aa r X i v : . [ m a t h . N T ] J un GEOMETRIC AND p -ADIC MODULAR FORMS OFHALF-INTEGRAL WEIGHT by Nick Ramsey
Contents
1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Notation and Conventions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43. Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64. Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75. The Modular Units Θ m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106. Hecke Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137. p -adic modular forms of half-integral weight. . . . . . . . . . . . . . . . . . . . . 178. p -adic Hecke operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1. Introduction
The aim of this article is to generalize the notion of a classical (holomorphic)modular form of half-integral weight to rings other than C . In particular we define,for any positive integers N and k , the space M k/ (4 N, R ) of modular forms of weight k/ N over the ring R , for any ring R in which the level 4 N is invertible.The definition is geometric in nature in the sense that it involves the modular curves X (4 N ), and admits a modification to a p -adic theory by “deleting the supersingularlocus”.The author’s primary motivation for studying such objects lies in their relationto (central) special values of modular L -functions. This remarkable relationship, dis-covered by Waldspurger, expresses these values (appropriately normalized) in termsof squares of the Fourier coefficients of modular forms of half-integral weight. In-vestigating the p -adic nature of modular forms of half-integral, and in particular thenature of p -adic families of modular forms of half-integral weight, represents a firststep in one approach to understanding the p -adic variation of (square roots of) these NICK RAMSEY L -values. Eying the integral weight case, one expects the notion of overconvergenceto play a central role in questions about families of modular forms of half-integralweight. This, in turn, is currently best understood in geometric terms.In the integral weight case, a geometric theory of modular forms is achieved by con-sidering sections of tensor powers of the line bundle ω which is the push-forward of thesheaf of relative differentials from the universal elliptic curve over X ( N ) (supposing N ≥ k/ X (4 N ). This methodis reminiscent of the use of the Eisenstein family to define families of p -adic modularforms of integral weight (as in [ ] and [ ]).Let θ ( τ ) = X n ∈ Z q n , q = e πiτ denote the usual Jacobi theta function. It is well-known that θ obeys a transformationlaw with respect to the groupΓ (4) = (cid:26) (cid:18) a bc d (cid:19) ∈ SL ( Z ) (cid:12)(cid:12)(cid:12)(cid:12) | c (cid:27) , namely θ (cid:18) aτ + bcτ + d (cid:19) = ε − d (cid:16) cd (cid:17) ( cτ + d ) / θ ( τ ) , for all (cid:18) a bc d (cid:19) ∈ Γ (4) , where(1) ǫ d = (cid:26) d ≡ i d ≡ − − π/ , π/ ].We will take the θ k as the “well-understood” forms by which we divide as indicatedabove. Indeed, in section 3 we single out a family of divisors Σ N,k on the curves X (4 N ) with the property that the rational functions F on X (4 N ) such that F θ k is a holomorphic modular forms of weight k/ F for which ( F ) ≥− Σ N,k . The caveat is that Σ N,k has non-integral (rational) coefficients. The divisorΣ N,k is supported on the cusps and makes perfect sense as a relative Q -divisor on X (4 N ) R over any base R in which 4 N is invertible. We will accordingly identify M k/ (4 N, R ) with the R -module of rational functions F on X (4 N ) R with ( F ) ≥− Σ N,k . In other words, if ⌊ Σ N,k ⌋ is the relative effective Cartier divisor obtainedby taking the floor of the coefficients of Σ N,k , then M k/ (4 N, R ) = H ( X (4 N ) R , O ( ⌊ Σ N,k ⌋ )) . In what follows, we will adopt the suggestive notation H ( X (4 N ) R , Σ N,k )for the right-hand side of this equation.
EOMETRIC AND p -ADIC MODULAR FORMS OF HALF-INTEGRAL WEIGHT If this definition is to be a useful one, it must enjoy some the host of propertiespossessed by the integral weight analog. To this end, we will define q -expansions, provea q -expansion principle, prove some basic “change-of-ring” compatibilities, and furnisha geometric construction of the Hecke operators. Because of its central importance,we briefly expound on this last point.Let X ( N, m ) denote the modular curve over Z [1 / ( N m ))] (roughly) classifyingelliptic curves with a point of order N and a cyclic subgroup of order m having trivialintersection with that generated by the point. Let π , π : X ( N, m ) −→ X ( N )denote the degeneracy maps which forget and quotient by, respectively, the cyclicsubgroup of order m . These pairs of maps form the usual Hecke correspondences on X ( N ), and the geometric manifestation of the integral weight Hecke operator T m (for, say, m prime) is H ( X ( N ) , ω k ) −→ H ( X ( N ) , ω k ) f π ∗ ( π ∗ f ) , where we have exploited the canonical identification π ∗ ω ∼ = π ∗ ω. A little reflection on the definition of a modular form of half-integral weight givenabove leads one to look for a map of the form H ( X (4 N ) , Σ N,k ) −→ H ( X (4 N ) , Σ N,k ) F π ∗ ( π ∗ F · Θ)for some rational function Θ on X (4 N, m ) satisfying(2) div(Θ) ≥ π ∗ Σ N,k − π ∗ Σ N,k . Note that this last divisor is of degree 0, so if such a Θ exists, then one has equalityabove. It then follows that Θ is necessarily unique up to a constant factor, andthe divisor π ∗ Σ N,k − π ∗ Σ N,k actually has integral coefficients. The following isproven in section 6, and involves a slightly refined space of forms M k/ (4 N, χ, R )with Nebentypus character χ : ( Z / N Z ) × → R × . Theorem 1.1 . —
Suppose gcd(4
N, m ) = 1 . Then there exists a Θ satisfying (2) ifand only if m is a square. If m = l for a prime l not dividing N , the correspondingendomorphism T l of M k/ (4 N, χ, R ) (appropriately normalized) has the followingeffect: if P a n q n is the q -expansion of F at ∞ , then the q -expansion of T l F at ∞ is P b n q n where b n = a l n + (cid:18) − l (cid:19) (1 − k ) / l ( k − / − (cid:16) nl (cid:17) χ ( l ) a n + l k − χ ( l ) a n/l . This theorem furnishes a geometric explanation of the classical fact that there areno good (i.e. prime to the level) Hecke operators of non-square index, and shows thatthe operators obtained by the procedure above at square indices are the same as theclassical ones (as in [ ]). NICK RAMSEY
Let K be a discretely valued subfield of C p . In section 7 we define K -Banachspace M k/ (4 N, K, r ) of p -adic modular forms of weight k/
2, level 4 N , and growthcondition r ∈ [0 , Theorem 1.2 . —
Let p be a prime number not dividing N and let r furthersatisfy r > p − /p (1+ p ) . There is a completely continuous endomorphism U p of M k/ (4 N, K, r ) having the effect X a n q n X a p n q n on q -expansions at ∞ . Moreover, the norm of U p is at most p . The definition of the space M k/ (4 N, K, r ) is a straightforward modification ofthat of M k/ (4 N, R ) achieved by deleting part of the supersingular locus. In par-ticular, it still relies on properties of the forms θ k (equivalently, the divisors Σ N,k ).We note that it is perhaps more natural in this p -adic context to use Eisensteinseries of half-integral weight in place of θ , since much of their divisors lie outsideof the range of consideration . This point of view is adopted in a forthcoming workinvestigating p -adic families of modular forms of half-integral weight (see [ ] and [ ]) . Acknowledgments
This work was done while the author was a graduate student Harvard University.He extends his thanks to his advisor Barry Mazur for suggesting a problem which ledto this work and for being helpful and encouraging throughout. He would also liketo thank the referee for pointing out some typos, as well as providing some helpfulsuggestions about the level of detail.
2. Notation and Conventions
For an integer N ≥ R in which N is invertible, Y ( N ) will denotethe fine moduli scheme classifying pairs ( E/S, P ) where E is an elliptic curve over an R -algebra S and P is a point of order N (as in [ ]). For such N , X ( N ) R will denotethe compactified moduli scheme, which is a fine moduli scheme for the generalized Γ ( N ) problem if N ≥
5, and coarse moduli scheme if N = 4. We will have nooccasion to make use of generalized elliptic curves in what follows.Choosing another positive integer m which is invertible in R , we may also formthe moduli scheme Y ( N, m ) R classifying triples ( E/S, P, C ) consisting of an ellipticcurve E over an R -algebra S , a point P of order N , and a cyclic subgroup C of order m not meeting that generated by P , as well as its compactification X ( N, m ) R . Allof these moduli schemes are smooth over R .If K is the complex field C or a p -adic field, we will denote by X ( N ) an K (andsimilarly X ( N, m ) an K , etc.) the associated complex-analytic or rigid-analytic space, EOMETRIC AND p -ADIC MODULAR FORMS OF HALF-INTEGRAL WEIGHT respectively. Over C , we choose to uniformize Y ( N ) via the mapΓ ( N ) \ H ∼ −→ Y ( N ) an C τ (cid:18) E τ , N (cid:19) where H denotes the complex upper half-plane and E τ denote the elliptic curve C / h , τ i over C . Similarly, we choose to uniformize Y ( N, m ) via the map(Γ ( N ) ∩ Γ ( m )) \ H ∼ −→ Y ( N, m ) an C τ (cid:18) E τ , N , D τm E(cid:19) . If Γ ⊆ SL ( Z ) is any subgroup, we will also use the notation X (Γ) an C for the compact-ification of the quotient Γ \ H , ignoring any algebraic model of this Riemann surface(i.e. this is not intended to denote the analytification of anything in this case).We will also need to consider the moduli spaces X ( N, p n ) R in the case that p is not invertible in R (but N is). By such a space we will always mean the Katz-Mazur model (see [ ]). In particular, X ( N, p n ) R is not smooth over R and willeven have non-reduced fibers when n >
1. We note for later use that X ( N, p n ) R isCohen-Macaulay over R as its singularities are all local complete intersections.We will denote the Tate elliptic curve over Z (( q )) by Tate( q ) (see [ ]). Our con-ventions concerning the Tate curve differ from the standard ones as follows. In thepresence of, for example, level N structure, previous authors (e.g., [ ]) have preferredto consider the curve Tate( q N ) over the base Z (( q )). Points of order N on this curveare used to characterize the behavior of a modular form at the cusps, and are alldefined over the fixed ring Z (( q ))[ ζ N ] (where ζ N is some primitive N th root of 1). Weprefer to fix the curve Tate( q ) and instead consider extensions of the base. Thus, inthe presence of level N structure, we introduce the formal variable q N , and define q = q NN . Then the curve Tate( q ) is defined over the sub-ring Z (( q )) of Z (( q N )) andall of its N -torsion is defined over the ring Z (( q N ))[ ζ N ]. To be precise, the N -torsionis given by ζ iN q jN , ≤ i, j ≤ N − . The reason for this formal relabeling is that, with these conventions, the q -expansions of modular forms look like they do complex-analytically if we identify q N with e πiτ/N . In particular, the “ q -order” of vanishing (i.e. the smallest, generallyfractional, power of q appearing in the expansion) at a cusp does not change whenone pulls back through a degeneracy map, even in the presence of ramification.For a prime l and an l th root of unity ζ we will denote the associated quadraticGauss sum by g l ( ζ ) := l − X a =1 (cid:16) al (cid:17) ζ a . NICK RAMSEY
3. Definitions
Given that θ is non-vanishing in H , we must examine the behavior of θ at the cuspsof X (4) an C in order to determine the divisor Σ N,k mentioned in the introduction.There are three such cusps, and using standard formulas for transforming “thetafunctions with characteristics” one finds that the q -expansions of θ at these cusps areas follows (defined up to a constant involving powers of 2 and roots of unity, exceptin the first case).(3) cusp on X (4) an q -expansion ∞ θ ( q ) := P n ∈ Z q n / θ ( q ) := q P n ∈ Z q n + n θ ( q ) := P n ∈ Z q n Recall that in the complex-analytic setting, q h is shorthand for e πiτ/h .If F is a rational function on X (4 N ) an C , it follows that F · θ k is holomorphic if andonly if ( F ) ≥ − Σ N,k , where(4) Σ N,k = X c ∼ / kw c · c, the sum is taken over all cusps c on X (4 N ) an C mapping to 1 / X (4) an C under thedegeneracy map which multiplies the point of order 4 N by N , and w c denotes thewidth of the cusp c (which coincides with the ramification index of this map at c since1 / ∈ X (4) an C has width 1).To determine the appropriate replacement for Σ N,k on the algebraic curve X (4 N ) R ( R a Z [1 / (4 N )]-algebra), we consider the form θ . By the transformationformula (1), this is a modular form of weight 2 for Γ (4). By GAGA and the q -expansion principle, θ furnishes a section of ω on the curve X (4) Z [1 / . Thus wemay evaluate the corresponding rule on pairs (Tate( q ) , P ) where P runs through thepoints of order 4 on Tate( q ) (recall the conventions Section 2). There are six suchpairs up to isomorphism, and the values are as follows,(5) point on Tate( q ) cusp on X (4) an C q -expansion ζ ∞ (cid:16)P n ∈ Z q n (cid:17) ζ q / q (cid:16)P n ∈ Z q n + n (cid:17) q − (cid:16)P n ∈ Z q n (cid:17) ζ q − (cid:16)P n ∈ Z ( ζ q ) n (cid:17) ζ q = − q − (cid:16)P n ∈ Z ( − q ) n (cid:17) ζ q = − ζ q − (cid:16)P n ∈ Z ( ζ q ) n (cid:17) where we have filled in the middle column by comparison with table (3).It is now clear from the table that the cusp associated to the pair (Tate( q ) , ζ q )is the correct replacement for the cusp 1 / EOMETRIC AND p -ADIC MODULAR FORMS OF HALF-INTEGRAL WEIGHT simply denote the cusp on X (4) R corresponding to this pair by the symbol 1 /
2, andaccordingly define a divisor Σ N,k on the curve X (4 N ) R by the formula (4) for anyring R in which 4 N is invertible.Now that we have a good version of Σ N,k on X (4 N ) R , we enshrine the identifi-cation of the Introduction in a definition . Definition 3.1 . — Let k be an odd positive integer, let N be any positive integer,and let R be a ring in which 4 N is invertible. A modular form of weight k/ N is an element of the R -module M k/ (4 N, R ) := H ( X (4 N ) R , Σ N,k ) . We recall from the Introduction that this notation is shorthand for the collection ofrational functions F on X (4 N ) R with ( F ) ≥ − Σ N,k . In other words, this is the spaceof section of the sheaf O ( ⌊ Σ N,k ⌋ ) associated to the relative effective Cartier divisor ⌊ Σ N,k ⌋ on X (4 N ) R . The reason for carrying around the Q -divisor Σ N,k instead ofuniformly resorting to its floor ⌊ Σ N,k ⌋ (or equivalently the associated sheaf) is thatthe latter do not behave well with respect to degeneracy maps and the former does.For example, if N | M and π : X (4 M ) → X (4 N )denotes the map which multiplies the point by M/N , then π ∗ Σ N,k = Σ M,k . The precise behavior of Σ N,k under various degeneracy maps will be of critical im-portance in the sequel.For d ∈ ( Z / (4 N ) Z ) × , let h d i denote the diamond automorphism of X (4 N ) corre-sponding to d . Note that h d i ∗ Σ N,k = Σ N,k , for all d ∈ ( Z / (4 N ) Z ) × . It follows that the group ( Z / (4 N ) Z ) × acts on the space M k/ (4 N, R ). For a character χ : ( Z / (4 N ) Z ) × → R × , we will denote the χ -isotypic component of M k/ (4 N, R ) by M k/ (4 N, χ, R ), and referto such forms as having “nebentypus χ ”.
4. Properties
As in the integral weight case, it will be useful to reinterpret the elements of M k/ (4 N, R ) as “rules” which take test objects as input and give some output subjectto a few restraints. In particular, we will take as test objects pairs (
E/S, P ) where E is an elliptic curve over an R -algebra S and P is a point of order 4 N on E . Anelement F of M k/ (4 N, R ) can then be identified with a rule (also called F ) whichtakes as input a test object ( E/S, P ) and outputs an element of S such that – if the test objects ( E/S, P ) and ( E ′ /S, P ′ ) are isomorphic over S , then F ( E, P ) = F ( E ′ , P ′ ), NICK RAMSEY – if φ : S → S ′ is any map of R -algebras and ( E/S, P ) is a test object over S ,then F (( E, P ) × S S ′ ) = φ ( F ( E, P )) , and – if P is a point of order 4 N on Tate( q ), then F (Tate( q ) , P ) ∈ q − e N Z [[ q N ]]where e = 4 Nw c ord c Σ N,k = (cid:26) N k c ∼ / c ≁ / c is the cusp associated to (Tate( q ) , P ).With this in mind, we can define define the q -expansions of a modular form of half-integral weight by evaluating the corresponding rule on pairs (Tate( q ) , P ) for variouspoints P of order 4 N . By the definition of the space M k/ (4 N, R ), these q -expansionswill not agree with the classical ones, and will in fact be off exactly by a factor of thecorresponding q -expansion of θ k , and we must adjust accordingly.To be precise, let P be a point of order 4 on Tate( q ). Then, up to isomorphism, P is one of the points listed in the first column of Table 5. Up to roots of unity, thecorresponding q -expansions of θ are as follows (see Table 3).(6) P q -expansion of θζ P n ∈ Z q n ζ q q P n ∈ Z q n + n ζ k q , ≤ k ≤ ζ k P n ∈ Z ζ kn q n Definition 4.1 . — Let F ∈ M k/ (4 N, R ) and let P be a point of order 4 N onTate( q ). Let θ P be the q -expansion of θ in the above table corresponding to the point N P of order 4. The q -expansion of F at (Tate( q ) , P ) is F (Tate( q ) , P ) θ kP ∈ Z (( q N )) ⊗ R [ ζ N ] . We remark in particular that the definition of Σ N,k exactly suffices to ensure thatall of the q -expansions of an F ∈ M k/ (4 N, R ) are in fact in the subring Z [[ q N ]] ⊗ R [ ζ N ] . That is, such an F is “holomorphic at the cusps.” We will often fix a root of unity ζ N ,and refer to the q -expansion at (Tate( q ) , ζ N ) as the “ q -expansion at ∞ .” The reasonis that if R = C and we have fixed ζ N = e πi/N , then under the uniformizationsdetailed in Section 2 we recover the classical q -expansion at “ i ∞ .”Half-integral weight modular forms enjoy a q -expansion principle much like theintegral-weight version. As in that case, one must first define modular forms over modules as opposed to rings . For a Z [1 / (4 N )]-module K , one defines the space ofmodular forms of weight k/ N with coefficients in K as the collectionof rules which associate to each test datum ( E, P ) over a Z [1 / (4 N )]-algebra R , anelement of R ⊗ Z [1 / (4 N )] K , subject to the usual compatibilities and conditions on theTate curve. Note that, in the case that K is in fact a ring, the space so defined incanonically isomorphic to the space we have already defined. However, in contrast to EOMETRIC AND p -ADIC MODULAR FORMS OF HALF-INTEGRAL WEIGHT the case of rings, the q -expansions of elements of M k/ (4 N, K ) for a Z [1 / (4 N )]-module K , lie in the ring Z [[ q N ]] ⊗ Z Z [1 / (4 N ) , ζ N ] ⊗ Z [1 / (4 N )] K, In particular, these q -expansion do not agree with the ones defined for rings (evenwhen K is a ring), though one can still deduce useful information about the “ring” q -expansions from the “module” q -expansions. At any rate, it is in the context of the“module” q -expansion that the q -expansion principle is most naturally stated. Theorem 4.2 . —
Let K ⊂ L be an inclusion of Z [1 / (4 N )] -modules and let F ∈ M k/ (4 N, L ) . Let P be a points of order N on Tate( q ) and suppose that the q -expansion of F at (Tate( q ) , P ) has coefficients in Z [1 / (4 N ) , ζ N ] ⊗ Z [1 / (4 N )] K . Then F ∈ M k/ (4 N, K ) .Proof . — Referring to Table 6, we see that each of θ P and θ − P has coefficients in Z [1 / (4 N ) , ζ N ]. It follows that the q -expansion of F at (Tate( q ) , P ) has coefficientsin Z [1 / (4 N ) , ζ N ] ⊗ Z [1 / (4 N )] K if and only if F (Tate( q ) , P ) does. If the latter is true,then F ∈ M k/ (4 N, K ) by the ordinary q -expansion principle in weight 0 (the polesare of no consequence).We wrap up this section with a basic result about changes-of-ring. Any map R → S of Z [1 / (4 N )]-algebras induces a map M k/ (4 N, R ) ⊗ R S −→ M k/ (4 N, S ) . We would like conditions under which this map is an isomorphism. As these aresections of the sheaf O ( ⌊ Σ N,k ⌋ ), it suffices by standard base-changing results to seethat H ( X (4 N ) R , O ( ⌊ Σ N,k ⌋ ) = 0 . Theorem 4.3 . —
This vanishing of H holds whenever k ≥ .Proof . — By Serre duality it suffices to show thatdeg( ⌊ Σ N,k ⌋ ) > deg(Ω X (4 N ) R ) . Note that the form θ furnishes a section of ω on X (4 N ) R with divisor Σ N, . Also,by the Kodaira-Spencer isomorphism,Ω X (4 N ) ∼ = ω ( − C )where C is the divisor of cusps. Thusdeg(Ω X (4 N ) ) = deg( ω ) − | C | < deg(Σ N, ) ≤ deg( ⌊ Σ N,k ⌋ )whenever k ≥ Remark 4.4 . — In weights 1 / / NICK RAMSEY
5. The Modular Units Θ m Let m be an odd positive integer. In this section we introduce a family of rationalfunctions on X (4 m ) whose divisors are supposed on the cusps. These functions willlater be used to construct the Hecke operators.Let Θ m be the holomorphic function on H defined byΘ m ( τ ) = θ ( τ /m ) θ ( τ ) . Note that, for γ = (cid:18) a bc d (cid:19) ∈ Γ (4) ∩ Γ ( m ) , we have Θ m | γ = (cid:16) md (cid:17) Θ m . Thus Θ m furnishes a holomorphic function on (Γ (4) ∩ Γ ( m )) \ H which descendsthrough the natural map(7) (Γ (4) ∩ Γ ( m )) \ H −→ (Γ (4) ∩ Γ ( m )) \ H exactly when m is a square. Let d denote the extension of this map to X (Γ (4) ∩ Γ ( m )) an C −→ X (Γ (4) ∩ Γ ( m )) an C = X (4 , m ) an C . For N ≥ π i : Y (4 N, m ) −→ Y (4 N )by the following two transformations of moduli functors π : ( E, P, C ) ( E, P ) π : ( E, P, C ) ( E/C, P/C ) . We remark that in the complex-analytic uniformizations of these curves given inSection 2, these maps are given by π ( τ ) = τ and π ( τ ) = τ /m . Proposition 5.1 . —
The divisor of Θ m as a meromorphic function on X (Γ (4) ∩ Γ ( m )) an C is div(Θ m ) = ( π ◦ d ) ∗ Σ , − ( π ◦ d ) ∗ Σ , Proof . — It is tempting to deduce this directly from Table (3) and the “formula”Θ m = ( π ◦ d ) ∗ θ/ ( π ◦ d ) ∗ θ . The problem is that θ has not been realized as the sectionof any bundle. One way around this is to raise both sides to the fourth power anduse the fact that θ ∈ H ( X (4) an C , ω ) = M (Γ (4) , C ) . We have Θ m ( τ ) = θ ( τ /m ) θ ( τ ) , so that Θ m = ( π ◦ d ) ∗ θ ⊗ (( π ◦ d ) ∗ θ ) − p -ADIC MODULAR FORMS OF HALF-INTEGRAL WEIGHT where we have used the canonical identification O = ω ⊗ ( ω ) − . It follows that 4div O Θ m = div O Θ m = div ω ( π ◦ d ) ∗ θ − div ω ( π ◦ d ) ∗ θ = ( π ◦ d ) ∗ div ω θ − ( π ◦ d ) ∗ div ω θ . The desired result will then follow from the fact that div ω θ = 4Σ , upon divisionby 4. This fact is easily read off from Table (5).Since gcd(4 , m ) = 1, the groups Γ (4) ∩ Γ ( m ) and Γ (4 m ) are conjugate, and thecurves X (Γ (4) ∩ Γ ( m )) an C and X (4 m ) an C are isomorphic. One way to realize thisisomorphism explicitly is via the moduli interpretation of the latter curve and themap(8) X (Γ (4) ∩ Γ ( m )) an C −→ X (4 m ) an C given (on noncuspidal points) by τ ( E τ , P )where P is the unique point on E τ with mP = 1 / P = τ /m . Translating thenatural map d into these terms one arrives at the map d ′ : X (4 m ) an C → X (4 , m ) an C given on noncuspidal points by( E, P ) ( E, mP, h P i ) . We denote the pullback of Θ m through the (inverse of) the isomorphism (8) by thesame name. It’s divisor is thendiv(Θ m ) = ( π ◦ d ′ ) ∗ Σ , − ( π ◦ d ′ ) ∗ Σ , . Recall that the divisor Σ N,k is defined more generally on the algebraic curve X (4 N ) R for any R in which 4 N is invertible. Applying GAGA to Θ m thought of asa section of O X (4 m ) an C (( π ◦ d ′ ) ∗ Σ , − ( π ◦ d ′ ) ∗ Σ , ) , we see that Θ m comes from a section (also denoted Θ m ) of the corresponding sheafon the algebraic curve X (4 m ) C . Repeating this argument with Θ − m as a section of O X (4 m ) an C (( π ◦ d ′ ) ∗ Σ , − ( π ◦ d ′ ) ∗ Σ , )furnishes the opposite inequality of the divisor of Θ m and shows that Θ m is a rationalfunction on the algebraic curve X (4 m ) C with(9) div(Θ m ) = ( π ◦ d ′ ) ∗ Σ , − ( π ◦ d ′ ) ∗ Σ , . Finally, we note that, as in the analytic case, Θ m descends through the map d ′ : X (4 m ) −→ X (4 , m ) NICK RAMSEY (given by the same transformation of moduli functors as above) exactly when m is asquare. To see this, note that the group( Z / m Z ) × ∼ = ( Z / Z ) × × ( Z /m Z ) × acts on the curve X (4 m ) C in the usual way by scaling the point of order 4 m . Onesees by passage to the analytic world that the action of ( Z /m Z ) × on Θ m by pullbackis again given by the character ( m/ · ).In the next section, we will need a few q -expansions of Θ m in terms of Tate curves.To compute these expansions, one must translate this data into complex-analytic dataand compute with standard transformation formulas for θ . To do this, we proceed asfollows. Let R denote the subring C (( q m ))) consisting of q -expansions of holomorphicfunctions in the upper half-plane H which are invariant under translation by 4 m , andfor τ ∈ H let φ τ denote the map R −→ C X a n q n m X a n e πinτ m . For any point P of order 4 m on Tate( q ), the pair (Tate( q ) , P ) is defined over Z [1 / (4 m ) , ζ , ζ m ](( q m )), and if we pick an embedding of the coefficient ring ι : Z [1 / (4 m ) , ζ , ζ m ] ֒ → C and base change (Tate( q ) , P ) to C (( q m )) accordingly, the result is actually definedover R (simply look at the defining equations of Tate( q )). Moreover, if we then basechange to C via the map φ τ , we arrive at a pair ( E τ , P ′ ) for some point P ′ of order4 m on E τ . Thus φ τ ( ι (Θ m (Tate( q ) , P ))) = Θ m ( E τ , P ′ ) = Θ m ( γτ )where γ = (cid:18) a bc d (cid:19) ∈ SL ( Z )is chosen so that the isomorphism E τ ∼ −→ E γτ z zcτ + d carries the point P ′ to the unique point P ′′ on E γτ such that 4 P ′′ = γτ /m and mP ′′ = 1 /
4. Now Θ m ( γτ ) can be worked out explicitly from the definition of Θ m ,and the desired expansion is easily read off.We illustrate this technique by computing one of these q -expansions. Fix a primi-tive 4 th root of 1, ζ . If ζ m is any m th root of 1 (not necessarily primitive), and P isthe unique point of order 4 m with 4 P = ζ m q m and mP = ζ , then we claim thatΘ m (Tate( q ) , P ) = P n ∈ Z ζ n m q n m P n ∈ Z q n . We specify the embedding ι by sending ζ to i and ζ m to e πik/m for some integer k . Thus φ τ ( ι (Θ m (Tate( q ) , P ))) = Θ m ( E τ , P ′ ) EOMETRIC AND p -ADIC MODULAR FORMS OF HALF-INTEGRAL WEIGHT where P ′ is the unique point on E τ such that 4 P ′ = ( k + τ ) /m and mP ′ = 1 /
4. Onethen notes that matrix γ = (cid:18) k (cid:19) has the desired property. That is, that the isomorphism E τ ∼ −→ E τ + k z z sends P ′ to the unique point on E τ + k with 4 P ′ = ( k + τ ) /m = γτ /m and mP ′ = 1 / m ( τ + k ) = θ ( τ + km ) θ ( τ + k ) = P n ∈ Z e πin τ + km P n ∈ Z e πin ( τ + k ) = P n ∈ Z ( e πik/m ) n ( e πiτ/m ) n P n ∈ Z ( e πiτ ) n . This implies that the q -expansion is as claimed, by the definitions of ι and φ τ .Note in particular that this q -expansion with ζ m = 1 corresponds to a Z [1 / (4 m )]-rational cusp and has rational coefficients. By the q -expansion principle (in weightzero), we see that Θ m is in fact defined over Z [1 / (4 m )]. That is, Θ m extends to arational function on X (4 m ) Z [1 / (4 m )] with divisor given by (9).In the special case m = l with l and odd prime, we have two more q -expansions.The procedure used to derive these is exactly as illustrated above. We omit the detailsfor brevity. Recall that Θ l descends to X (4 , l ) in this case. Let ζ l be a primitive( l ) th root of unity. Then we haveΘ l (Tate( q ) , ζ , h ζ l i ) = l P n ∈ Z q l n P n ∈ Z q n and Θ l (Tate( q ) , ζ , h ζ l q l i ) = (cid:18) − l (cid:19) g l ( ζ ll ) P n ∈ Z ζ ln l q n P n ∈ Z q n Finally, we will need one more expansion in the special case m = l , and odd prime.Let ζ l be a primitive l th root of unity, and P be the unique point of order 4 l with4 P = ζ l and lP = ζ , thenΘ l (Tate( q ) , P ) = g l ( ζ l ) P n ∈ Z q ln P n ∈ Z q n .
6. Hecke Operators
As explained in the Introduction, in order to construct Hecke operators on thespace M k/ (4 N, R ), we look for rational functions Θ on X (4 N, m ) with the propertythat div(Θ) = π ∗ Σ N,k − π ∗ Σ N,k . The following theorem is the manifestation in this formalism of the classical fact thatin half-integral weight, good Hecke operators occur only for square indices. NICK RAMSEY
Theorem 6.1 . —
Let m be a positive integer with gcd(4 N, m ) = 1 , and let R bea ring in which N m is invertible. Then there exists a rational function Θ on X (4 N, m ) R with div(Θ) = π ∗ Σ N,k − π ∗ Σ N,k if and only if m is a square. In case m is a square, such a Θ is unique up tomultiplication by a constant in R × .Proof . — The uniqueness statement is clear. Suppose that Θ has the specified divisor.Then d ∗ Θ is a rational function on X (4 N m ) with divisor d ∗ ( π ∗ Σ N,k − π ∗ Σ N,k ) = ( π ◦ d ) ∗ Σ N,k − ( π ◦ d ) ∗ Σ N,k . By Proposition 5.1 and the compatibility of Σ N,k with pullback through the standardmap X (4 N m ) −→ X (4 m ) , this is the divisor of Θ km on X (4 N m ). Thus Θ and Θ km agree up to a constant, andthe “only if” statement follows from the fact that Θ km descends through d if and onlyif m is a square (note that we are using that k is odd here, as we must). The “if”statement follows simply because Θ km furnishes a function on X (4 N, m ) with thedesired divisor when m is a square.Let l be a prime not dividing 4 N and let R be a ring in which 4 N l is invertible.We define T l to be the endomorphism of M k/ (4 N, R ) defined by F l π ∗ ( π ∗ F · Θ kl ) . Note that this endomorphism preserves the subspaces M k/ (4 N, R, χ ) for each char-acter χ . The following result says that this definition agrees with the classical one, asin ([ ]). Theorem 6.2 . —
Fix a primitive (4 N ) t h root of unity ζ N and let ∞ denote thecorresponding cusp. Let F ∈ M k/ (4 N, R, χ ) have q -expansion P a n q n at ∞ . Thenthe modular form T l F has q -expansion P b n q n at ∞ where b n = a l n + χ ( l ) (cid:18) − l (cid:19) k − l k − − (cid:16) nl (cid:17) a n + χ ( l ) l k − a n/l . Proof . — We are given that F (Tate( q ) , ζ N ) · X n ∈ Z q n ! k = X a n q n . The quantity π ∗ ( π ∗ F · Θ kl )(Tate( q ) , ζ N )is a sum over the cyclic subgroups of order l on Tate( q ). These subgroups are ofthree types, and we shall compute the corresponding contributions to the above sumone by one. Choose a primitive root ζ l as above. EOMETRIC AND p -ADIC MODULAR FORMS OF HALF-INTEGRAL WEIGHT Firstly, there is the lone subgroup h ζ l i . Passing to the quotient we find(Tate( q ) / h ζ l i , ζ N / h l i ) ∼ = (Tate( q l ) , ζ l N )so that the contribution of this subgroup to the sum is F (Tate( q l ) , ζ l N )Θ l (Tate( q ) , ζ N , h ζ l i ) k = F (Tate( q l ) , ζ l N ) l P n ∈ Z q l n P n ∈ Z q n ! k = χ ( l ) l k P a n q l n ( P n ∈ Z q n ) k where we have used the q -expansion formulas for Θ l form the previous section.Secondly, there are the subgroups h ζ il q l i for 0 ≤ i ≤ l −
1. Note that(Tate( q ) / h ζ il q l i , ζ N / h ζ i N q l i ) ∼ = (Tate( ζ i N q l ) , ζ N ) , so that these subgroups collectively contribute l − X i =0 F (Tate( ζ il q l ) , ζ N )Θ l (Tate( q ) , ζ N , h ζ il q l i ) k = l − X i =0 F (Tate( ζ il q l ) , ζ N ) P n ∈ Z ζ in l q n l P n ∈ Z q n ! k = l − X i =0 P a n ( ζ il q l ) n ( P n ∈ Z q n ) k = l P a l n q n ( P n ∈ Z q n ) k . Lastly, there are the subgroups h ζ jl q l i for 1 ≤ j ≤ l −
1. Note that(Tate( q ) / h ζ jl q l i , ζ N / h ζ jl q l i ) ∼ = (Tate( ζ jl q ) , ζ l N ) . Thus the collective contribution of these terms is l − X j =1 F (Tate( ζ jl q ) , ζ l N )Θ l (Tate( q ) , ζ N , h ζ jl q l i ) k = l − X j =1 F (Tate( ζ jl q ) , ζ l N ) (cid:18) − l (cid:19) g l ( ζ jl ) P n ∈ Z ζ jn l q n P n ∈ Z q n ! k Note that g l ( ζ jl ) = (cid:18) jl (cid:19) g l ( ζ l ) , NICK RAMSEY so that the above continues as χ ( l ) (cid:18) − l (cid:19) (cid:18) g l ( ζ l ) P n ∈ Z q n (cid:19) k l − X j =1 (cid:18) jl (cid:19) F (Tate( ζ l q ) .ζ N ) X n ∈ Z ( ζ jl q ) n = χ ( l ) (cid:18) − l (cid:19) (cid:18) g l ( ζ l ) P n ∈ Z q n (cid:19) k l − X j =1 (cid:18) jl (cid:19) X a n ( ζ jl q ) n = χ ( l ) (cid:18) − l (cid:19) (cid:18) g l ( ζ l ) P n ∈ Z q n (cid:19) k X n a n l − X j =1 (cid:18) jl (cid:19) ζ jnl q n = χ ( l ) (cid:18) − l (cid:19) g l ( ζ l ) k +1 (cid:0)P n ∈ Z q n (cid:1) k X n (cid:16) nl (cid:17) a n q n . It is well known that the Gauss sum above squares to g l ( ζ l ) = (cid:18) − l (cid:19) l, so we may further continue the above as χ ( l ) (cid:18) − l (cid:19) k − l k +12 (cid:0)P n ∈ Z q n (cid:1) k X n (cid:16) nl (cid:17) a n q n . Adding these three expressions together and dividing by l we arrive at the desired q -expansion for T l F .Suppose now that l is an odd prime dividing N . Consider the map e defined asfollows. e : X (4 N, l ) −→ X (4 l )( E, P, C ) ( E/C, ( N/l ) P/C ) . By chasing degeneracy maps around, one verifies that e ∗ (div(Θ − l )) = π ∗ Σ N, − π ∗ Σ N, , so that we get a well-defined endomorphism of M k/ (4 N, R ), namely, U l ( F ) = 1 l π ∗ ( π ∗ F · e ∗ Θ − kl ) Theorem 6.3 . —
Let l be an odd prime dividing N . Fix a primitive (4 N ) th rootof unity ζ N and let ∞ denote the corresponding cusp. The endomorphism U l of M k/ (4 N, R ) has the effect X a n q n g l ( ζ N/l N ) X a ln q n on q -expansions at ∞ . EOMETRIC AND p -ADIC MODULAR FORMS OF HALF-INTEGRAL WEIGHT Proof . — Note that the collection of subgroups of Tate( q ) of order l not meeting thesubgroup generated by ζ N is h ζ il q l i , for 0 ≤ i < l −
1. We compute π ∗ ( π ∗ F · e ∗ Θ − kl )(Tate( q ) , ζ N )= l − X i =0 ( π ∗ F · e ∗ Θ − kl )(Tate( q ) , ζ N , h ζ il q l i )= l − X i =0 F (Tate( q ) / h ζ il q l i , ζ N / h ζ il q l i )Θ(Tate( q ) / h ζ il q l i , ζ N/l N / h ζ il q l i ) − k = l − X i =0 F (Tate( ζ il q l ) , ζ N )Θ(Tate( ζ il q l ) , ζ N/l N ) − k = l − X i =0 F (Tate( ζ il q l ) , ζ N ) g l ( ζ N/l N ) P n ∈ Z q n P n ∈ Z ( ζ il q l ) n ! − k = g l ( ζ N/l N ) − k l − X i =0 P a n ( ζ il q l ) n ( P n ∈ Z q n ) k = l g l ( ζ N/l N )( P n ∈ Z q n ) k X a ln q n dividing by l we get the desired result.Note in particular that the effect of the map U l on q -expansions depends on which ∞ is chosen, whereas that of its square ( U l ) does not (since the square of the Gausssum is independent of the root of unity used). On a related note, U l (unlike ( U l ) or T l ) does not preserve the subspaces M k/ (4 N, R, χ ) but rather induces maps U l : M k/ (4 N, R, χ ) −→ M k/ (4 N, R, χ · ( l/ · ))where ( l/ · ) is the usual Legendre character. Remark 6.4 . — One can also construct operators U and (if N is even), U . Pre-dictably, the operator U multiplies the Nebentypus by (2 / · ). One can also showthat Kohnen’s +-space makes good sense in this more general setting. That is, onecan construct a natural operator (closely related to U ) on the space of forms, ofwhich Kohnen’s +-space is an eigenspace. The details of these constructions are inthe author’s thesis ([ ]). p -adic modular forms of half-integral weight In this section we fix a rational prime p ≥
5. Let K be a complete subfield of C p and let r ∈ Q ∩ [0 , M k/ (4 N, K, r )of r -overconvergent p -adic modular forms of weight k/ N , as well as anumber of natural operators on this space. For the sake of simplicity of exposition,we restrict ourselves to the case p ∤ N .By our choice of p , the weight p − E p − furnishesa lifting the Hasse invariant to characteristic zero. It is well-known that E p − hasFourier coefficients in Q ∩ Z p and therefore furnishes a section of ω p − on X ( N ) Z p NICK RAMSEY for any N ≥
4. Note that this makes sense even in case N = 4 because the bundle ω does descend from the moduli stack for the Γ (4) problem to the coarse modulischeme X (4) (even though ω does not ).For N ≥
4, let X (4 N ) an K denote the rigid analytic space over Q p associated to thecurve X (4 N ) K . For any r as above, we may also consider the region in this curvedefined by the inequality | E p − | ≥ r . This is known to be a connected affinoid subdo-main of X (4 N ) an K , and will be denoted by X (4 N ) an ≥ r . Thus, in particular, X (4 N ) an ≥ is the “ordinary locus” in X (4 N ) an K . Similarly, one may consider X (4 N, m ) an ≥ r . Thisspace is connected exactly when gcd( m, p ) = 1 (recall that we are assuming that p ∤ N ). For these facts and others which we will freely use in the sequel, see [ ] and[ ]. Definition 7.1 . — The space of p -adic modular forms of weight k/
2, level 4 N ( p ∤ N ), and growth condition r over K is M k/ (4 N, K, r ) = H ( X (4 N ) an ≥ r , Σ N,k ) . Each of the spaces M k/ (4 N, K, r ) is a (generally infinite-dimensional) Banachspace over K . As usual, forms in the spaces above with r < overcon-vergent . The space of all overconvergent forms is accordingly the inductive limit M † k/ (4 N, K ) = lim r → M k/ (4 N, K, r ) . The curve Tate( q ) can also be thought of as a p -adic analytic family of ellipticcurves over the punctured unit disk 0 < | q | <
1. As usual, in the presence of level 4 N structure, we instead consider a parameter q N and define q = q N N . Then we mayconsider the family Tate( q ) over the punctured disk 0 < | q N | <
1, so that all of the4 N -torsion is rational after adjoining the 4 N th roots of unity.If F ∈ M k/ (4 N, K, r ) and P is a point of order 4 N on Tate( q ) then we may evaluate F on the pair (Tate( q ) , P ) to obtain a meromorphic function on the punctured disk0 < | q N | < K (( q N )). Just as in the algebraiccase, we can define the q -expansion of F at this pair (Tate( q ) , P ) be adjusting thisexpansion by the corresponding expansion of θ from Table 6. Thus we find, just as inthe algebraic case, that a meromorphic function on X (4 N ) an ≥ r is in M k/ (4 N, K, r )if and only if it is analytic away from the cusps and all of these q -expansions are in K [[ q N ]]. p -adic Hecke operators Our aim is to define a Hecke action on the spaces M k/ (4 N, K, r ). The situation israther different depending on whether or not we are dealing with the Hecke operator“at p ”, but in all cases we use a variant of the “pull-back, multiply by a unit, andpush forward” construction above relative to the usual correspondence π , π : X (4 N, m ) an K ⇒ X (4 N ) an K . EOMETRIC AND p -ADIC MODULAR FORMS OF HALF-INTEGRAL WEIGHT Suppose Z and Z are admissible opens in X (4 N ) an K and W is an admissible openin X (4 N, m ) an K such that the π i restrict to a pair of maps W π ~ ~ |||||||| π BBBBBBBB Z Z Suppose further that π − ( Z ) ⊆ π − ( Z ) and Θ is a meromorphic function on W with div(Θ) ≥ π ∗ Σ N,k − π ∗ Σ N,k . These data furnish us with a map H ( Z , Σ N,k ) −→ H ( Z , Σ N,k )defined as the composition H ( Z , Σ N,k ) π ∗ / / H ( π − ( Z ) , π ∗ Σ N,k ) · Θ / / H ( π − ( Z ) , π ∗ Σ N,k ) EDBCGF@A / / H ( π − ( Z ) , π ∗ Σ N,k ) π ∗ / / H ( Z , Σ N,k )where the long arrow is restriction.This setup suffices for the case m = l with gcd( l, N p ) = 1. Then we may take Z = Z = X (4 N ) an ≥ r , W = X (4 N, l ) an ≥ r , and Θ = Θ kl . The two conditions thatmust be satisfied, namely that the π i restrict as above and that π ( Z ) ⊆ π − ( Z ),together are equivalent to the assertion that if C is a cyclic subgroup of E of order l , then | E p − ( E ) | ≥ r if and only if | E p − ( E/C ) | ≥ r . Of course, it suffices to verifythe “only if” part since we may then apply the result to the pair ( E/C, E [ l ] /C ) toverify the other implication. At any rate, the result is well known and holds for cyclicsubgroups of all orders prime to p . Theorem 8.1 . —
There exists a continuous endomorphism T l of M k/ (4 N, K, χ, r ) having the effect X a n q n X b n q n where b n = a l n + χ ( l ) (cid:18) − l (cid:19) k − l k − − (cid:16) nl (cid:17) a n + χ ( l ) l k − a n/l on q -expansions at (any choice of ) ∞ .Proof . — Dividing the endomorphism of M k/ (4 N, K, r ) obtained from the abovechoice of Θ by l we arrive at a new endomorphism denoted T l . The effect of T l on q -expansions at ∞ is exactly as in the algebraic case, as the same computationverifies. NICK RAMSEY
We claim that k T l k ≤
1, so that, in particular, T l is continuous. That the normsof π ∗ and π ∗ are at most one are generalities. It therefore suffices to show thatmultiplication by Θ kl as a map H ( π − ( Z ) , π ∗ Σ N,k ) −→ H ( π − ( Z ) , π ∗ Σ N,k )is bounded by 1.This follows immediately from the fact that Θ kl is an integral section of the sheaf O X (4 N,l ) ( π ∗ Σ N,k − π ∗ Σ N,k )on the algebraic curve X (4 N, l ) K (i.e., that it extends to a section over all of X (4 N, l ) O K ) as the q -expansion principle readily demonstrates.The more interesting case to consider is that of m = p . If r > p − /p (1+ p ) , thenthe existence of the canonical subgroup of order p (see [ ],[ ]) furnishes a map X (4 N ) an ≥ r −→ X (4 N, p ) an ≥ r which is an isomorphism onto the connected component of right-hand space containingthe cusp associated to (Tate( q ) , ζ N , µ p ) (for any primitive choice of ζ N ), the inversebeing π . We will denote this component by W ≥ r . Standard results on quotientingby canonical subgroups show that, for r as above, π restricts to a map π : W ≥ r −→ X (4 N ) an ≥ r p . We will denote the restrictions of the various modular units and divisors on X (4 N, p ) an K to W ≥ r by their usual names. Then we have, as usual, thatdiv(Θ − kp ) = π ∗ Σ N,k − π ∗ Σ N,k . Thus we have a well-defined map M k/ (4 N, K, r ) −→ M k/ (4 N, K, r p ) F p i ∗ π ∗ ( π ∗ F · ( p Θ p ) − k )Let us determine the effect of this map on q -expansions. To compute the trace π ∗ we note that the fiber of π above the pair ( E, P ) consists of all (isomor-phism classes of) triple ( E ′ , P ′ , C ) such that C is the canonical subgroup of E ′ of order p , and ( E ′ /C, P ′ /C ) ∼ = ( E, P ). This is the same as the set of triples(
E/C ′ , mP/C ′ , E [ p ] /C ′ ) where C ′ runs through all order p cyclic subgroups of E which have trivial intersection with its canonical subgroup of order p (i.e. that donot contain the order p canonical subgroup of E ), and m is an inverse of p mod 4 N . EOMETRIC AND p -ADIC MODULAR FORMS OF HALF-INTEGRAL WEIGHT So we may compute( h p i ∗ π ∗ ( π ∗ h p i ∗ F · ( p Θ p ) − k ))(Tate( q ) , ζ N )= p − X i =0 ( π ∗ F · ( p Θ p ) − k )(Tate( q ) / h ζ ip q p i , ζ N / h ζ ip q p i , Tate( q )[ p ] / h ζ ip q p i )= p − X i =0 F (Tate( ζ ip q p ) , ζ N )( p Θ p (Tate( ζ ip q p ) , ζ N , µ p )) − k = p − X i =0 F (Tate( ζ ip q p ) , ζ N ) P n ∈ Z q n P n ∈ Z ( ζ ip q p ) n ! − k = p P a p n q n ( P n ∈ Z q n ) k , where P a n q n is the q -expansion of F at (Tate( q ) , ζ N ). Theorem 8.2 . —
Suppose that p − /p (1+ p ) < r . Then there exists a continuous en-domorphism U p of M k/ (4 N, K, r ) of norm at most p having the effect X a n q n a p n q n on q -expansions at (any choice of ) ∞ . Moreover, in case r < , U p is in factcompletely continuous.Proof . — The endomorphism we seek is the map M k/ (4 N, K, r ) −→ M k/ (4 N, K, r p )defined above divided by p and post composed with the natural inclusion M k/ (4 N, K, r p ) ⊆ M k/ (4 N, K, r ) . As this latter map is completely continuous if r <
1, it suffices to show that the map M k/ (4 N, K, r ) −→ M k/ (4 N, K, r p )is continuous. As in the case for T l above, it suffices to see that multiplication bythe unit ( p Θ p ) − k is bounded as a map H ( W ≥ r , π ∗ Σ N,k ) −→ H ( W ≥ r , π ∗ Σ N,k ) . This is more subtle than the case of T l because of the bad reduction of X (4 N, p ).Consider the modular unit ( p Θ p ) − on the algebraic curve X (4 N, p ) K , and let X (4 N, p ) O K denote the Katz-Mazur model of this curve over the ring of integers O K in K . We claim that ( p Θ p ) − extends to all of X (4 N, p ) O K as a section of thesheaf O X (4 N,p ) O K ( π ∗ Σ N, − π ∗ Σ N, ) . Since X (4 N, p ) O K is Cohen-Macaulay over O K it suffices by the q -expansion princi-ple to show that ( p Θ p ) − has integral q -expansions at a collection of cusps which, to-gether, meet all three components of the special fiber of X (4 N, p ) in characteristic p . NICK RAMSEY
In particular, we fix primitive roots ζ N and ζ p and take the three cusps correspond-ing to (Tate( q ) , ζ N , h ζ p i ), (Tate( q ) , ζ N , h ζ p q p i ), and (Tate( q ) , ζ N , h q p i ). That theindicated q -expansions are integral follows readily from the results of Section 5.It follows that multiplication by ( p Θ p ) − k is of norm at most 1, and thereforecontinuous. Taking into account the division by p , we see that k U p k ≤ p . References [1]
K. Buzzard – Analytic continuation of overconvergent eigenforms,
J. Amer. Math. Soc. (2003), no. 1, p. 29–55 (electronic).[2] R. Coleman & B. Mazur – The eigencurve, in
Galois representations in arithmeticalgebraic geometry (Durham, 1996) , London Math. Soc. Lecture Note Ser., vol. 254, Cam-bridge Univ. Press, Cambridge, 1998, p. 1–113.[3]
R. F. Coleman – p -adic Banach spaces and families of modular forms, Invent. Math. (1997), no. 3, p. 417–479.[4]
N. M. Katz – p -adic properties of modular schemes and modular forms, in Modularfunctions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp,1972) , Springer, Berlin, 1973, p. 69–190. Lecture Notes in Mathematics, Vol. 350.[5]
N. M. Katz & B. Mazur – Arithmetic moduli of elliptic curves , Annals of MathematicsStudies, vol. 108, Princeton University Press, Princeton, NJ, 1985.[6]
N. Ramsey – Geometric and p -adic modular forms of half-integral weight, HarvardUniversity Thesis (2004).[7] , The half-integral weight eigencurve,
Algebra Number Theory (2008), no. 7,p. 755–808.[8] , The overconvergent Shimura lifting, Int. Math. Res. Not. IMRN (2009), no. 2,p. 193–220.[9]
G. Shimura – On modular forms of half integral weight,
Ann. of Math. (2) (1973),p. 440–481. Nick Ramsey , Department of Mathematics, University of Michigan