aa r X i v : . [ m a t h . N T ] J un THE OVERCONVERGENT SHIMURA LIFTING by Nick Ramsey
Abstract . —
We construct a rigid-analytic map from the the author’s half-integralweight cuspidal eigencurve (see [ ]) to its integral weight counterpart that interpo-lates the classical Shimura lifting. Contents
1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Modular forms of half-integral weight. . . . . . . . . . . . . . . . . . . . . . . . . . . . 23. The eigencurves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54. Some density results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95. Interpolation of the Shimura lifting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116. Properties of Sh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1. Introduction
In [ ], Shimura discovered the following remarkable connection between holomor-phic eigenforms of half-integral weight and their integral weight counterparts. Theorem 1.1 . —
Let F be a nonzero holomorphic cusp form of level N , weight k/ ≥ / and nebentypus χ . Assume that F is an eigenform for T ℓ and U ℓ for allprimes ℓ with eigenvalues α ℓ . Then there exists a nonzero holomorphic cusp form f of weight k − , level N , and character χ that is an eigenform for all T ℓ and U ℓ with eigenvalues α ℓ . This research is supported in part by NSF Grant DMS-0503264.
NICK RAMSEY
Strictly speaking, Shimura had only conjectured that f is of level 2 N , but this wasproven shortly thereafter by Niwa in [ ] for weights at least 7 / / ].Let p be an odd prime and let N be a positive integer with p ∤ N . In [ ] the authorconstructed a rigid analytic space e D (denoted e D there) parameterizing all finite-slopesystems of eigenvalues of Hecke operators acting on overconvergent cuspidal p -adicmodular forms of half-integral weight and tame level 4 N . Let D denote the integralweight cuspidal eigencurve of tame level 2 N constructed, for example, in [ ]. Inthis paper we construct a rigid-analytic map Sh : e D red −→ D red that interpolatesthe Shimura lifting in the sense that if x ∈ e D is a system of eigenvalues occurringon a classical cusp form F of half-integral weight (and such points are shown to beZariski-dense in e D ) then Sh( x ) is the system of eigenvalues associated to the classicalShimura lifting of F .
2. Modular forms of half-integral weight
Fix an odd prime p and let W denote p -adic weight space over Q p . We brieflyrecall a few facts and bits of notation concerning W . See Section 2.4 of [ ] for moredetails. The K -valued points of W (for a complete extension K/ Q p ) correspond tocontinuous characters κ : Z p −→ K × . Each κ ∈ W ( K ) factors uniquely as κ = τ i · κ ′ where τ is the Teichmuller character, i is an integer well-defined modulo p −
1, and κ ′ is trivial on µ p − ⊆ Z × p . The space W is accordingly the admissible disjoint unionof p − W i for 0 ≤ i < p −
1. Each W i is isomorphic to the open unit ball B (1 ,
1) around 1 under the map κ κ (1 + p ) . Also, W is the rising union of the nested sequence of admissible open affinoids { W n } whose points are those κ with | κ (1+ p ) p n − − | ≤ | p | . For an integer i with 0 ≤ i < p − n we set W in = W i ∩ W n . Finally, if λ ≥ Q p -valued point x x λ of W simply by λ .Let N be a positive integer not divisible by the odd prime p . Given a p -adicweight κ ∈ W ( K ), with K and complete and discretely-valued extension of Q p , andan r ∈ [0 , r n ] ∩ Q , we introduced in [ ] the Banach space f M κ (4 N, K, p − r ) of half-integral weight p -adic modular forms of tame level 4 N and weight κ defined over K . Here { r n } is the decreasing sequence of positive rational numbers introducedin [ ] and [ ], the details of which will be of no importance to us in this paper.This space is endowed with a continuous action of the Hecke operators T ℓ ( ℓ ∤ N p )and U ℓ ( ℓ | N p ), as well as the tame diamond operators h d i N , ( d ∈ ( Z / N Z ) × ).A cuspidal subspace e S κ (4 N, K, p − r ) is also defined, and is equipped with all of thesame operators. In this section we will define a space of classical modular forms ofhalf-integral weight and use it and results of [ ] to define the classical subspaces of f M κ (4 N, K, p − r ) and its cuspidal analog. We will then recast the classical Shimuralifting (Theorem 1.1) in these terms. HE OVERCONVERGENT SHIMURA LIFTING Remark 2.1 . — The classical subspaces considered in this paper are limited in thesense that we restrict our attention to classical forms of level 4
N p . One can alsoinclude classical forms of higher level 4
N p m into the above spaces of p -adic forms.We omit these forms here in part because we have no real need for them, and in partbecause we have not proven an analog of our control theorem (Theorem 2.4) for suchforms (though we expect such a result to hold).For any positive integer M , let Σ M be the Q -divisor on the algebraic curve X (4 M ) Q given by Σ M = 14 π ∗ [ c ]where c is the cusp on (the coarse moduli scheme) X (4) Q corresponding to the pair(Tate( q ) , ζ q ) and π : X (4 M ) Q −→ X (4) Q is the natural map. This divisor Σ M is set up to look like the divisor of zeros of thepullback of the Jacobi theta-function θ to X (4 M ) Q . Indeed, if F is a meromorphicfunction on X (4 M ) an C , then F θ k is a holomorphic modular form of weight k/ F ) ≥ − k Σ M .Let C M be the divisor on X (4 M ) Q given by the sum of the cusps at which Σ M has integral coefficients (this includes, in particular, all cusps outside of the support ofΣ M ). If F is a meromorphic function on X (4 M ) an C , then F θ k is a cuspidal modularform of weight k/ F ) ≥ − k Σ M + C M . The reason for omittingthe cusps at which Σ M has non-integral coefficients is that, since div( F ) has integralcoefficients, F θ k automatically vanishes at such a cusp as soon as it is holomorphicthere. Definition 2.2 . — Let k be an odd positive integer. The space of classical modularforms of weight k/ M over K is defined by f M cl k/ (4 M, K ) = H ( X (4 M ) an K , O ( k Σ M )) , and the subspace of cusp forms is defined by e S cl k/ (4 M, K ) = H ( X (4 M ) an K , O ( k Σ M − C M )) . Both of these spaces are endowed with a geometrically defined action of the Heckeoperators T ℓ ( ℓ ∤ M ) and U ℓ ( ℓ | M ) as well as the diamond operators h d i ( d ∈ ( Z / M Z ) × ). The construction of the Hecke operators is a “twisted” versionof the usual pull-back/push-forward through the Hecke correspondence where onemust multiply by a well-chosen rational function (essentially the ratio of the pull-backs of θ k through the maps defining the correspondence) on the source space of thecorrespondence. This construction is carried out in Section 6 of [ ] and Section 5 of[ ] (where it is applied to a slightly different space of forms). The diamond operatorsare simply given by pull-back with respect to the corresponding automorphisms of X (4 M ) Q .Suppose now that M = N p with p and odd prime not dividing N . For reasons of p -adic weight character book-keeping we separate the diamond action into two kinds of NICK RAMSEY diamond operators using the Chinese remainder theorem. For d ∈ ( Z /p Z ) × we define h d i p to be h d ′ i where d ′ is chosen so that d ′ ≡ d (mod p ) and d ′ ≡ N ). Theoperators h d i N for d ∈ ( Z / N Z ) × are defined similarly, and for any d prime to 4 N p there is a factorization h d i = h d i p ◦ h d i N .Let k be an odd positive integer and define λ = ( k − /
2. In Section 6 of [ ] wedefined an injection f M cl k/ (4 N p, K ) τ j −→ f M λτ j (4 N, K, p − r )for any rational number r with 0 ≤ r ≤ r , where () τ j indicates the eigenspace of the j th power of the Teichmuller character τ for the action of the h d i p . This injection isequivariant with respect to all of the Hecke operators and tame diamonds operators hi N . It is also compatible with varying r and in particular furnishes an injection f M cl k/ (4 N p, K ) τ j −→ f M † λτ j (4 N, K )into the space of all overconvergent forms. If we further restrict to the eigenspace ofa Dirichlet character χ modulo 4 N (valued in K ) for the h d i N , then we also get anembedding f M cl k/ (4 N p, K, χτ j ) −→ f M † λτ j (4 N, K, χ )of the space of classical forms with nebentypus character χτ j for the entire groupof diamond operators ( Z / N p Z ) × into the space of overconvergent forms of tamenebentypus χ . The image of any of these injections will be referred to as the classical subspace of the target. Note that this definition is consistent with the definition of aclassical form given in [ ]. Remark 2.3 . — Everything in the previous paragraph goes through verbatim whenrestricted to the respective spaces of cusp forms. However, we caution that it ispossible for a noncuspidal classical form to be mapped to the space of p -adic cuspforms e S † λτ j (4 N, K ) under the above inclusions; the form need only vanish at the cuspsin the connected component X (4 N p ) an ≥ of the ordinary locus in X (4 N p ) an K . Still,by the classical subspace of e S † λτ j (4 N, K ) we mean the image of the map e S cl k/ (4 N p, K ) τ j −→ e S † λτ j (4 N, K ) . Theorem 2.4 . —
Let F be an element of f M † λτ j (4 N, K ) or e S † λτ j (4 N, K ) and supposethat there exists a monic polynomial P ( T ) ∈ K [ T ] all of whose roots have valuationless than λ − such that P ( U p ) F = 0 . Then F is classical.Proof . — The case of F ∈ f M † λτ j (4 N, K ) was settled in [ ] (Theorem 6.1). The prooffollows Kassaei’s approach in [ ] and builds the classical form by analytic continuationand and gluing. In particular, one writes down an explicit sequence of forms on the“other” component of the ordinary locus of X (4 N p ) an K that converges to the analyticcontinuation of F . It is easy to see that these forms vanish at all cusps as long as F does, so the proof of carries over to the cuspidal case verbatim.For later ease of use, we translate Theorem 1.1 into the world of p -adic coefficients. HE OVERCONVERGENT SHIMURA LIFTING Theorem 2.5 . —
Suppose that k ≥ and F ∈ e S cl k/ (4 N p, K, χτ j ) is a nonzero clas-sical eigenform for all Hecke operators T ℓ and U ℓ with eigenvalues α ℓ ∈ K . Thenthere exists a unique nonzero normalized classical cuspidal modular form f of weight k − , level N p , and nebentypus τ j χ defined over K that is an eigenform for allHecke operators T ℓ and U ℓ with eigenvalues α ℓ .Proof . — Fix an embedding i : K ֒ → C . By p -adic GAGA, any F ∈ f M cl k/ (4 N p, K )is the analytification of an element of H ( X (4 N p ) K , O (Σ Np )). Pulling back viathe embedding i and passing to the complex analytic space we arrive at an element F i ∈ H ( X (4 N p ) an C , O (Σ Np )) depending on i . The condition on the divisor of F i exactly guarantees that the meromorphic modular form F i θ k of weight k/ F F i θ k is equivariant for the action ofHecke and diamond operators on both sides. This can be seen as a formal consequenceof the (entirely parallel) construction of these operators on both spaces. Alternatively,in case of the Hecke operators, this can be deduced by examining their effect on q -expansions. Replacing the divisor Σ Np with Σ Np − C Np we see that the association F F i θ k also preserves the condition of cuspidality.Suppose that F ∈ e S cl k/ (4 N p, K ) τ j satisfies T ℓ F = α ℓ F for all ℓ ∤ N pU ℓ F = α ℓ F for all ℓ | N p h d i N F = χ ( d ) F for all d ∈ ( Z / N Z ) × for some Dirichlet character χ mod 4 N , with α ℓ , χ ( d ) ∈ K for all ℓ and d . It followsthat the holomorphic cusp form F i θ k is of weight k/
2, level 4
N p , nebentypus character i ◦ ( τ j χ ), and is an eigenform for all T ℓ and U ℓ with eigenvalues i ( α ℓ ). By theclassical lifting theorem (Theorem 1.1), we can associate to this form a cuspidalmodular form f i of weight k −
1, level 2
N p , and nebentypus character i ◦ ( τ j χ )that is an eigenfunction for all T ℓ and U ℓ with eigenvalues i ( α ℓ ). By complex-analyticGAGA, this form is actually an algebraic modular form defined over C with all thesame properties. The q -expansion coefficients of f i at the cusp (Tate( q ) , e πi/ Np ) arethe leading coefficient a ( f i ) times polynomials in the Hecke eigenvalues i ( α ℓ ). Since f i = 0, a ( f i ) = 0 as well and we may normalize f i so that a ( f i ) = 1. It now followsfrom the q -expansion principle that f i is in fact an algebraic modular form definedover the field K of weight k −
1, level 4
N p , and nebentypus τ j χ that is an eigenformfor all the T ℓ and U ℓ with eigenvalues α ℓ . Moreover, f i is completely determined bythe eigenvalues α ℓ for all ℓ and is therefore unique (and in particular independent of i ).
3. The eigencurves
As the details of the construction of the relevant eigencurves will be used extensivelyin the sequel, we briefly recall them here. The construction uses various Banachmodules of modular forms equipped with a Hecke action. We refer the reader toSections 6 and 7 of [ ] for the integral weight definitions and to Sections 4 and 5 NICK RAMSEY of [ ] for the half-integral weight definitions. We also refer the reader to [ ] forfoundational details concerning the Fredholm theory that goes into the constructionof eigenvarieties in general.For the moment, let W be any reduced rigid space over a complete and discretely-valued extension field K of Q p . Fix a set T with a distinguished element φ ∈ T .Suppose that we are given, for each admissible affinoid open X ⊆ W , an O ( X )-Banachmodule M X satisfying property ( Pr ) of [ ], equipped with map T −→ End O ( X ) ( M X ) t t X whose image consists of commuting continuous endomorphisms and such that φ X iscompact for each X . Suppose also that for each pair X ⊆ X ⊆ W of admissibleaffinoid opens we are given a continuous injective map α : M X −→ M X b ⊗ O ( X ) O ( X )of O ( X )-modules that is a “link” in the sense of [ ]. Finally, suppose that these linkscommute with T in the sense that α ◦ t X = ( t X b ⊗ ◦ α for each t ∈ T and thatthey satisfy the cocycle condition α = α ◦ α for any triple X ⊆ X ⊆ X ⊆ W of admissible open affinoids.Out of this data one can use the machinery of [ ] to construct rigid analytic spacesspaces D and Z over K called the eigenvariety and spectral variety , respectively,equipped with canonical maps D −→ Z −→ W . The points of D correspond to systems of eigenvalues of T acting on the modules { M X } such that the φ -eigenvalue is nonzero, in a sense made precise below in Lemma3.3, and the map D −→ Z simply records the reciprocal of the φ -eigenvalue and apoint in W .The space Z is easy to define. For any admissible affinoid X ⊆ W we define Z X tobe the zero locus of the Fredholm determinant P X ( T ) = det(1 − φ X T | M X )in X × A . The links guarantee that this determinant is independent of X in thesense that if X ⊆ X ⊆ W are two admissible open affinoids, then P X ( T ) is theimage of P X ( T ) under the natural restriction map on the coefficients. It follows thatwe can glue the Z X for varying X covering W to obtain a space Z equipped with amap Z −→ W .The construction of D is more complicated, and involves first finding a nice ad-missible cover of Z and constructing the part of D over each piece separately andthen gluing these pieces together. This cover is furnished by the following theorem(Theorem 4.6 of [ ]). Theorem 3.1 . —
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 HE OVERCONVERGENT SHIMURA LIFTING – 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).Fix an admissible open affinoid X ⊆ W and fix Y ∈ C ( Z X ) with connected image Y ′ ⊆ X . Let(1) P Y ′ ( T ) = det(1 − ( φ X b ⊗ T | M X b ⊗ O ( X ) O ( Y ′ ))Note that this is not in conflict with the existing notation P Y ′ (for an arbitraryconnected admissible affinoid open Y ′ ⊆ W ) by Lemma 2.13 of [ ] and the abovecomments about the independence of P X on X . When an ambient X ⊆ W is fixed,we prefer to use the definition (1) instead so as to avoid using the links.As explained in Section 5 of [ ], we can associate to the choice of Y a factorization P Y ′ ( T ) = Q ( T ) Q ′ ( T ) into relatively prime factors with constant term 1, where Q is apolynomial of degree equal to the degree of the projection π : Y −→ Y ′ whose leadingcoefficient is a unit. Geometrically speaking, Y is the zero locus of the polynomial Q in π − ( Y ′ ) while its complement π − ( Y ′ ) \ Y is cut out by the Fredholm series Q ′ .By the Fredholm theory of [ ] there is a unique decomposition M X b ⊗ O ( X ) O ( Y ′ ) ∼ = N ⊕ F into closed φ -invariant submodules with the property that Q ∗ ( φ ) vanishes on N andis invertible on F . Moreover, N is projective of rank equal to the degree of Q andthe characteristic power series of φ on N is Q . The projectors onto to the submod-ules N and F are in the closure of O ( Y ′ )[ φ ], so by the commutativity assumptionthese submodules are preserved under all of the endomorphisms associated to ele-ments of T . Let T ( Y ) denote the O ( Y ′ )-subalgebra of End O ( Y ′ ) ( N ) generated bythe endomorphisms t X b ⊗ t ∈ T . This algebra is finite over O ( Y ′ ) and thereforeaffinoid.Since the polynomial Q is the characteristic power series of φ on N and has aunit for a leading coefficient, φ is invertible on N . Moreover, since Q ∗ ( φ ) = 0 on N , Q ( φ − ) = 0 on N as well. Thus we have well-defined map O ( Y ) ∼ = O ( Y ′ )[ T ] / ( Q ( T )) −→ T ( Y ) T φ − which is to say that the affinoid D Y = Sp( T ( Y )) is equipped with a natural finite map D Y −→ Y . These affinoids and maps can be glued together for varying Y ∈ C ( Z X )to obtain a space D X equipped with a map D X −→ Z X . Finally, using the links α ij we can glue over varying X ⊆ W to obtain a space D and canonical maps D −→ Z −→ W . NICK RAMSEY
These spaces and maps have a particularly nice interpretation on the level of points.Let L be a complete and discretely-valued extension of K . Definition 3.2 . — A pair ( κ, γ ) consisting of an L -valued point κ ∈ W ( L ) and amap of sets γ : T −→ L is an L -valued system of eigenvalues of T acting on the { M X } if there exists an admissible open affinoid X ⊆ W containing κ and a nonzeroelement m ∈ M X b ⊗ O ( X ) ,κ L such that ( t b ⊗ m = γ ( t ) m for all t ∈ T . This system of eigenvalues is called φ -finite if γ ( φ ) = 0.Let x be an L -valued point of D . Then x lives over a point κ x in some admissibleaffinoid open X ⊆ W and moreover lies in D Y for some Y ∈ C ( Z X ). The associated K -algebra map T ( Y ) −→ L gives a map γ x : T −→ L of sets. Lemma 3.3 . —
The association x ( κ x , γ x ) is a well-defined bijection betweenthe set of L -valued points of D and the set of L -valued φ -finite systems of eigenvaluesof T acting on the { M X } . The map D −→ Z is given by x ( κ x , γ x ( φ ) − ) on L -valued points.Proof . — The first assertion is proven in [ ]. The second is obvious from the thedefinition of the map D −→ Z .For the remainder of the paper, p will denote an odd prime and W will denote p -adic weight space over Q p . Fix a positive integer N prime to p . For each admissibleaffinoid open X ⊆ W and each rational number r ∈ [0 , r n ], we define M X ( N, Q p , p − r )to be the O ( X )-Banach module of families of (integral weight) modular forms of tamelevel N and growth condition p − r on X . This module has been defined, for example,in [ ] and [ ] where an action of the Hecke operators T ℓ ( ℓ ∤ N p ) and U ℓ ( ℓ | N p ) andthe tame diamond operators h d i N ( d ∈ ( Z /N Z ) × ) is also defined. These operatorsare continuous and the operator U p is compact whenever r >
0. Similarly, we define f M X (4 N, Q p , p − r ) to be the O ( X )-Banach module of families of half-integral weightmodular forms of tame level 4 N and growth condition p − r on X . This module wasintroduced in [ ] where an action of the Hecke operators T ℓ ( ℓ ∤ N p ) and U ℓ ( ℓ | N p ) and the tame diamond operators h d i N ( d ∈ ( Z / N Z ) × ) is also defined.These operators are continuous and U p is compact whenever r >
0. Each of thesemodules has a cuspidal submodule having all of the same operators and propertiesand will be denoted by replacing the letter M by the letter S . The tilde will be usedthroughout the paper to distinguish half-integral weight objects from their integralweight counterparts.All of these modules of forms satisfy ( Pr ) of [ ]. The system of Banach modules { S X } , where S X = S X ( N, Q p , p − r n ) and n is chosen to be the smallest integer suchthat X ⊆ W n , carries canonical links defined in [ ] (strictly speaking on the entirespace of forms, but they can be defined in exactly the same manner on the cuspidal HE OVERCONVERGENT SHIMURA LIFTING submodule). In [ ] we construct canonical links for the system { e S X } where e S X = e S X (4 N, Q p , p − r n ). The Hecke operators and tame diamond operators in each casefurnish commuting endomorphisms that are compatible with these links, and U p and U p are compact, so we may apply the above construction in both the integral and half-integral weight cases. Note that the eigenvarieties and spectral varieties so obtainedare equidimensional of dimension 1 by Lemma 5.8 of [ ]. We denote the eigencurveand spectral curve associated to { S X ( N, Q p , p − r n ) } by D and Z , respectively, andrefer to them as the cuspidal integral weight eigencurve and spectral curve of tamelevel N , respectively. Similarly we denote the eigencurve and spectral curve associatedto { e S X (4 N, Q p , p − r n ) } by e D and e Z , respectively, and refer to them as the cuspidalhalf-integral weight eigencurve and spectral curve of tame level 4 N , respectively. Remark 3.4 . — In the interest of notational brevity we have chosen not to adornthese spacing so as to indicate that we are working only with the cuspidal part. Thisputs the notation in conflict with, for example, the author’s previous notation in [ ].Hopefully no confusion will arise from this conflict.
4. Some density results
We will need some lemmas on the density of certain sets of classical points in whatfollows. Following Chenevier ([ ]), we call a subset Σ ⊆ W ( C p ) very Zariski-dense iffor each κ ∈ Σ and each irreducible (equivalently, connected) admissible affinoid open V ⊆ W containing κ , V ( C p ) ∩ Σ is Zariski-dense in V . Lemma 4.1 . —
The set Σ of weights of the form λτ j for λ ≥ and ≤ j < p − is very Zariski-dense in W .Proof . — Fix λ τ j ∈ Σ and suppose moreover that λ τ j ∈ W i . Let V be anyconnected admissible affinoid containing λ , so V ⊆ W i . Recall that W i is isomorphicto the open unit ball B (1 ,
1) about 1, the isomorphism being κ κ (1 + p ) . Note that the points in Σ are all Q p -valued. Since V ( Q p ) is open in the p -adictopology, there exists an ǫ > B ( λ τ j , ǫ ] of radius ǫ about λ τ j is contained in V . Admit for the moment that B ( λ τ j , ǫ ]( Q p ) ∩ Σ is Zariskidense in B ( λ τ j , ǫ ]. We claim that this implies that V ( Q p ) ∩ Σ is Zariski-dense in V . Let T ⊂ V be an analytic subset containing V ( Q p ) ∩ Σ. Then T ∩ B ( λ τ j , ǫ ] isan analytic subset of B ( λ τ j , ǫ ] containing B ( λ τ j , ǫ ]( Q p ) ∩ Σ and by assumptionwe conclude that T contains B ( λ τ j , ǫ ]. It now follows from Lemma 2.2.3 of [ ] that T = V .So we are reduced proving Zariski-density in the special case of B ( λ τ j , ǫ ], whichamounts to proving that this ball contains infinitely many elements of Σ. But | λτ j − λ τ j | = | (1 + p ) λ − (1 + p ) λ | = | (1 + p ) λ − λ − | and we can find infinitely many λτ j ∈ B ( λ τ j , ǫ ] which make this as small as we likeby choosing j = j and λ ≡ λ (mod ( p − p N ) for sufficiently large N . NICK RAMSEY
Let X ⊆ W be an admissible affinoid open. For any polynomial e h over O ( X ) inthe Hecke operators T ℓ , U ℓ and the diamond operators h d i N , we denote by e Z e hX thesubspace of X × A defined as the zero locus of e P e hX ( T ) = det(1 − e hU p T | e S X )in X × A . Definition 4.2 . — Let K/ Q p be a finite extension. – We call a K -valued point ( κ, α ) ∈ e Z e hX classical if κ ∈ Σ and there exists anonzero classical form F ∈ e S † κ (4 N, K ) such that e hU p F = α − F . – Let x ∈ e D ( K ) and let ( κ, γ ) be the K -valued system of eigenvalues correspond-ing to x by Lemma 3.3. Then x is called classical if κ ∈ Σ and there exists anonzero classical form F ∈ e S † κ (4 N, K ) on which the the operators T ℓ , U ℓ , and h d i N act through γ . Lemma 4.3 . —
Let X ⊆ W be a connected admissible open affinoid containing anelement of Σ . Then the classical points are Zariski-dense in e Z e hX .Proof . — Let C be an irreducible component of e Z e hX . By Theorem 4.2.2 of [ ], C is aFredholm hypersurface and hence has Zariski-open in image in X . It follows that C contains a point x mapping to an element of Σ by Lemma 4.1. Let Y be an elementof the canonical cover C ( e Z e hX ) containing x with connected image Y ′ ⊆ X . By Lemma4.1, Σ ∩ Y ′ is Zariski-dense in Y ′ . Moreover, the proof of the lemma shows that thisis also true after omitting any finite collection of points in Σ.Let N denote the direct summand of e S X b ⊗ O ( X ) O ( Y ′ ) corresponding to the choiceof Y . The O ( Y ′ )-module N is projective of rank equal to the degree of Y −→ Y ′ and is stable under the endomorphism U p . Moreover U p acts invertibly on N since e hU p does. It follows that the eigenvalues of U p are bounded away from 0 on N .That is, there exists a positive integer M such that for all finite extensions K/ Q p andall K -valued points O ( Y ′ ) −→ K , the roots of the characteristic polynomial of U p acting on the fiber of N ⊗ O ( Y ′ ) K have absolute value at least p − M . This can be seen,for example, by examining the variation of the Newton polygon of the characteristicpolynomial(2) R ( T ) = det( T − U p | N )over Y ′ . Let Σ ′ denote the complement of the finite collection of λτ j ∈ Σ with2 λ − ≤ M , and note that Σ ′ ∩ Y ′ is still Zariski dense in Y ′ by the above comments.Since Y −→ Y ′ is finite and flat, each irreducible component of Y surjects onto Y ′ ,and it follows that the preimage of Σ ′ in Y is Zariski-dense in every component of Y .If ( λτ j , α ) is a K -valued point in this preimage, then there is a nonzero overconvergentform F ∈ N ⊗ O ( Y ′ ) K ⊆ e S † λτ j (4 N, K )with e hU p F = α − F . Since F is annihilated by R λτ j ( U p ), the characteristic polyno-mial (2) with coefficients evaluated at λτ j , it follows from Theorem 2.4 that F is a HE OVERCONVERGENT SHIMURA LIFTING classical cusp form. Thus the classical locus is also dense in each component of Y . ByCorollary 2.2.9 of [ ], Y ∩ C is a nonempty (since it contains x ) union of irreduciblecomponents of Y . It follows easily from Lemma 2.2.3 of [ ] that a Zariski-dense sub-set of an admissible open in an irreducible space is in fact Zariski dense in the wholespace, so we conclude that the classical points are Zariski-dense in C for each C , andthe lemma follows. Remark 4.4 . — In case e h = 1, this proof also shows that the set of points in e Z X ofthe form ( λτ j , α ) with v ( α − ) < λ − e Z X . Since W is admissiblycovered by a collection of connected admissible affinoid subdomains { X } each of whichmeets Σ (such as { W in } ) we conclude that the set of ( λτ j , α ) ∈ e Z with v ( α − ) < λ − e Z . Corollary 4.5 . —
The classical points are Zariski-dense in e D .Proof . — By Lemma 5.8 of [ ], the finite map e D −→ e Z carries every irreduciblecomponent of e D surjectively onto an irreducible component of e Z . It follows from thisand Remark 4.4 that the set of points x ∈ e D such that the corresponding systemof eigenvalues ( κ, α ) has κ ∈ Σ and v ( α − ) < λ − e D . But byTheorem 2.4, all such points are in fact classical.
5. Interpolation of the Shimura lifting
Let N be a positive integer not divisible by the odd prime p . For the remainderof the paper all spaces of modular forms of half-integral weight will be taken at tamelevel 4 N and all spaces of modular forms of integral weight will be taken at tame level2 N . Let : W −→ W denote the finite map given by ( κ ) = κ on the level of points. We wish to constructmaps Sh : e D red −→ D red and e Z red −→ Z red fitting into the diagram(3) e D red Sh / / (cid:15) (cid:15) D red (cid:15) (cid:15) e Z red / / (cid:15) (cid:15) Z red (cid:15) (cid:15) W / / W whose vertical arrows are the canonical ones arising from the constructions of theeigencurves and such that if x is a classical point then Sh( x ) is the system of eigen-values of the classical Shimura lift of a classical form corresponding to x . The map e Z red −→ Z red will simply be given on points by( κ, α ) ( κ , α ) , though it is not yet at all clear that this map is well-defined. NICK RAMSEY
We record a couple of lemmas concerning nilreductions of Fredholm varieties thatwe will need in the sequel.
Lemma 5.1 . —
With notation as in Theorem 3.1, assume moreover that R is rela-tively factorial. Then the map C ( Z ) −→ C ( Z red ) Y Y red is a bijection.Proof . — First note that Z red is Fredholm by Theorem 4.2.2 of [ ]. Let Y ∈ C ( Z )with image Y ′ ⊆ Sp( R ). That Y −→ Y ′ is finite implies that Y red −→ Y ′ red = Y ′ isfinite. To get an idempotent that cuts out Y red , simply pull back the idempotent thatcuts out Y through the canonical reduction map, so the proposed map at least makessense.Let X ∈ C ( Z red ) with image Y ′ ⊆ Sp( R ). By the proof of A1.1 in [ ], the map Z red −→ Z is a homeomorphism of Grothendieck topologies. In particular the under-lying open set of X is also an admissible open in Z . As such, it inherits the structureof a rigid space by restricting the structure sheaf of Z to X . Let Y denote the rigidspace so obtained. I claim that Y ∈ C ( Z ) and that the map X Y is the inverseto the above map.Since reduction and passing to an admissible open commute, Y red = X . NowTheorem A1.1 of [ ] implies that Y −→ Y ′ is finite, so the comments followingTheorem 3.1 imply that Y ∈ C ( Z ). That these two maps are inverse to each other isclear.If F is a Fredholm series over a relatively factorial affinoid, let F red denote theunique Fredholm series such that Z ( F red ) = Z ( F ) red (the existence and uniquenessof such a series is guaranteed by Theorem 4.2.2 of [ ]). Lemma 5.2 . —
Let A and B be relatively factorial affinoid algebras over K with A an integral domain and let f : Sp( A ) −→ Sp( B ) be a map of rigid spaces over K . Let F and G be Fredholm series over A and B respectively. The following are equivalent. (a) F red divides ( f ∗ G ) red in the ring of entire series over A . (b) f ∗ G vanishes on the zero locus of F . (c) Z ( F ) red is a union of irreducible components of ( Z ( G ) × Sp( B ) Sp( A )) red = Z ( f ∗ G ) red . (d) there exists a unique map Z ( F ) red −→ Z ( G ) red that is given by ( x, α ) ( f ( x ) , α ) on points.Proof . — Since passing to the reduction does not change the zero locus, (a) implies(b) trivially. Suppose that (b) holds. Then ( f ∗ G ) red also vanishes on Z ( F ), and (c)follows from Lemma 4.1.1 of [ ]. Now suppose that (c) holds. By Corollary 2.2.6 of[ ], Z ( F ) red is a “component part” of Z ( f ∗ G ) red in the sense of [ ]. By Proposition HE OVERCONVERGENT SHIMURA LIFTING ], F red divides ( f ∗ G ) red , and we have (a). It remains to see that (d) isequivalent to these first three conditions. Suppose that that (c) holds. The compositemap Z ( F ) red ֒ → Z ( f ∗ G ) red −→ Z ( f ∗ G ) = Z ( G ) × Sp( B ) Sp( A ) −→ Z ( G )factors through a unique map Z ( F ) red −→ Z ( G ) red by the universal property ofreduction. This map has the desired effect on points and is the unique one with thisproperty since these spaces are reduced. Finally, that (d) implies (b) is clear. Proposition 5.3 . —
Let X and e X be connected admissible affinoid subdomains of W such that restricts to a map : e X −→ X , and suppose that e X contains a pointof Σ . Let h be a polynomial over O ( X ) in the symbols T ℓ , U ℓ , and h d i N and let e h denote the corresponding polynomial obtained by replacing these symbols by T ℓ , U ℓ ,and h d i N , respectively, and pulling back the coefficients to O ( e X ) . Let P hX ( T ) = det(1 − hU p T | S X ) and e P e h e X ( T ) = det(1 − e hU p T | e S e X ) . Then e P e h e X ( T ) red divides ∗ P hX ( T ) red .Proof . — By Lemma 5.2 it suffices to check that ∗ P hX ( T ) vanishes on the zero locusof e P e h e X . By Lemma 4.3, the classical points are dense in this locus. The same is trueof the set of classical points of weight λτ j with λ ≥ λτ j , α ) be a point in the zero locus of e P e h e X with λ ≥ α in a finite extension K of Q p and define k = 2 λ + 1. Then the space V = { F ∈ e S cl k/ (4 N, K ) τ j | e h λτ j U p F = α − F } , where e h λτ j denotes the polynomial e h with coefficients evaluated at λτ j , is nonzero.Since all of the operators T ℓ , U ℓ , and h d i N commute with e h λτ j U p , they act onthe space V and hence there exists a finite extension L/K and a nonzero element F ∈ e S cl k/ (4 N, L ) τ j that is a simultaneous eigenform for all of these operators. Inparticular it has a character χ for the action of the h d i N , and by Theorem 2.5 thereexists a nonzero classical form f of level 2 N and weight k − K that liesin the τ j -eigenspace for h d i p such that h (2 λ ) τ j U p f = α − f. Since α, τ ( d ) ∈ K for all d ∈ ( Z /p Z ) × , there must also be a form f defined over K with these properties. Thus ((2 λ ) τ j , α ) is a root of P hX , which is to say that ( λτ j , α )is a root of ∗ P hX .Let X ⊆ W be an admissible affinoid open and let κ ∈ W ( K ) be a point. Forany module M over O ( X ) we denote by M κ the vector space M ⊗ O ( X ) ,κ K and forany power series P over O ( X ) we denote by P κ the power series over K obtained byevaluating the coefficients of P at κ . NICK RAMSEY
Corollary 5.4 . —
Let κ ∈ W ( K ) , let h be a polynomial over K in the symbols T ℓ , U ℓ , and h d i N , and let e h denote the polynomial obtained by replacing these symbolsby T ℓ , U ℓ , and h d i N , respectively. Pick a connected admissible affinoid open e X containing κ and let X = ( e X ) . Then det(1 − e h U p T | ( e S e X ) κ ) red | det(1 − h U p T | ( S X ) κ ) red Proof . — Note that X is necessarily a connected admissible affinoid (and moreover : e X −→ X is an isomorphism), so the assertion at least makes sense. By enlarging e X if necessary we may assume that e X contains an element of Σ, since the links α ij ensure that this enlargement does not affect the claimed divisibility. Let h be anypolynomial over O ( X ) in the symbols T ℓ , U ℓ , and h d i N with h κ = h , and let e h denote the polynomial over O ( e X ) obtained by replacing these symbols by T ℓ , U ℓ ,and h d i N , respectively, and pulling back the coefficients via ∗ . Clearly we have e h κ = e h . By Proposition 5.3 we havedet(1 − ˜ hU p T | e S e X ) red | ( ∗ det(1 − hU p T | S X )) red . The result now follows from Lemma 2.13 of [ ] and Lemma 5.2 by specializing to κ . Corollary 5.5 . —
Let e X ⊆ W be a connected admissible affinoid open and let X = ( e X ) . There is a unique finite map e Z e X, red −→ Z X, red having the effect ( κ, α ) ( κ , α ) on points.Proof . — Choose integers i and n with e X ⊆ W in , so that X ⊆ W in . By Proposition5.3 (with e h = 1) and Lemma 5.2, ∗ P W in ( T ) vanishes on the zero locus of e P W in ( T ).Let ι and e ι denote the inclusions of X and e X into W in and W in , respectively. Then e ι ∗ ∗ P W in ( T ) = ∗ ι ∗ P W in ( T )vanishes on the zero locus of e ι ∗ P W in ( T ). Lemma 2.13 and the links α ij ensure that ι ∗ P W in ( T ) = P X ( T ) and e ι ∗ e P W in ( T ) = e P e X ( T ) . The existence of a map having the indicated effect on points now follows from Lemma5.2, and it remains to see that this map is finite. But by Lemma 5.2 the map isthe composition of the inclusion of the union of irreducible components e Z e X, red into( Z X × X e X ) red and the (nilreduction of) the projection Z X × X e X −→ Z X . Theformer is obviously finite and the latter is finite because the map : e X −→ X is anisomorphism.For each i , the W in form a nested sequence, so we may glue the maps furnished byCorollary 5.5 applied to e X = W in and X = W in over increasing n to obtain a diagram e Z W i , red / / (cid:15) (cid:15) Z W i , red (cid:15) (cid:15) W i / / W i HE OVERCONVERGENT SHIMURA LIFTING for each 0 ≤ i < p −
1. Now since W is the disjoint union of the W i , we obtain thebottom square in the diagram (3). Let g : e Z red −→ Z red be the map so obtained. Onthe level of points, g is simply given by g ( κ, α ) = ( κ , α ), as desired. The next lemmashows that g interacts well with the canonical covers of its source and target. Lemma 5.6 . —
Let e X ⊆ W be a connected admissible affinoid open and let X = ( e X ) . Let Y be an element of the canonical cover C ( Z X ) of Z X with connected image Y ′ ⊆ X . Then g − ( Y red ) is either empty or is an element of C ( e Z e X, red ) with connectedimage − ( Y ′ ) .Proof . — Since g and are finite, g − ( Y red ) and − ( Y ′ ) are affinoids. The latter isconnected since Y ′ is connected and : e X −→ X is an isomorphism.By the construction of g (see the proof of Lemma 5.2), g − ( Y red ) is the intersectioninside Z X, red × X e X of the admissible affinoid Y red × X e X and the union of irreduciblecomponents e Z e X, red . This intersection is a (possibly empty) union of irreducible com-ponents of the admissible affinoid Y red × X e X by Corollary 2.2.9 of [ ]. But Y red × X e X ∼ = Y red × Y ′ − ( Y ′ )is finite over − ( Y ′ ) since Y red is finite over Y ′ , and hence so is any nonemptysubspace of components, such as g − ( Y red ) (if nonempty).That g − ( Y red ) is disconnected from its complement in the full preimage of − ( Y ′ )follows from the analogous property of Y red ; one simply pulls back the idempotentthat cuts out Y red though g to get one that cuts out g − ( Y red ).Let X , e X , and Y be as in the previous lemma and assume that g − ( Y ) = ∅ so that g − ( Y red ) is in C ( e Z e X, red ) with connected image − ( Y ′ ). Let P Y ′ ( T ) = Q ( T ) Q ′ ( T )and e P − ( Y ′ ) ( T ) = e Q ( T ) e Q ′ ( T )denote the factorizations arising from the choice of Y and the e Y ∈ C ( e Z e X ) of which g − ( Y red ) is the underlying reduced affinoid (this well-defined by Lemma 5.1). Since g restricts to maps g − ( Y red ) −→ Y red and e Z − ( Y ′ ) , red \ g − ( Y red ) −→ Z Y ′ , red \ Y red , Lemma 5.2 ensures that e Q red | ( ∗ Q ) red and f Q ′ red | ( ∗ Q ′ ) red . Let A and B denote theaffinoid algebras of Y ′ and − ( S ), respectively. Let S X b ⊗ O ( X ) A = N ⊕ F and e S O ( e X ) b ⊗ B ∼ = e N ⊕ e F denote the corresponding decompositions of the spaces of families of cusp forms.To construct the p -adic Shimura lift Sh and complete the diagram (3), we willconstruct the part covering g : g − ( Y red ) −→ Y red for each Y and glue. Let T ( Y )denote the A -subalgebra of End R ( N ) generated by T ℓ , U ℓ , and h d i N and let e T ( e Y ) NICK RAMSEY denote the B -subalgebra of End S ( e N ) generated by T ℓ , U ℓ , and h d i N . We wish toshow that the map T ( Y ) red −→ e T ( e Y ) red (4) T ℓ T ℓ U ℓ U ℓ h d i N d i N that is given by ∗ : A −→ B on coefficients, is well-defined. The following lemmafurnishes the key divisibility needed to prove this well-definedness. Lemma 5.7 . —
Let h be a polynomial over A in the symbols T ℓ , U ℓ , and h d i N , andlet e h be the polynomial over B obtained by replacing these symbols by T ℓ , U ℓ , and h d i N , respectively, and applying the map ∗ : A −→ B to the coefficients. Then det(1 − e hU p T | e N ) red | ( ∗ det(1 − hU p T | N )) red . Proof . — By Lemma 5.2, the divisibility claimed in the statement of the lemma canbe checked after specializing to each κ ∈ − ( Y ′ ). Define e S κ = e S e X b ⊗ O ( e X ) ,κ K and S κ = S X b ⊗ O ( X ) ,κ K and define e N κ and N κ similarly. By Lemma 2.13 of [ ], this amounts to checkingthat(5) det(1 − e h κ U p T | e N κ ) red | det(1 − h κ U p T | N κ ) red for each κ ∈ − ( Y ′ ).For any σ ∈ R , we define e S σκ ( S σκ , . . . , etc.) to be the slope σ subspace forthe relevant operator ( U p for integral weight spaces and U p for half-integral weightspaces). These spaces are all finite-dimensional and e N κ is moreover the direct sumof the subspaces e N σκ that are nonzero since U p is invertible on e N κ , and similarlyfor N κ . Let h be any polynomial over K in the symbols T ℓ , U ℓ , and h d i N andlet e h be the polynomial obtained by replacing the symbols by T ℓ , U ℓ , and h d i N ,respectively (no need to pull back the coefficients). We claim that(6) det( T − e h | e S σκ ) red | det( T − h | S σκ ) red for all σ . Since the endomorphisms of N and e N associated to h and e h , respectively,are continuous, their eigenvalues are bounded and we can find a single nonzero x ∈ K × such that both h ′ = 1 + xh and e h ′ = 1 + x e h have all eigenvalues of absolute value1. It follows that det(1 − e h ′ U p T | e S κ ) σ = det(1 − e h ′ U p T | e S σκ )and det(1 − h ′ U p T | S κ ) σ = det(1 − h ′ U p T | S σκ )where by F ( T ) σ for a Fredholm determinant F ( T ) over K we mean the unique (poly-nomial) Fredholm factor with the property that the Newton polygon of F ( T ) σ has HE OVERCONVERGENT SHIMURA LIFTING pure slope σ and the Newton polygon of F ( T ) /F ( T ) σ has no sides of slope σ . Butnow it follows from Corollary 5.4 thatdet(1 − e h ′ U p T | e S σκ ) red | det(1 − h ′ U p T | S σκ ) red and since e h ′ U p and h ′ U p are invertible on these spaces, we can deduce the analogousdivisibility for characteristic polynomials,det( T − e h ′ U p | e S σκ ) red | det( T − h ′ U p | S σκ ) red as well. By the same reasoning, this divisibility holds after replacing h ′ and e h ′ by h ′ + ǫ and e h ′ + ǫ all sufficiently small ǫ ∈ K . Letdet( T − ( e h ′ + X ) U p | e S σκ ) = Y i ( T − e a i X − e b i )and det( T − ( h ′ + X ) U p | S σκ ) = Y j ( T − a j X − b j ) . Then we have shown that for infinitely many ǫ ∈ K it is the case that for each i thereexists j such that e a i ǫ + e b i = a j ǫ + b j . It follows easily that for each i there exists j such that e a i = a j and e b i = b j . Simul-taneously upper-triangularizing the commuting endomorphisms e h ′ and U p , we seethat the e b i / e a i are exactly the eigenvalues of e h ′ on e S σκ . Similarly, the b j /a j are theeigenvalues of h ′ on S σκ , and we conclude thatdet( T − e h ′ | e S σκ ) red | det( T − h ′ | S σκ ) red . Now (6) follows from a linear change of variables.We claim that det(1 − e h κ U p | e N σκ ) red | det(1 − h κ U p T | N σκ ) red for each σ ∈ R . In particular, this establishes the desired divisibility (5) by thecomments that follow it. Let α be a root ofdet(1 − e h κ U p T | e N σκ ) . By enlarging K if necessary (the only requirement on K in the preceding argumentsis that it be finite over Q p and contain the residue field of κ ) we may assume that α ∈ K . Let H denote the free commutative K -algebra generated by the symbols T ℓ , U ℓ , and h d i N . The finite-dimensional K -vector space S σκ ⊕ e S σκ is a finite lengthalgebra over H , where H acts in the obvious way on S σκ and on e S σκ we agree that T ℓ acts by T ℓ , U ℓ acts by U ℓ , and h d i N acts by h d i N . Let f W be a simple (over H )constituent of { F ∈ e N σκ | e h κ U p F = α − F } ss = 0 , where for a (finite length) H -module M , M ss denotes the semisimplification of M as an H -module. By general facts about semisimple algebras, there exists h ∈ H such that h acts via the identity on f W and h = 0 on any simple constituent of NICK RAMSEY S σ, ss κ ⊕ e S σ, ss κ that is not isomorphic to W . Divisibility (6) implies that 1 is a root ofdet( T − h | S σκ ), so there must be a simple constituent W of S σ, ss κ isomorphic over H to f W . I claim that moreover W is a simple constituent of N σ, ss κ . To see this, notethat since e Q κ, red | Q κ , red and these are polynomials, there exists a positive integer M such that e Q κ | Q Mκ . Thus the fact that e Q ∗ κ ( U p ) is zero on e N σκ (and therefore on f W )implies that Q ∗ κ ( U p ) M is zero on W . But Q ∗ κ ( U p ) is invertible on S σκ /N σκ ∼ = F σκ ,so W must occur in N σ, ss κ . Since W and f W are isomorphic over H , h κ U p w = α − w for all w ∈ W = 0, so α is a root ofdet(1 − hU p T | N σ, ss κ ) = det(1 − hU p T | N σκ )and the claimed divisibility follows.We now return to proving that the map (4) is well-defined. In other words, wemust show that if h is a polynomial over A in the symbols T ℓ , U ℓ , and h d i N thatis nilpotent on N , and e h is the corresponding polynomial over B as usual, then e h isnilpotent on e N . But if h is nilpotent on N then det(1 − hU p T | N ) = 1, so by Lemma5.7, det(1 − e hU p T | e N ) = 1 as well. It follows that e hU p is nilpotent on e N since thismodule is projective of finite rank, and hence so is e h since U p acts invertibly on e N .The (nonempty) admissible opens g − ( Y red ) form an admissible cover of e D e X, red ,so in order to glue the maps we have defined to a map e D e X, red −→ D X, red we mustcheck that they agree on the overlaps. Since : e X −→ X is an isomorphism, thisis immediate from the characterization on points afforded by the definition (4) andLemma 3.3 since these spaces are reduced. Now note that, for each i , the spaces W i and W i are covered by the nested families of affinoids { W in } and { W in } , respectively,and : W in −→ W in is an isomorphism for each n . Thus we may glue over increasing n to obtain a diagram e D i red Sh / / (cid:15) (cid:15) D i red (cid:15) (cid:15) e Z i red g / / (cid:15) (cid:15) Z i red (cid:15) (cid:15) W i / / W i where the superscript i on a space mapping to W denotes the preimage of the con-nected component W i . Finally, since W is the disjoint union of the W i we obtain thedesired diagram (3).We now come to the main result of this paper. Theorem 5.8 . —
Let N be a positive integer and let p be an odd prime not dividing N . Let D and Z denote the integral weight cuspidal eigencurve and spectral curveof level N , respectively, and let e D and e Z denote the half-integral weight cuspidal HE OVERCONVERGENT SHIMURA LIFTING eigencurve and spectral curve of level N , respectively. There exists a unique diagram e D red Sh / / (cid:15) (cid:15) D red (cid:15) (cid:15) e Z red g / / (cid:15) (cid:15) Z red (cid:15) (cid:15) W / / W where and g are characterized on points by ( κ ) = κ and g ( κ, α ) = ( κ , α ) and Sh has the property that if x ∈ e D ( L ) corresponds to a system of eigenvalues oc-curring on a nonzero classical form F ∈ e S cl k/ (4 N p, K, χτ j ) with k ≥ , then Sh( x ) corresponds to the system of eigenvalues associated to the image of F under the clas-sical Shimura lifting.Proof . — That the maps Sh and g that we have constructed have the indicated prop-erties is clear from their construction and Theorem 2.5. Uniqueness follows fromLemma 4.5 since these spaces are reduced (the omission of the finitely many classicalweights with λ ≤
6. Properties of
ShIn this section we determine the nature of the image and fibers of the map Sh.For each i , the map : W i −→ W i is and isomorphism, and it follows easily fromLemma 3.3 and the definition (4) that the restriction Sh : e D i red −→ D i red is injective. Proposition 6.1 . —
The map Sh carries e D i red isomorphically onto a union of irre-ducible components of D i red .Proof . — Let X ⊆ W i , e X ⊆ W i , and Y be as in Lemma 5.6. Then by that lemma g − ( Y red ) is either empty or in C ( e Z e X, red ). Accordingly, Sh − ( D ( Y ) red ) is either emptyor equal to e D ( e Y ) red , where e Y is the element of C ( e Z e X ) of which g − ( Y red ) is theunderlying reduced affinoid. In the latter case, the mapSh ∗ : T ( Y ) red −→ e T ( e Y ) red is clearly surjective from its definition (4). Thus the image of Sh is locally cut out bya coherent ideal (since the rings T ( Y ) are Noetherian), and is therefore an analyticset in D i red . Since these spaces are reduced, Sh is an isomorphism of e D i red onto thisanalytic set. Both e D i red and D i red are equidimensional of dimension one by Lemma 5.8of [ ], so Corollary 2.2.7 of [ ] ensures that the image of Sh is a union of irreduciblecomponents. NICK RAMSEY
Note that for each i , there are exactly two connected components of W that mapvia to W i , namely W i and W i ′ where i ′ = i + ( p − /
2. We will construct acanonical isomorphism e D i ∼ −→ e D i ′ fitting into a commutative diagram(7) e D i red Sh ' ' NNNNNNNNNNNNN ∼ (cid:15) (cid:15) D i red e D i ′ red Sh ppppppppppppp In particular, it will follows that the diagonal arrows have the same union of irreduciblecomponents as image, and the map Sh is everywhere two-to-one. The existence ofsuch an isomorphism stems from the existence of a Hecke operator U p on families ofoverconvergent forms that is a kind of “square-root” of the operator U p .In [ ] we constructed operators U ℓ (on the spaces considered in that paper) havingthe effect P a n q n P a ℓn q n on q -expansions, for all primes ℓ dividing the level.These operators were found to commute with all other Hecke operators, but onlycommute with the diamond operators up to a factor of the quadratic character ( ℓ/ · ).In our case, if ℓ = p , then such a map would in fact alter the weight since the p -part of the nebentypus is part of the p -adic weight character. Note that we have afactorization (cid:16) p · (cid:17) = (cid:18) − · (cid:19) ( p − / (cid:18) ( − ( p − / p · (cid:19) = (cid:18) − · (cid:19) ( p − / τ ( p − / . Let ǫ denote the involution of W given by ǫ ( κ ) = κ · τ ( p − / Proposition 6.2 . —
Fix a primitive (4 N p ) th root of unity ζ Np . Let X ⊆ W bea connected admissible affinoid open and let r ∈ [0 , r n ] ∩ Q . There is a compact O ( X ) -linear map U p : f M ǫ ( X ) (4 N, Q p , p − r ) b ⊗ O ( ǫ ( X )) O ( X ) −→ f M X (4 N, Q p , p − r ) having the effect P a n q n P a pn q n on q -expansions at (Tate( q ) , ζ Np ) . This mapcommutes with the operators T ℓ and U ℓ for all ℓ and satisfies U p ◦ h d i N = (cid:18) − d (cid:19) ( p − / h d i N ◦ U p . Proof . — The construction of U p , like the all of the operators T ℓ and U ℓ follows thegeneral procedure set up in Section 5 of [ ]. We will omit the details as they wouldtake up considerable space and are very similar to the constructions of the operators T ℓ and U ℓ , and content ourselves with commenting that the nontrivial commutationrelation with the diamond operators arises (as it does in the construction of U ℓ givenin [ ]) because the “twisting” function H on X (4 N p, p ) an Q p used in the constructionis not fixed by the diamond operators as it is in the case of T ℓ and U ℓ . HE OVERCONVERGENT SHIMURA LIFTING Extending scalars to O ( ǫ ( X )) and replacing X by ǫ ( X ) we arrive at a map in theopposite direction f M X (4 N, Q p , p − r ) −→ f M ǫ ( X ) (4 N, Q p , p − r ) b ⊗ O ( ǫ ( X )) O ( X )that has the same effect on q -expansions. It follows that the composition of thesemaps in either order is U p (or its scalar extension). Everything we have said about U p thus far holds equally well for cusps forms, and Lemmas 2.12 and 2.13 of [ ] nowimply thatdet(1 − U p T | e S X ) = det(1 − ( U p b ⊗ T | e S ǫ ( X ) b ⊗ O ( ǫ ( X )) O ( X ))= ǫ ∗ det(1 − U p T | e S ǫ ( X ) ) . Since ǫ is an isomorphism we get a diagram e Z X ∼ / / (cid:15) (cid:15) e Z ǫ ( X ) (cid:15) (cid:15) X ǫ / / ǫ ( X )in which the horizontal arrows square to the identity map in the evident fashion. Notethat this diagram establishes a bijection between the covers C ( e Z X ) and C ( e Z ǫ ( X ) ).Using the links, we see that it is also compatible with enlarging X and hence oneobtains an involution of the whole space e Z covering the involution ǫ .Let Y ∈ C ( e Z X ) with connected image Y ′ ⊆ X and note that the correspondingelement Y ǫ ∈ C ( e X ǫ ( X ) ) has connected image ǫ ( Y ′ ) ⊆ ǫ ( X ). Corresponding to Y and Y ǫ we obtain decompositions e S X b ⊗ O ( X ) O ( Y ′ ) ∼ = e N ⊕ e F and e S ǫ ( X ) b ⊗ O ( ǫ ( X )) O ( ǫ ( Y ′ )) ∼ = e N ǫ ⊕ e F ǫ respectively. Note that by extending scalars to O ( Y ′ ), U p induces a map( e S ǫ ( X ) b ⊗ O ( ǫ ( X )) O ( ǫ ( Y ′ ))) b ⊗ O ( ǫ ( Y ′ )) O ( Y ′ ) −→ e S X b ⊗ O ( X ) O ( Y ′ ) . Lemma 6.3 . —
This extension of scalars restricts to an isomorphism U p : e N ǫ b ⊗ O ( ǫ ( Y ′ )) O ( Y ′ ) ∼ −→ e N .
Proof . — Let Q ∈ O ( Y ′ )[ T ] be the polynomial factor ofdet(1 − U p T | e S X b ⊗ O ( X ) O ( Y ′ ))associated to the choice of Y , so that ǫ ∗ Q ∈ O ( ǫ ( Y ′ ))[ T ] is the polynomial associatedto Y ǫ . Thus the summand e N ǫ b ⊗ O ( ǫ ( Y ′ )) O ( Y ′ ) of( e S ǫ ( X ) b ⊗ O ( ǫ ( X )) O ( ǫ ( Y ′ ))) b ⊗ O ( ǫ ( Y ′ )) O ( Y ′ )is precisely the kernel of ǫ ∗ ( ǫ ∗ Q ∗ ( U p )) = Q ∗ ( U p ) NICK RAMSEY and since the map U p commutes with the action of U p , this summand maps to thekernel of Q ∗ ( U p ) in e S X b ⊗ O ( X ) O ( Y ′ ) under U p . The latter is simply e N , so U p at leastrestricts to some map e N ǫ b ⊗ O ( ǫ ( Y ′ )) O ( Y ′ ) −→ e N. As above we can extend scalars to O ( ǫ ( Y ′ )) and reverse the roles of Y ′ and ǫ ( Y ′ )to get a map in the other direction with the property that both compositions aresimply U p (or a scalar extension thereof) on the respective spaces. That these mapsare isomorphisms now follows from the fact that U p is invertible on the modules e N and e N ǫ .It follows from Lemma 6.3 and the commutation relations in Proposition 6.2 thatthe map e T ( e Y ǫ ) −→ e T ( Y )given by ǫ ∗ : O ( ǫ ( Y ′ )) −→ O ( Y ′ ) on coefficients and T ℓ T ℓ U ℓ U ℓ (8) h d i N (cid:18) − d (cid:19) ( p − / h d i N on the generators is in fact well-defined. Thus we obtain a map e D Y −→ e D Y ǫ coveringthe map Y −→ Y ǫ . This construction readily glues over Y ∈ C ( e Z X ). Moreover,the maps U p are compatible with the canonical links used in the construction of e D (because these links are simply induced by restriction to smaller admissible opens in X (4 N p ) an K ), and it follows that these maps glue to a map e D i −→ e D i ′ . When thismap is composed with the one obtained by reversing the roles of i and i ′ (in eitherdirection), one obtains the identity, as is evident from the definition (8). In particularit is an isomorphism and extends to an involution of the whole space e D . Finally, thatthe diagram (7) commutes can be checked on points since these spaces are reduced.But then it follows immediately from the characterization of these points in terms ofsystems of eigenvalues in Lemma 3.3 and the definition (8) Example 6.4 . — The restriction k ≥ / /
2. There is no meaningful modular liftingof the theta series of weight 1 / ]). However, one can lift cuspidal thetafunctions of weight 3 /
2, but one obtains Eisenstein series of weight 2 instead of cuspforms. Let ψ be a primitive Dirichlet character modulo a positive integer r such that ψ ( −
1) = −
1. Then θ ψ ( q ) = 12 X n ∈ Z ψ ( n ) nq n is a classical cusp form of weight 3 /
2, level 4 r and nebentypus character ψ = ψ · ( − / · ). The form θ ψ is an eigenform for all T ℓ ( ℓ ∤ r ) and U ℓ ( ℓ | r ) with HE OVERCONVERGENT SHIMURA LIFTING eigenvalues (1 + ℓ ) ψ ( ℓ ). The Shimura lift of this form is the Eisenstein series E ψ ( q ) = ∞ X n =1 ψ ( n ) σ ( n ) q n of weight 2, where σ ( n ) = P d | n d , as it easy to check from the explicit formulas in[ ].Let p be an odd prime. If p | r , then θ ψ is in the kernel of U p and does not furnisha point on the eigencurve e D . If on the other hand p ∤ r , then θ ψ thought of in level4 r p is in fact a U p eigenform with eigenvalue ψ ( p ) p = 0. Thus θ ψ furnishes a pointon e D and the existence of the map Sh implies that there exists a cuspidal p -adiceigenform with the same eigenvalues. It is easy to check that the form E ∗ ψ = E ψ − ψ ( p ) V p E ψ has the expected eigenvalues, so this must be the image form. In fact it is easy tocheck using the explicit formulas in [ ] that this is in fact the classical Shimura liftapplied to θ ψ thought of at level 4 r p .Thus we a lead to the conclusion that E ∗ ψ , while not a classical cusp form, is acuspidal p -adic modular form. That is, E ∗ ψ vanishes at the cusps in the connectedcomponent X (4 r p ) an ≥ of the ordinary locus. This fact also follows from Theorem3.4 of [ ], where Cipra computes the value of E ∗ ψ at every cusp. We also remark thatthis is only possible since E ∗ ψ is of critical slope (in this case, slope 1) since if it wereof low slope then the technique of Kassaei [ ] implies that a low-slope form that is p -adically cuspidal is in fact a classical cuspidal modular form. References [1]
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B. A. Cipra – On the Niwa-Shintani theta-kernel lifting of modular forms,
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Nagoya Math. J. (1975), p. 147–161.[9] N. Ramsey – Geometric and p -adic modular forms of half-integral weight, Ann. Inst.Fourier (Grenoble) (2006), no. 3, p. 599–624. NICK RAMSEY [10] , The half-integral weight eigencurve,
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Ann. of Math. (2) (1973),p. 440–481. Nick Ramsey , Department of Mathematics, University of Michigan