +1 P, h m N P i ) p = 2Note that the preimage under g of the locus that reduces to the other componentof X (4 N, p ) F p (or X ( N, F if p = 2) is U m − , so in particular { U m − , V } is anadmissible cover of X (4 N p m +1 / q ) an Q p and U m − ∩ V is the supersingular locus.For any subinterval I ⊆ ( p − p/ (1+ p ) ,
1] let V I (respectively U m − I ) denote theadmissible open in V (respectively U m − ) defined by the condition p − v ( E ) ∈ I . Notethat the complement of U m − in X (4 N p m +1 / q ) an K is V [1 , U p -eigenformof suitably low slope we will define a function on V [1 ,
1] and use the gluing techniquesof [ ] to glue it to the analytic continuation of our eigenform to U m − guaranteedby Proposition 6.6. These techniques rely heavily on the norms introduced in Section2.3. The use of Lemma 2.5 to reduce these norms to the supremum norm on thecomplement of the residue disks around the cusps will be implicit in many of theestimates that follow.Over V ( p − /p (1+ p ) ,
1] we have a section h to π given on noncuspidal points by h : V ( p − /p (1+ p ) , −→ X (4 N p m +1 / q , p ) an K ( E, P ) ( E, P, H p )By standard results on quotienting by the canonical subgroup ([ ], Theorem 3.3), thecomposition π ◦ h restricts to a map(9) Q : V ( p − r , −→ V ( p − p r , ≤ r ≤ /p (1 + p ). Note that since Q preserves the property of havingordinary or supersingular reduction, Q restricts to a map V ( p − r , → V ( p − p r , ϑ on V ( p − /p (1+ p ) ,
1] by ϑ = h ∗ Θ p , and note that(10) div( ϑ ) = h ∗ ( π ∗ Σ Np m +1 / q − π ∗ Σ Np m +1 / q ) = Q ∗ Σ Np m +1 / q − Σ Np m +1 / q . Let F ∈ H ( U m − , O ( k Σ Np m +1 / q )) and suppose that U p F = αF + H NICK RAMSEY on U m − for some classical form H and some α = 0. Note that this condition makessense because π − ( U m − ) ⊆ π − ( U m − ) by Lemma 6.7. For a pair ( E, P ) ∈ U m − corresponding to a noncuspidal point, we have(11) F ( E, P ) = 1 αp X C F ( E/C, P/C )Θ kp ( E, P, C ) − α H ( E, P )where the sum is over the cyclic subgroups of order p having trivial intersectionwith the group generated by P . Suppose that ( E, P ) corresponds to a point in V ( p − /p (1+ p ) , P has trivial intersection withthe canonical subgroup H p , and thus the canonical subgroup is among the sub-groups occurring in the sum above. One can check using Theorem 3.3 of [ ] that( E/H p , P/H p ) corresponds to a point of V ( p − p/ (1+ p ) , C = H p is a cyclicsubgroup of order p with trivial intersection with h P i , then ( E/C, P/C ) correspondsto a point of U m − ( p − /p (1+ p ) , F on V ( p − /p (1+ p ) ,
1) by F = F − αp ϑ k Q ∗ ( F | V ( p − p/ (1+ p ) , ) . Lemma 6.9 . —
The function F on V ( p − /p (1+ p ) , extends to an element of H ( V ( p − /p (1+ p ) , , O ( k Σ Np m +1 / q )) .Proof . — Equation (11) and the comments that follow it show how to define theextension e F of F , at least on noncuspidal points. For a pair ( E, P ) correspondingto a noncuspidal point of V ( p − /p (1+ p ) , e F ( E, P ) = 1 αp X C F ( E/C, P/C )Θ kp ( E, P, C ) − α H ( E, P )where the sum is over the cyclic subgroups of order p of E not meeting h P i and notequal to H p ( E ). We can formalize this as follows.The canonical subgroup of order p furnishes a section to the finite map π − ( V ( p − /p (1+ p ) , π −→ V ( p − /p (1+ p ) , π − ( V ( p − /p (1+ p ) , Z denote the compliment of this connected component. Then π restricts to afinite and flat map Z −→ V ( p − /p (1+ p ) , . Note that Z = π − ( V ( p − /p (1+ p ) , ∩ Z ⊆ π − ( U m − ( p − /p (1+ p ) , ∩ Z as can be checked on noncuspidal points (see the comments following Equation (11)).Now we may apply the general construction of Section 5 with this Z and define e F = 1 αp π ∗ ( π ∗ F · Θ kp ) − α H. Then e F ∈ H ( V ( p − /p (1+ p ) , , O ( k Σ Np m +1 / q ))and Equation (11) shows that e F extends F . HE HALF-INTEGRAL WEIGHT EIGENCURVE For n ≥ F n of H ( V ( p − /p n − (1+ p ) , , O ( k Σ Np m +1 / q ))inductively, where F is as above and for n ≥ F n +1 = F + 1 αp ϑ k Q ∗ ( F n | V ( p − /p n +1(1+ p ) , ) . Note that (9) and (10) show that the F n do indeed lie in the spaces indicated. Ourgoal is to show that the sequence { F n } , when restricted to V [1 , G of H ( V [1 , , O ( k Σ Np m +1 / q )) that glues to F in the sense that thereexists a global section of O ( k Σ Np m +1 / q ) that restricts to F and G on U m − and V [1 , ]. The following lemmas furnish some necessary norm estimates. Lemma 6.10 . —
The function Θ p on Y (4 , p ) Q p is integral. That is, it extends toa regular function on the fine moduli scheme Y (4 , p ) Z p .Proof . — Each Γ (4) ∩ Γ ( p ) structure on the elliptic curve Tate( q ) / Q p (( q )) liftstrivially to one over the Tate curve thought of over Z p (( q )). Since the Tate curveis ordinary, such a structure specializes to a unique component of the special fiber Y (4 , p ) F p . Since Y (4 , p ) Z p is Cohen-Macaulay, the usual argument used to provethe q -expansion principal (as in the proof of Corollary 1.6.2 of [ ]) shows that Θ p is integral as long as it has integral q -expansion associated to a level structure spe-cializing to each component of the special fiber. In fact, all q -expansion of Θ p arecomputed explicitly in Section 5 of [ ], and are all integral. Lemma 6.11 . —
Let R be an F p -algebra, let E be an elliptic curve over R , and let E ( p ) denote the base change of E via the absolute Frobenius morphism on Spec( R ) .Let Fr : E −→ E ( p ) denote the relative Frobenius morphism. Then for any point P of order on E wehave Θ p ( E, P, ker(Fr )) = 0 Proof . — In characteristic p , the forgetful map Y (4 , p ) F p −→ Y (4) F p has a section given on noncuspidal points by s : ( E, P ) ( E, P, ker(Fr )) . By Lemma 6.10, we may pull back (the reduction of) Θ p through this section toarrive at a regular function on the smooth curve Y (4) F p .The q -expansion of s ∗ Θ p at the cusp associated to (Tate( q ) , ζ ) is s ∗ Θ p (Tate( q ) , ζ ) = Θ p (Tate( q ) , ζ , (ker(Fr ))) . Recall that the map Tate( q ) −→ Tate( q p ) NICK RAMSEY given by quotienting by µ p is a lifting of Fr to characteristic zero (more specifically,to the ring Z (( q ))). Thus the q -expansion we seek is the reduction ofΘ p (Tate( q ) , ζ , µ p ) = p P n ∈ Z q p n P n ∈ Z q n modulo p , which is clearly zero. We refer the reader to Section 5 of [ ] for thecomputation of the above q -expansion in characteristic zero. It follows from the q -expansion principle that s ∗ Θ p = 0, which implies our claim. Lemma 6.12 . —
Let ≤ r < /p (1 + p ) . Then the section ϑ of O (Σ Np m +1 / q − Q ∗ Σ Np m +1 / q ) satisfies k ϑ k V [ p − r , ≤ p pr − . Proof . — By Lemma 2.5, we may ignore points reducing to cusps in computing thenorm. Let x ∈ V [ p − r ,
1] be outside of this collection of points, so x corresponds toa pair ( E, P ) with good reduction. Let H p i denote the canonical subgroup of E oforder p i (for whichever i this is defined). Let E be a smooth model of E over O L andlet P and H p be the extensions of P and H p to E , respectively (these E and H should not be confused with the functions by the same name introduced in Section3).By Theorem 3.10 of [ ], H p reduces modulo p/p v ( E ) to ker(Fr). Applying this to E / H p we see that H p / H p reduces modulo p/p v ( E/H p ) to ker(Fr) on the correspondingreduction of E/H p . By Theorem 3.3 of [ ], we know that v ( E/H p ) = pv ( E ), so p − v ( E/H p ) | p − v ( E ) and we may combine these statements to conclude that H p reduces modulo p − pv ( E ) to ker(Fr ) on the reduction of E .Combining this with the integrality of Θ p (from Lemma 6.10), we have h ( x ) = Θ p ( E, P, H p ) ≡ Θ p ( E, P, ker(Fr )) (mod p − pv ( E ) ) . This is zero by Lemma 6.11, so | h ( x ) | ≤ | p − pv ( E ) | = p pv ( E ) − ≤ p pr − as desired. Proposition 6.13 . —
Let F ∈ H ( U m − , O ( k Σ Np m +1 / q )) and suppose that U p F − αF is classical for some α ∈ K with v ( α ) < λ − . Then F is classical as well.Proof . — Define F n as above. We first show that the sequence F n | V [1 , converges.Note that over V [1 ,
1] we have F n +2 − F n +1 = (cid:18) F + 1 αp ϑ k Q ∗ F n +1 (cid:19) − (cid:18) F + 1 αp ϑ k Q ∗ F n (cid:19) = 1 αp ϑ k Q ∗ ( F n +1 − F n ) . HE HALF-INTEGRAL WEIGHT EIGENCURVE By Lemma 6.12 (with r = 0) we have k F n +2 − F n +1 k V [1 , ≤ p − k | α | k F n +1 − F n k V [1 , . The hypothesis on α ensures that (cid:18) p − k | α | (cid:19) n −→ n −→ ∞ and hence that the sequence has successive differences that tend to zero. As H ( V [1 , , O ( k Σ Np m +1 / q )) is a Banach algebra with respect to k·k V [1 , by Lemma2.1, it follows that the sequence converges. Set G = lim n →∞ F n | V [1 , . Next we apply Kassaei’s gluing lemma (Lemma 2.3 of [ ]) to glue G to F assections of the line bundle O ( ⌊ k Σ Np m +1 / q ⌋ ). So that we are gluing over an affinoid asrequired in the hypotheses of the gluing lemma, we first restrict F to V [ p − /p (1+ p ) , G to this restriction to get a section over the smooth affinoid V [ p − /p (1+ p ) , { V [ p − /p (1+ p ) , , U m − } is an admissible cover of X (4 N p m +1 / q ) an K ,this section glues to F to give a global section.The “auxiliary” approximating sections that are required in the hypotheses ofthis lemma (denoted F n in [ ]) are the F n introduced above. So that the F n liveon affinoids (as in the hypotheses of the gluing lemma) we simply restrict F n to V [ p − /p n (1+ p ) , k F n − F k V [ p − /p n (1+ p ) , → k F n − G k V [1 , → . The second of these is simply the definition of G . As for the first, it is not even clearthat the indicated norms are finite (since the norms are over non-affinoids). To seethat these norms are finite and that the ensuing estimates make sense, we must showthat F has finite norm over V [ p − /p (1+ p ) , F over the affinoids V n = V [ p − /p n (1+ p ) , p − /p n +2 (1+ p ) ]are uniformly bounded for n ≥
1. The key is that the map Q restricts to a map Q : V n −→ V n +1 NICK RAMSEY for each n ≥
1. Since F extends to the affinoid V [ p − /p (1+ p ) , V n are certainly uniformly bounded, say, by M . We have k F k V n ≤ max k F k V n , (cid:13)(cid:13)(cid:13)(cid:13) αp ϑ k Q ∗ F (cid:13)(cid:13)(cid:13)(cid:13) V n ! ≤ max (cid:18) M, p | α | k ϑ k k V n k Q ∗ F k V n (cid:19) ≤ max (cid:18) M, p | α | (cid:16) p p n − p ) − (cid:17) k k Q ∗ F k V n (cid:19) ≤ max (cid:18) M, p − k | α | p kp n − p ) k F k V n − (cid:19) Iterating this, we see that k F k V n does not exceed the maximum ofmax ≤ m ≤ n − M (cid:18) p − k | α | (cid:19) m p k p „ p n − + ··· + p n − m )+1 « ! and (cid:18) p − k | α | (cid:19) n − p k p “ p n − + ··· + p ” k F k V . The sums in the exponents of are geometric and do not exceed 1 / ( p − p ). Moreover,the hypothesis on α ensures that p − k / | α | <
1. Thus we have k F k V n ≤ max (cid:16) M p k p p − p , p k p p − p k F k V (cid:17) , which is independent of n , as desired. This ensures that all of the norms encounteredbelow are indeed finite.From the definition of the F n , we have F n +1 − F = F + 1 αp ϑ k Q ∗ F n − F = F − αp ϑ k Q ∗ F + 1 αp ϑ k Q ∗ F n − F = 1 αp ϑ k Q ∗ ( F n − F ) . Taking supremum norms over the appropriate admissible opens, we see k F n +1 − F k V [ p − /p n +2(1+ p ) , ≤ p | α | k ϑ k k V [ p − /p n +2(1+ p ) , k Q ∗ ( F n − F ) k V [ p − /p n +2(1+ p ) , ≤ p | α | (cid:16) p p n +1(1+ p ) − (cid:17) k k F n − F k V [ p − /p n (1+ p ) , = p − k | α | p kp n +1(1+ p ) k F n − F k V [ p − /p n (1+ p ) , Iterating this we find that k F n − F k V [ p − /p n (1+ p ) , ≤ (cid:18) p − k | α | (cid:19) n − p k p “ p + p + ··· + p n − ” k F − F k V [ p − /p p ) , . Again the sum in the exponent is less than 1 / ( p − p ) for all n , so the hypothesis on α ensures that the above norm tends to zero as n → ∞ , as desiredWe are now ready to prove the main result, which is a mild generalization ofTheorem 6.1 stated at the beginning of this section. Theorem 6.14 . —
Let m be a positive integer, let ψ : ( Z / q p m − Z ) × −→ K × be acharacter, and define κ ( x ) = x λ ψ ( x ) . Let P ( T ) ∈ K [ T ] be a monic polynomial allroots of which have valuation less than λ − . If F ∈ f M † κ (4 N, K ) and P ( U p ) F isclassical, then F is classical as well.Proof . — Pick 0 < r < r m such that F ∈ f M κ (4 N, K, p − r ) and let let F ′ ∈ f M k/ (4 N p m +1 / q , K, p − r ) be the form corresponding to F under the isomorphismof Proposition 6.2. We must show that F ′ is classical in the sense that it an-alytically continues to all of X (4 N p m +1 / q ) an K . Note that P (0) = 0 for sucha polynomial, so by Proposition 6.8, F ′ analytically continues to an element of H ( U m − , O ( k Σ Np m +1 / q )). Now we proceed by induction on the degree d of P . Thecase d = 1 is Proposition 6.13. Suppose the result holds for some degree d ≥ P ( T ) be a polynomial of degree d + 1 as above. We may pass to a finite extensionand write P ( T ) = ( T − α ) · · · ( T − α d +1 ) . The condition that P ( U p ) F ′ is classical implies by the inductive hypothesis that( U p − α d +1 ) F ′ is classical. This implies that F ′ is classical by the case d = 1. Remark 6.15 . — The results of this section likely also follow from the very generalclassicality machinery developed in the recent paper [ ] of Kassaei, though we havenot checked the details.
7. The half-integral weight eigencurve
To construct our eigencurve, we will use the axiomatic version of Coleman andMazur’s Hecke algebra construction, as set up by Buzzard in his paper [ ]. We brieflyrecall some relevant details.Let us for the moment allow W to be any reduced rigid space over K . Let T bea set with a distinguished element φ . Suppose that, for each admissible affinoid open X ⊆ W , we are given a Banach module M X over O ( X ) satisfying a certain technicalhypothesis (called ( Pr ) in [ ]) and a map T −→ End O ( X ) ( M X ) t t X NICK RAMSEY whose image consists of commuting endomorphisms and such that φ X is compactfor each X . Assume that, for admissible affinoids X ⊆ X ⊆ W , we are given acontinuous injective O ( X )-linear map α : M X −→ M X b ⊗ O ( X ) O ( X )that is a “link” in the sense of [ ] and such that ( t X b ⊗ ◦ α = α ◦ t X . Assumemoreover that, if X ⊆ X ⊆ X ⊆ W are admissible affinoids, then α = α ◦ α with the obvious notation. Note that the link condition ensures that the characteristicpower series P X ( T ) of φ X acting on M X is independent of X in the sense that theimage of P X ( T ) under the natural map O ( X )[[ T ]] → O ( X )[[ T ]] is P X ( T ) (see [ ]).Out of this data, Buzzard constructs rigid analytic spaces D and Z , called the eigenvariety and spectral variety , respectively, equipped with canonical maps(12) D −→ Z −→ W . The points of D parameterize systems of eigenvalues of T acting on the { M X } forwhich the eigenvalue of φ is nonzero, in a sense that will be made precise in Lemma7.3, while the image of such a point in Z simply records the inverse of the φ eigenvalueand a point of W . If W is equidimensional of dimension d , then the same is true ofboth of the spaces D and Z .As the details of this construction will be required in the next section, we recallthem here. The following is Theorem 4.6 of [ ], and is the deepest part of the con-struction. Theorem 7.1 . —
Let R be a reduced affinoid algebra over K , let P ( T ) be a Fredholmseries over R , and let Z ⊂ Sp( R ) × A denote the hypersurface cut out by P ( T ) equipped with the projection π : Z −→ Sp( R ) . Define C ( Z ) to be the collection ofadmissible affinoid opens Y in Z such that – Y ′ = π ( Y ) is an admissible affinoid open in Sp( R ) , – π : Y −→ Y ′ is finite, and – there exists e ∈ O ( π − ( Y ′ )) such that e = e and Y is the zero locus of e .Then C ( Z ) is an admissible cover of Z . We will generally take Y ′ to be connected in what follows. This is not a seriousrestriction, since Y is the disjoint union of the parts lying over the various connectedcomponents of Y ′ . We also remark that the third of the above conditions follows fromthe first two (this is observed in [ ] where references to the proof are supplied).To construct D , first fix an admissible affinoid open X ⊆ W . Let Z X denote thezero locus of P X ( T ) = det(1 − φ X T | M X ) in X × A and let π : Z X → X denotethe projection onto the first factor. Let Y ∈ C ( Z X ) and let Y ′ = π ( Y ) as aboveand assume that Y ′ is connected. We wish to associate to Y a polynomial factor of P Y ′ ( T ) = det(1 − ( φ X b ⊗ T | M X b ⊗ O ( X ) O ( Y ′ )). Since the algebra O ( Y ) is a finite andlocally free module over O ( Y ′ ), we may consider the characteristic polynomial Q ′ of T ∈ O ( Y ). Since T is a root of its characteristic polynomial, we have a map(13) O ( Y ′ )[ T ] / ( Q ′ ( T )) −→ O ( Y ) . HE HALF-INTEGRAL WEIGHT EIGENCURVE It is shown in Section 5 of [ ] that this map is surjective and therefore an isomorphismsince both sides are locally free of the same rank.Now since the natural map O ( Y ′ )[ T ] / ( Q ′ ( T )) −→ O ( Y ′ ) / ( Q ′ ( T ))is an isomorphism, it follows that Q ′ ( T ) divides P Y ′ ( T ) in O ( Y ′ ) . If a isthe constant term of Q ′ ( T ), then this divisibility implies that a is a unit. We set Q ( T ) = a − Q ′ ( T ). The spectral theory of compact operators on Banach modules (seeTheorem 3.3 of [ ]) furnishes a unique decomposition M X b ⊗ O ( X ) O ( Y ′ ) ∼ = N ⊕ F into closed φ -invariant O ( Y ′ )-submodules such that Q ∗ ( φ ) is zero on N and invertibleon F . Moreover, N is projective of rank equal to the degree of Q and the characteristicpower series of φ on N is Q ( T ). The projector M X b ⊗ O ( X ) O ( Y ′ ) −→ N is in the closureof O ( Y ′ )[ φ ], so N is stable under all of the endomorphisms associated to elementsof T . Let T ( Y ) denote the O ( Y ′ )-subalgebra of End O ( Y ′ ) ( N ) generated by theseendomorphisms. Then T ( Y ) is finite over O ( Y ′ ) and hence affinoid, so we we may set D Y = Sp( T ( Y )). Because the leading coefficient of Q (= the constant term of Q ∗ ) isa unit there is an isomorphism O ( Y ′ )[ T ] / ( Q ( T )) −→ O ( Y ′ )[ S ] / ( Q ∗ ( S )) T S − Thus we obtain a canonical map D Y −→ Y , namely, the one corresponding to themap O ( Y ) ∼ = O ( Y ′ )[ T ] / ( Q ( T )) ∼ = O ( Y ′ )[ S ] / ( Q ∗ ( S )) S φ −→ T ( Y )of affinoid algebras.For general Y ∈ C ( Z X ), we define D Y be the disjoint union of the affinoids definedabove from the various connected components of Y ′ . We then glue the affinoids D Y for Y ∈ C ( Z X ) to obtain a rigid space D X equipped with maps D X −→ Z X −→ X. Finally, we vary X and glue the desired spaces and maps above to obtain the spacesand maps in (12). This final step is where the links α ij above come into play. Werefer the reader to [ ] for further details. Definition 7.2 . — Let L be a complete discretely-valued extension of K . An L -valued system of eigenvalues of T acting on { M X } X is a pair ( κ, γ ) consisting of a mapof sets γ : T −→ L and a point κ ∈ W ( L ) such that there exists an affinoid X ⊆ W containing κ and a nonzero element m ∈ M X b ⊗ O ( X ) ,κ L such that ( t X b ⊗ m = γ ( t ) m for all t ∈ T . Such a system of eigenvalues is called φ -finite if γ ( φ ) = 0.Let x be an L -valued point of D . Then x lies over a point in κ x ∈ W ( L ) which liesin X for some affinoid X , and x moreover lies in D Y ( L ) for some Y ∈ C ( Z X ). Thusto x and the choice of X and Y corresponds a map T ( Y ) −→ L , and in particulara map of sets λ x : T −→ L . In [ ], Buzzard proves the following characterization ofthe points of D . NICK RAMSEY
Lemma 7.3 . —
The correspondence x ( κ x , λ x ) is a well-defined bijective corre-spondence between L -valued points of D and φ -finite L -valued systems of eigenvaluesof T acting on the { M X } . In our case, we let W be weight space over Q p as in Section 2.4, and let T be theset of symbols (cid:26) { T ℓ } ℓ Np ∪ { U ℓ } ℓ | Np ∪ {h d i N } d ∈ ( Z / N Z ) × p = 2 { T ℓ } ℓ N ∪ { U ℓ } ℓ | N ∪ {h d i N } d ∈ ( Z /N Z ) × p = 2For an admissible affinoid open X ⊆ W we let M X = f M X (4 N, Q p , p − r n )where n is the smallest positive integer such that X ⊆ W n . This module is a directsummand of the Q p -Banach space ( H ( X (4 N p ) an ≥ p − rn , O (Σ Np )) b ⊗ Q p O ( X ) p = 2 H ( X (4 N ) an ≥ − rn , O (Σ N )) b ⊗ Q p O ( X ) p = 2and therefore satisfies property ( Pr ) since this latter space is potentially orthonor-malizable in the terminology of [ ] by the discussion in Section 1 of [ ]. We take themap T −→ End O ( X ) ( M X )to be the one sending each symbol to the endomorphism by that name defined inSection 5.Let X ⊆ X ⊆ W be admissible affinoids and let n i be the smallest positive integerwith X i ⊆ W n i . Then n ≤ n so that r n ≤ r n and we have an inclusion f M X (4 N, Q p , p − r n ) −→ f M X (4 N, Q p , p − r n )given by restriction. We define the required continuous injection α via the diagram f M X (4 N, Q p , p − r n ) / / α * * VVVVVVVVVVVVVVVVV f M X (4 N, Q p , p − r n ) f M X (4 N, Q p , p − r n ) b ⊗ O ( X ) O ( X ) ∼ O O and note that the required compatibility condition is satisfied. To see that these mapsare links, choose numbers r n = s ≥ s > s > · · · > s k − ≥ s k = r n with the property that p s i +1 > s i for all i . Then the map α factors as the compo-sition the maps f M X (4 N, Q p , p − s i ) −→ f M X (4 N, Q p , p − s i +1 )for 0 ≤ i ≤ k − f M X (4 N, Q p , p − s k − ) −→ f M X (4 N, Q p , p − s k ) b ⊗ O ( X ) O ( X ) . Each of these maps is easily seen to be a primitive link from the construction of U p . HE HALF-INTEGRAL WEIGHT EIGENCURVE The result is that we obtain rigid analytic spaces e D and e Z which we call the half-integral weight eigencurve and the half-integral weight spectral curve , respectively, aswell as canonical maps e D −→ e Z −→ W . As usual, the tilde serves to distinguish these spaces from their integral weight coun-terparts first constructed in level 1 by Coleman and Mazur and later constructed forgeneral level by Buzzard in [ ].If instead of using the full spaces of forms we use only the cuspidal subspaces ev-erywhere, then we obtain cuspidal versions of all of the above spaces, which we willdelineate with a superscript 0. Thus we have e D and e Z with the usual maps, andthe points of these spaces parameterize systems of eigenvalues of the Hecke opera-tors acting on the spaces of cusp forms by Lemma 7.3. We remark that there is acommutative diagram e D / / (cid:15) (cid:15) e D (cid:15) (cid:15) e Z AAAAAAAA / / e Z (cid:127) (cid:127) ~~~~~~~~ W where the horizontal maps are injections that identify the cuspidal spaces on the leftwith unions of irreducible components of the spaces on the right. This is an exercisein the linear algebra that goes into the construction of these eigenvarieties and basicfacts about irreducible components of rigid spaces found in [ ], and is left to thereader.For κ ∈ W ( K ), let e D κ and e D κ denote the fibers e D and e D over κ . The followingtheorem summarizes the basic properties of these eigencurves. Theorem 7.4 . —
Let κ ∈ W ( K ) . For a complete extension L/K , the correspon-dence x λ x is a bijection between the L -valued points of the fiber e D κ ( L ) and theset of finite-slope systems of eigenvalues of the Hecke operators and tame diamond op-erators occurring on the space f M † κ (4 N, L ) of overconvergent forms of weight κ definedover L . The same statement holds with e D replaced by e D and f M † κ (4 N, L ) replaced by e S † κ (4 N, L ) .Proof . — We prove the statement for the full space of forms. The proof for cuspidalforms is identical. Fix κ ∈ W ( K ). Once we establish that the L -valued systems ofeigenvalues of the form ( κ, γ ) occurring on the { M X } X as defined above are exactlythe systems of eigenvalues the Hecke and tame diamond operators that occur on f M † κ (4 N, L ), the result is simply Lemma 7.3 “collated by weight.” To see this onesimply notes that, for any f ∈ f M † κ (4 N, L ), we have both f ∈ f M κ (4 N, L, p − r n ) and κ ∈ W n for n sufficiently large. In particular, if f is a nonzero eigenform for the Heckeand tame diamond operators, then the system of eigenvalues associated to f occursin the module M W n for n sufficiently large. NICK RAMSEY
We remark that the classicality result of Section 6 has the expected consequencethat the collection of points of e D corresponding to systems of eigenvalues occurringon classical forms is Zariski-dense in e D . This result is contained in the forthcomingpaper [ ]. Appendix AProperties of the stack X ( M p, p ) over Z ( p ) by Brian ConradIn this appendix, we establish some geometric properties concerning the cuspidallocus in compactified moduli spaces for level structures on elliptic curves. We areespecially interested in the case of non-´etale p -level structures in characteristic p ,so it is not sufficient to cite the work in [ ] (which requires ´etale level structuresin the treatment of moduli problems for generalized elliptic curves) or [ ] (whichworks with Drinfeld structures over arbitrary base schemes but avoids non-smoothgeneralized elliptic curves). The viewpoints of these works were synthesized in thestudy of moduli stacks for Drinfeld structures on generalized elliptic curves in [ ], andwe will use that as our foundation in what follows.Motivated by needs in the main text, for a prime p and an integer M ≥ p we wish to consider the moduli stack X ( M p r , p e ) over Z ( p ) that classifiestriples ( E, P, C ) where E is a generalized elliptic curve over a Z ( p ) -scheme S , P ∈ E sm ( S ) is a Drinfeld Z /M p r Z -structure on E sm , and C ⊆ E sm is a cyclic subgroupwith order p e such that some reasonable ampleness and compatibility properties for P and C are satisfied. (See Definition A.1 for a precise formulation of these additionalproperties.) The relevant case for applications to p -adic modular forms with half-integer weight is e = 2, but unfortunately such moduli stacks were only considered in[ ] when either r ≥ e or r = 0. (This is sufficient for applications to Hecke operators,and avoids some complications.) We now need to allow 1 ≤ r < e , and the purposeof this appendix is to explain how to include such r and to record some consequencesconcerning the cusps in these cases. The consequence that is relevant the main textis Theorem A.11. To carry out the proofs in this appendix we simply have to adaptsome proofs in [ ] rather than develop any essentially new ideas. For the convenienceof the reader we will usually use the single paper [ ] as a reference, though it mustbe stressed that many of the key notions were first introduced in the earlier work [ ]and [ ]. In the context of subgroups of the smooth locus on a generalized ellipticcurve, we will refer to a Drinfeld Z /N Z -structure (resp. a Drinfeld Z /N Z -basis) as a Z /N Z -structure (resp. Z /N Z -basis) unless some confusion is possible. A.1. Definitions. —
We refer the reader to [ , § f : E → S over a scheme S and of the closed subscheme S ∞ ⊆ S that is the “locus of degenerate fibers” for such an object. (It would be more accu-rate to write S ∞ ,f , but the abuse of notation should not cause confusion.) Roughlyspeaking, E → S is a proper flat family of geometrically connected and semistablecurves of arithmetic genus 1 that are either smooth or are so-called N´eron polygons, HE HALF-INTEGRAL WEIGHT EIGENCURVE and the relative smooth locus E sm is endowed with a commutative S -group struc-ture that extends (necessarily uniquely) to an action on E such that whenever E s is a polygon the action of E sm s on E s is via rotations of the polygon. Also, S ∞ isa scheme structure on the set of s ∈ S such that E s is not smooth. The definitionof the degeneracy locus S ∞ (as given in [ , 2.1.8]) makes sense for any proper flatand finitely presented map C → S with fibers of pure dimension 1, and if S ′ is any S -scheme and then there is an inclusion S ′ × S S ∞ ⊆ S ′∞ as closed subschemes of S ′ (with S ′∞ corresponding to the S ′ -curve C × S S ′ ), but this inclusion can fail to be anequality even when each geometric fiber C s is smooth of genus 1 or a N´eron polygon[ , Ex. 2.1.11]. Fortunately, if C admits a structure of generalized elliptic curve over S then this inclusion is always an equality [ , 2.1.12], so the degeneracy locus makessense on moduli stacks for generalized elliptic curves (where it defines the cusps).We wish to study moduli spaces for generalized elliptic curves E /S equipped withcertain ample level structures defined by subgroups of E sm . Of particular interestare those subgroup schemes G ⊆ E sm that are not only finite locally free over thebase with some constant order n but are even cyclic in the sense that fppf -locallyon the base we can write G = h P i := P j ∈ Z /n Z [ jP ] in E sm as Cartier divisors forsome n -torsion point P of E sm . By [ , 2.3.5], if P and P ′ are two such points for thesame G then for any d | n the points ( n/d ) P and ( n/d ) P ′ are Z / ( n/d ) Z -generators ofthe same S -subgroup of G , so by descent this naturally defines a cyclic S -subgroup G d ⊆ G of order d even if P does not exist over the given base scheme S . We call G d the standard cyclic subgroup of G with order d . For example, if d = d ′ d ′′ with d ′ , d ′′ ≥ d ′ , d ′′ ) = 1 then G d ′ × G d ′′ ≃ G d via the group law on G . Definition A.1 . — Let
N, n ≥ ( N ) -structure on a generalized elliptic curve E /S is an S -ample Z /N Z -structure on E sm , which is to say an N -torsion point P ∈ E sm ( S ) such that therelative effective Cartier divisor D = P j ∈ Z /N Z [ jP ] on E sm is an S -subgroup and D s is ample on E s for all s ∈ S .A Γ ( N, n ) -structure on E /S is a pair ( P, C ) where P is a Z /N Z -structure on E sm and C ⊆ E sm is a cyclic S -subgroup with order n such that the relative effectiveCartier divisor D = P j ∈ Z /N Z ( jP + C ) on E is S -ample and there is an equality ofclosed subschemes(14) X j ∈ Z /p ep Z ( j ( N/p e p ) P + C p ep ) = E sm [ p e p ]for all primes p | gcd( N, n ), with e p = ord p (gcd( N, n )) ≥ Example A.2 . — Obviously a Γ ( N, ( N )-structure. If N = 1 then we refer to Γ (1)-structures as Γ(1)-structures, and such astructure on a generalized elliptic curve E /S must be the identity section. Thus, bythe ampleness requirement, the geometric fibers E s must be irreducible. Hence, themoduli stack M Γ(1) of Γ(1)-structures on generalized elliptic curves classifies general-ized elliptic curves with geometrically irreducible fibers. NICK RAMSEY
In [ , 2.4.3] the notion of Γ ( N, n )-structure is defined as above, but with theadditional requirement that ord p ( n ) ≤ ord p ( N ) for all primes p | gcd( N, n ). Thisrequirement always holds when n = 1 and whenever it holds the standard subgroup C p ep in (14) is the p -part of C , but it turns out to be unnecessary for the proofs ofthe basic properties of Γ ( N, n )-structures and their moduli, as we shall explain in § A.2. For example, the proof of [ , 2.4.4] carries over to show that we can replace(14) with the requirement that X j ∈ Z /d Z ( j ( N/d ) P + C d ) = E sm [ d ]in E for d = gcd( N, n ). Another basic property that carries over to the general caseis that if (
P, C ) is a Γ ( N, n )-structure on E then the relative effective Cartier divisor P j ∈ Z /N Z ( jP + C ) on E sm is an S -subgroup; the proof is given in [ , 2.4.5] underthe assumption ord p ( n ) ≤ ord p ( N ) for every prime p | gcd( N, n ), but the argumentworks in general once it is observed that after making an fppf base change to acquire a Z /n Z -generator Q of C we can use symmetry in P and Q in the rest of the argumentso as to reduce to the case considered in [ ]. A.2. Moduli stacks. —
As in [ , 2.4.6], for N, n ≥ M Γ ( N,n ) to classify Γ ( N, n )-structures on generalized elliptic curves over arbitraryschemes, and we let M ∞ Γ ( N,n ) ֒ → M Γ ( N,n ) denote the closed substack given by thedegeneracy locus for the universal generalized elliptic curve. The arguments in [ , § § p ( n ) ≤ ord p ( N )for all primes p | gcd( N, n )) to prove the following result.
Theorem A.3 . —
The stack M Γ ( N,n ) is an Artin stack that is proper over Z . Itis smooth over Z [1 /N n ] , and it is Deligne–Mumford away from the open and closedsubstack in M ∞ Γ ( N,n ) classifying degenerate triples ( E, P, C ) in positive characteristics p such that the p -part of each geometric fiber of C is non-´etale and disconnected. The proof of [ , 3.3.4] does not use the condition ord p ( n ) ≤ ord p ( N ) for all primes p | gcd( N, n ) (although this condition is mentioned in the proof), so that argumentgives:
Lemma A.4 . —
The open substack M ( N,n ) = M Γ ( N,n ) − M ∞ Γ ( N,n ) classifying el-liptic curves endowed with a Γ ( N, n ) -structure is regular and Z -flat with pure relativedimension . We are interested in the structure of M Γ ( N,n ) around its cuspidal substack, espe-cially determining whether it is regular or a scheme near such points. Our analysis of M ∞ Γ ( N,n ) rests on the following theorem. Theorem A.5 . —
The map M Γ ( N,n ) → Spec( Z ) is flat and Cohen-Macaulay withpure relative dimension .Proof . — By Lemma A.4, we just have to work along the cusps. Also, it sufficesto check the result after localization at each prime p , and if p ∤ gcd( N, n ) or 1 ≤ HE HALF-INTEGRAL WEIGHT EIGENCURVE ord p ( n ) ≤ ord p ( N ) then [ , 3.3.1] gives the result over Z ( p ) . It therefore remainsto study the cusps in positive characteristic p when 1 ≤ ord p ( N ) < ord p ( n ). Asin the cases treated in [ ], the key is to study the deformation theory of a relatedlevel structure on generalized elliptic curves called a e Γ ( N, n ) -structure : this is apair ( P, Q ) where P is a Z /N Z -structure on the smooth locus and Q is a Z /n Z -structure on the smooth locus such that ( P, h Q i ) is a Γ ( N, n )-structure. The samedefinition is given in [ , 3.3.2] with the unnecessary restriction ord p ( n ) ≤ ord p ( N ) forall primes p | gcd( N, n ), and the argument in [ ] immediately following that definitionworks without such a restriction to show that the moduli stack M e Γ ( N,n ) of e Γ ( N, n )-structures is a Deligne–Mumford stack over Z that is a finite flat cover of the properArtin stack M Γ ( N,n ) .By the Deligne–Mumford property, any e Γ ( N, n )-structure x = ( E , P , Q ) overan algebraically closed field k admits a universal deformation ring. Since M e Γ ( N,n ) is a finite flat cover of M Γ ( N,n ) , as in the proof of [ , 3.3.1] it suffices to assumechar( k ) = p > x as a finite flat extensionof W ( k )[[ x ]] when E is a standard polygon, n = p e , and N = M p r with p ∤ M and e, r ≥
1. The case e ≤ r was settled in [ ], and we will adapt that argument tohandle the case 1 ≤ r < e . By the ampleness condition at least one of M P or Q generates the p -part of the component group of E sm0 , and moreover { M P , p e − r Q } is a Drinfeld Z /p r Z -basis of E sm0 [ p r ]. We shall break up the problem into three cases,and it is only in Case 3 that we will meet a situation essentially different from thatencountered in the proof for 1 ≤ e ≤ r in [ ]. Case 1 : We first assume that
M P generates the p -part of the component group,so by the Drinfeld Z /p r Z -basis hypothesis this point is a basis of E sm0 ( k )[ p ∞ ] over Z /p r Z (as we are in characteristic p and E is a polygon). Hence, Q = jM P for aunique j ∈ Z /p r Z (so p e − r Q = p e − r jM P ). Since n is a p -power, it also follows that h P i is ample. In particular, ( E , P ) is a Γ ( N )-structure. Thus, the formation of aninfinitesimal deformation ( E, P, Q ) of ( E , P , Q ) can be given in three steps: firstgive an infinitesimal deformation ( E, P ) of ( E , P ) as a Γ ( N )-structure, then givea Drinfeld Z /p r Z -basis ( M P, Q ′ ) of E sm [ p r ] with Q ′ deforming p e − r Q , and finallyspecify a p e − r th root Q of Q ′ lifting Q = jM P . The one aspect of this descriptionthat merits some explanation is to justify that such a p e − r th root Q of Q ′ must be a Z /p e Z -structure on E sm . The point Q is clearly killed by p e , so the Cartier divisor D = P j ∈ Z /p e Z [ jQ ] in E sm makes sense and we have to check that it is automaticallya subgroup scheme.The identification ( E sm0 ) [ p t ] = µ p t uniquely lifts to an isomorphism ( E sm ) [ p t ] ≃ µ p t for any t ≥
0. In particular, if p ν is the order of the p -part of the cyclic componentgroup of E sm0 (with ν ≥ r ) then E sm [ p e ] is an extension of Z /p j Z by µ p e where j = min( ν, e ). The image of h Q i in the component group can be uniquely identifiedwith Z /p i Z (for some i ≤ j ) such that Q
1, and this Z /p i Z has preimage G in E sm [ p e ] that is a p e -torsion commutative extension of Z /p i Z by µ p e with 0 ≤ i ≤ e .Since Q is a point of G over the (artin local) base, it follows from [ , 2.3.3] that Q is a Z /p e Z -structure on E sm if and only if the point p i Q in µ p e − i is a Z /p e − i Z -generatorof µ p e − i . The case i = e is therefore settled, so we can assume i < e (i.e., h Q i NICK RAMSEY is not ´etale, or equivalently p e − Q = 0). By hypothesis p e − r Q = Q ′ is a Z /p r Z -structure on E sm with 1 ≤ r < e , so p e − Q = p r − Q ′ is a Z /p Z -structure on E sm .This Z /p Z -structure must generate the subgroup µ p ⊆ E sm [ p e ] since p e − Q lies in( E sm ) (as p e − Q = 0). Hence, Q ′′ = p i Q is a point of µ p e − i such that p e − i − Q ′′ is a Z /p Z -generator of µ p . Since Z /m Z -generators of µ m are simply roots of thecyclotomic polynomial Φ m [ , 1.12.9], our problem is reduced to the assertion thatif s is a positive integer (such as e − i ) then an element ζ in a ring is a root of thecyclotomic polynomial Φ p s if ζ p s − is a root of Φ p . This assertion is obvious sinceΦ p s ( T ) = Φ p ( T p s − ), and so our description of the infinitesimal deformation theoryof ( E , P , Q ) is justified.The torsion subgroup E sm [ p r ] is uniquely an extension of Z /p r Z by µ p r deformingthe canonical such description for E sm0 [ p r ], so the condition on Q ′ is that it has theform ζ + p e − r jM P for a point ζ of the scheme of generators µ × p r of µ p r = ( E sm ) [ p r ].Thus, to give Q is to specify a p e − r th root of ζ in E sm deforming the identity, whichis to say a point of µ × p e . It is shown in the proof of [ , 3.3.1] that the universaldeformation ring A for ( E , P ) is finite flat over W ( k )[[ x ]], and the specification of ζ amounts to giving a root of the cyclotomic polynomial Φ p e , so the case when M P generates the p -part of the component group of E sm0 is settled (with deformation ring A [ T ] / (Φ p e ( T ))). Case 2 : Next assume that Q generates the p -part of the component group andthat h Q i is ´etale (i.e., Q ∈ E sm0 ( k ) has order p e ). The point Q must generate E sm0 ( k )[ p ∞ ] over Z /p e Z , and the ´etale hypothesis ensures that Q is a Z /p e Z -basis of E sm0 ( k )[ p ∞ ]. Thus, M P = p e − r jQ for some (unique) j ∈ Z /p r Z . By replacing P with P − M − p e − r jQ for any infinitesimal deformation ( E, P, Q ) of ( E , P , Q ) wecan assume that the p -part of P vanishes. The p -part of P must therefore be a pointof µ × p r . The Z /M Z -part of P together with Q constitutes a Γ ( M p e )-structure on E (in particular, the ampleness condition holds), and this is an ´etale level structuresince the cyclic subgroup h Q i in E sm0 is ´etale. Hence, the infinitesimal deformationfunctor of ( E , P , Q ) is pro-represented by µ × p r over the deformation ring of an ´etaleΓ ( M p e )-structure. For any R ≥
1, deformation rings for ´etale Γ ( R )-structures onpolygons over k have the form W ( k )[[ x ]] (as is explained near the end of the proof of[ , 3.3.1], using [ , II, 1.17]), so not only are we done but in this case the deformationring for ( E , P , Q ) is the ring W ( k )[[ x ]][ T ] / (Φ p r ( T )) that is visibly regular. Case 3 : Finally, assume Q generates the p -part of the component group butthat h Q i is not ´etale (i.e., Q ∈ E sm0 ( k ) has order strictly less than p e ), so p e − r Q ∈ E sm0 ( k ) has order strictly dividing p r . Since { M P , p e − r Q } is a Drinfeld Z /p r Z -basisof E sm0 [ p r ], the point M P must be a Z /p r Z -basis for E sm0 ( k )[ p r ]. Hence, if we write P = P ′ + P ′′ corresponding to the decomposition Z /N Z = ( Z /M Z ) × ( Z /p r Z ) then P ′′ has order exactly p r in E sm0 ( k ). We use P ′′ to identify E sm0 ( k )[ p r ] with Z /p r Z .It follows that if we make the analogous canonical decomposition P = P ′ + P ′′ foran infinitesimal deformation ( E, P, Q ) of ( E , P , Q ) then the p -part P ′′ deforms P ′′ and generates an ´etale subgroup of E sm with order p r . Thus, P ′ and Q together con-stitute a (non-´etale) Γ ( M p e )-structure on E (in particular, the ampleness conditionholds), and the data of P ′′ amounts to a section over 1 ∈ Z /p r Z with respect to the HE HALF-INTEGRAL WEIGHT EIGENCURVE unique quotient map E sm [ p r ] ։ Z /p r Z lifting the quotient map E sm0 [ p r ] ։ Z /p r Z defined by P ′′ . Since the specification of a Z /N Z -structure on E sm is the “same” asthe specification of a pair consisting of Z /M Z -structure and a Z /p r Z -structure [ ,1.7.3], we conclude that the universal deformation ring of ( E , P , Q ) classifies thefiber over 1 ∈ Z /p r Z in the connected-´etale sequence for the p r -torsion in infinitesimaldeformations of the underlying Γ ( M p e )-structure ( E , P ′ , Q ). Universal deforma-tion rings for Γ ( M p e )-structures over k are finite flat over W ( k )[[ x ]] (by the proof of[ , 3.3.1]), so we are therefore done. Corollary A.6 . —
The closed substack M ∞ Γ ( N,n ) ֒ → M Γ ( N,n ) is a relative effectiveCartier divisor over Z , and it has a reduced generic fiber over Q .Proof . — The reducedness over Q is shown in [ , 4.3.2], and the proof works withoutrestriction on gcd( N, n ). Likewise, the proof that M ∞ Γ ( N,n ) is a Z -flat Cartier divisoris part of [ , 4.1.1(1)] in case ord p ( n ) ≤ ord p ( N ) for all primes p | gcd( N, n ), butby using the above proof of Theorem A.5 we see that the method of proof works ingeneral.Using Lemma A.4, Theorem A.5, and Corollary A.6, Serre’s normality criterion canbe used to prove normality for M Γ ( N,n ) in general. (This is proved in [ , 4.1.4] subjectto the restrictions on gcd( N, n ) in the definition of Γ ( N, n )-structures in [ ], but theargument works in general by using the results that are stated above without any suchrestriction on gcd( N, n ).) However, the proof of regularity encounters complicationsat points of a certain locus of cusps in bad characteristics. This problematic locus isdefined as follows.
Definition A.7 . — Let Z Γ ( N,n ) ֒ → M ∞ Γ ( N,n ) be the 0-dimensional closed substackwith reduced structure consisting of geometric points ( E , P , C ) in characteristics p | gcd( N, n ) such that 1 ≤ ord p ( N ) < ord p ( n ), C is not ´etale, and ( N/p ord p ( N ) ) P does not generate the p -part of the component group of E sm0 .Note that if ord p ( n ) ≤ ord p ( N ) for all primes p | gcd( N, n ) (the situation consideredin [ ]) then Z Γ ( N,n ) is empty; this includes the case of Γ ( N )-structures for any N (take n = 1). In all other cases it is non-empty. The geometric points of Z Γ ( N,n ) correspond to precisely the points in Case 3 in the proof of Theorem A.5. Themethod in [ ] for analyzing regularity along the cusps assumes Z Γ ( N,n ) is empty, andby combining it with the modified arguments in the proof of Theorem A.5 (especiallythe regularity observation in Case 2) we obtain the following consequence. Theorem A.8 . —
The stack M Γ ( N,n ) is regular outside of the closed substack Z Γ ( N,n ) ⊆ M ∞ Γ ( N,n ) . A.3. Applications. —
Before we apply the preceding results, we record a usefullemma.
Lemma A.9 . —
Let S be a scheme and let X be an Artin stack over S . Assume X is S -separated. The locus of geometric points of X with trivial automorphism group NICK RAMSEY scheme is an open substack U ⊆ X that is an algebraic space. This algebraic space isa scheme if X is quasi-finite over a separated S -scheme.Proof . — The first part is [ , 2.2.5(2)], and the second part follows from the generalfact that an algebraic space that is quasi-finite and separated over a scheme is ascheme [ , Thm. A.2].In the setting of Lemma A.9, if X is quasi-finite over a separated S -scheme then wecall U the maximal open subscheme of X . The case of interest to us is X = M Γ ( N,n ) /S over any scheme S . This is quasi-finite over the S -proper stack M Γ(1) /S via fibralcontraction away from the identity component, and M Γ(1) /S is quasi-finite over P S via the j -invariant, so X is quasi-finite over the separated S -scheme P S .We wish to prove results concerning when certain components of M ∞ Γ ( N,n ) lie in themaximal open subscheme of M Γ ( N,n ) . To this end, we first record a general lemma. Lemma A.10 . —
Let Y be an irreducible Artin stack over F p , and let C be a finitelocally free commutative Y -group that is cyclic with order p e . If C has a multiplicativegeometric fiber over Y then all of its geometric fibers are connected. The abstract notion of cyclicity (with no ambient smooth curve group) is developedin [ , 1.5, 1.9, 1.10] over arbitrary base schemes, and the theory carries over whenthe base is an Artin stack. We will only need the lemma for situations that arisewithin torsion on generalized elliptic curves (over Artin stacks). Proof . — We can assume e ≥
1, and we may replace C with its standard subgroup C p of order p because it is obvious by group theory that a cyclic group scheme C of p -power order over an algebraically closed field of characteristic p is ´etale if and onlyif its standard subgroup of order p is ´etale. Hence, we can assume that C has order p . Our problem is therefore to rule out the existence of ´etale fibers. By openness ofthe locus of ´etale fibers and irreducibility of Y , if there is an ´etale fiber then there isa Zariski-dense open U ⊆ Y over which C has ´etale fibers. In particular, there is somegeometric point u of U that specializes to the geometric point y ∈ Y where we assumethe fiber is multiplicative, so after pullback to a suitable valuation ring we get an´etale group of order p in characteristic p specializing to a multiplicative one. Passingto Cartier duals gives a multiplicative group of order p having an ´etale specialization,and this is impossible since multiplicative groups of order p in characteristic p are not´etale. Theorem A.11 . —
Let p be a prime, and choose a positive integer M not divisibleby p such that M > . Also fix integers e, r ≥ . If e = 0 or r = 0 then assume M = 4 . Let x = ( E , P , C ) be a geometric point on the special fiber of the cuspidalsubstack in the proper Artin stack X = M Γ ( Mp r ,p e ) / Z ( p ) over Z ( p ) , and assume that C is ´etale.Let Y be the irreducible component of x in X F p . For every geometric cusp x =( E , P , C ) on Y the group C is ´etale and x lies in the maximal open subscheme of X . Moreover, if x ∈ X Q is a cusp specializing into Y then the Zariski closure D of x in X lies in the maximal open subscheme and D is Cartier in X . HE HALF-INTEGRAL WEIGHT EIGENCURVE The case e = 2 is required in the main text. It is necessary to avoid the cases M ≤ M, r ) = (4 ,
0) because in these cases there are cusps x in characteristic p as in the theorem such that x admits nontrivial automorphisms (and so x cannotlie in the maximal open subscheme of X ). Proof . — We first check that the ´etale assumption at x is inherited by all geometriccusps x ∈ Y . Let ( E , P , C ) be the pullback to Y of the universal family over X . Thegroup C is cyclic of order p e with e ≥
0, so applying Lemma A.10 to its Cartierdual gives the result (since at a cusp a connected subgroup of p -power order must bemultiplicative).Now we can rename x as x without loss of generality, so we have to check that x lies in the maximal open subscheme of X and that if x ∈ X Q is a geometriccusp specializing to x then the Zariski closure of x in X is Cartier. But the ´etalehypothesis on C ensures that x is not in the closed substack Z Γ ( Mp r ,p e ) / Z ( p ) , soby Theorem A.8 the stack X is regular at x . Hence, since X is Z ( p ) -flat with purerelative dimension 1 (by Theorem A.5), the desired properties of D at the end of thetheorem hold once we know that x is in the maximal open subscheme of X , which isto say that its automorphism group scheme G is trivial. To verify this triviality wewill make essential use of the property that C is ´etale. Let k be the algebraicallyclosed field over which x lives. Since E is d -gon over k for some d ≥ G is aclosed subgroup of the automorphism group µ d ⋊ h inv i of the d -gon. Since C is´etale with order p e in characteristic p it follows that C maps isomorphically into the p -part of the component group of E sm0 = G m × ( Z /d Z ). (In particular, p e | d .) If R is an artin local k -algebra with residue field k then any choice of generator Q of C must be carried to another generator of C by any g ∈ G ( R ) since C ( R ) → C ( k )is a bijection. But µ d ( R ) acts on ( E ) R in a manner that preserves the componentsof the smooth locus, and C meets each component of E sm0 in at most one point.Hence, G ∩ µ d acts as automorphisms of the Γ ( M p e )-structure on E defined by p r P and Q . Since M p e > M p e = 4 (due to the cases we are avoiding), suchan ample level structure on a d -gon has trivial automorphism group scheme. Thisshows that G ∩ µ d is trivial, so G injects into the group Z / Z of automorphisms ofthe identity component G m of E sm0 . Hence, the contraction operation on E awayfrom h P i is faithful on G since contraction does not affect the identity component.It follows that G is a subgroup of the automorphism group of the Γ ( M p r )-structureobtained by contraction away from h P i . But M p r
6∈ { , , } since we assume M >
M, r ) = (4 , ( M p r )-structures on polygons have trivial automorphismfunctor. Thus, G = { } as desired.We remark that, over the base Z ( p ) , the results of § ] concerning the prop-erties of the stack X ( N, n ) carry over if p ∤ n . In effect, the hypothesis on ord p ( n )imposed in [ ] only intervenes in the proofs when n is not invertible on the base. NICK RAMSEY
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