On -adic harmonic Maass functions
Modular and mock modular forms possess many striking -adic properties, as studied by Bringmann, Guerzhoy, Kane, Kent, Ono, and others. Candelori developed a geometric theory of harmonic Maass forms arising from the de Rham cohomology. In the setting of over-convergent -adic modular forms, Candellori and Castella showed this leads to -adic analogs of harmonic Maass forms.
In this paper we take an analytic approach to construct -adic analogs of harmonic Maass forms of weight with square free level. Although our approaches differ, where the two theories intersect the forms constructed are the same. However our analytic construction defines these functions on the full super singular locus.
As with classical harmonic Maass forms, these -adic analogs are connected to weight cusp forms and their modular derivatives are weight weakly holomorphic modular forms. Traces of their CM values also interpolate the coefficients of half integer weight modular and mock modular forms. We demonstrate this through the construction of -adic analogs of two families of theta lifts for these forms.
1. Introduction and statement of results
Serre  introduced the notion of a -adic modular form as the limit of a sequence of modular forms with -adically convergent -expansions. This theory has been expanded by Dwork , Katz , Hida  and many others, filling out a beautiful picture in terms of the analysis and geometry of the modular curve and the Hecke algebra.
The -adic properties of mock modular forms are less well-studied. However Bringmann, Guerzhoy, Kane, Kent, Ono, and others (see for instance [5, 6, 23, 24]) have demonstrated a number of striking examples. In his masters thesis, Candelori  defined a -harmonic differential for over-convergent -adic modular forms, and by means of the de Rham cohomology defined -adic analogs of -harmonic Maass forms of weight . In , Candelori presented a geometric theory of harmonic Maass forms. Later Candelori and Castella  considered certain -adic modular forms studied by Bringmann–Guerzhoy–Kane  associated to certain mock modular forms. They showed that these are over-convergent -adic modular forms arising from the de Rham cohomology in the same way as do harmonic Maass forms. From this geometric perspective, Candellori and Castella also reproduced results of Guerzhoy-Kent–Ono  on the -adic coupling of mock modular forms with their shadow.
Similar -adic properties are exhibited by both integer and half-integer weight mock modular forms. In fact, some evidence suggests many of theses properties carry through certain lifts from integer weight forms to half integer weight forms, including for instance the lifts studied by Zagier in , which give the Fourier coefficients of half-integer weight forms as twisted “traces” over CM points of fixed discriminants. Unfortunately, we cannot simply apply these traces to general -adic modular forms or over-convergent -adic modular forms, since for sufficiently large discriminants these traces will eventually involve points with super singular -invariants.
Here we construct -adic analogs of weight harmonic Maass forms from an analytic perspective, from sequences of classical modular functions. These functions converge on the full modular curve, including the super-singular locus. When the forms constructed have shadows which are ordinary for (see Section 5), then these functions align with those studied by Candelori and Castella. In other cases, the -adic harmonic Maass forms we construct are not standard -adic modular forms, as witnessed by the presence of unbounded denominators in the -series expansions.
If is a field over , we will use the notation and respectively to denote the space of weight holomorphic and weakly holomorphic modular forms on the modular curve (see Section 2). We also consider classical weight harmonic Maass forms as functions on Here and throughout, if is a number field we will use to denote an infinite place of , and to denote a finite place lying over a rational prime . We also denote the completion of at any place by . With this notation we have the following.
Suppose is a square free positive integer, a number field, and a prime of over a rational prime , not dividing . There exists a vector space of -adically continuous functions , satisfying the following properties:
We have that
Hecke operators and Atkin–Lehner involutions have well defined actions on and are endomorphisms.
Each function in has a well-defined -expansion in .
The usual modular differential operator acts on , and the following sequence is exact:
Each function in this space is then defined using two of sequences of modular functions which converge on overlapping regions which cover We construct these sequences using the Hecke algebra. The action of the Hecke operators, Atkin–Lehner operators and the differential operator are obtained by acting component-wise on the defining sequences.
A correspondence exists between certain functions in and in If is any embedding of fields , then extends naturally and uniquely to an embedding
so that Similarly, if and are two fields containing , then there is a natural equivalence relation between forms which satisfy
For instance, suppose is a number field. If has a -expansion in , then there is a unique corresponding form so that
We will establish a similar correspondence between a certain dense subspace of and respectively. This aim is complicated by the generally expected transcendence of the -series coefficients and values at algebraic points of these functions. However, this transcendence can be controlled in some key settings.
For instance, suppose is a level newform with -expansion given by . Then there is a subspace of harmonic Maass forms satisfying the following properties:
The differential operator defined in (3.1) acts on with
where is the usual Petersson norm of .
The holomorphic part of (see section 3.1) at each cusp has a -expansion in .
The principal parts of at each cusp is in
Of course we could have written instead of in part (2), but this notation suggests the direction in which we will generalize these properties for .
Suppose , with the -expansion of at given by
The -module containing the coefficients has rank at most : there is some so that
This method of controlling the transcendental part using the coefficients of the cusp form underlies the -adic coupling of mock-modular forms with their shadow as demonstrated by Guerzhoy–Kent–Ono [24, Theorem 1.2].
Similarly, in general we expect the values of at algebraic points to be transcendental, but there are important exceptions. The CM values of the function, known as singular moduli, are algebraic. Zagier  showed that the twisted traces of these singular moduli are coefficients of certain weight and weight modular forms. These results have been studied and generalized in several directions, including for other modular functions of higher level, and for harmonic Maass forms. Bruinier–Funke [9, 10] and Alfes  have realized these trace maps as theta lifts obtained by taking the inner product of modular functions against certain non-holomorphic theta kernels. In many cases it is not hard to see that certain -adic properties of -series are propagated through the lifts. Some of these properties have been explored by Bringmann–Guerzhoy–Kane .
Suppose and are fundamental discriminants, both squares modulo with . Let be chosen so that We define a twisted modular trace of values of at CM points of discriminant in equation (8.2). If is weakly holomorphic with rational coefficients, then this trace is literally a trace over Galois conjugates of CM values of . If is not weakly holomorphic, the algebraicity is connected to the twisted -function
By , we denote the derivative in the variable. Results of Bruinier–Ono and Alfes can be used to control the transcendence of the traces. We package these results in the following proposition.
For all and satisfying the conditions above and
we have that
More generally, there exist constants (dependent on , but independent of the choice of ), and some (independent of ) so that
In particular, when is weakly holomorphic, we can take The in the equation are coefficients of a certain weight modular form which corresponds to under the Shimura correspondence. The traces themselves can be given in terms of coefficients of a weight harmonic Maass forms, and so (1.4) strongly parallels (1.1). Equations (1.3) and (1.4) are tied together by Waldspurger’s theorem which shows that
If is square free, and not divisible by the prime , we define a space of -adic harmonic Maass forms which satisfy these same properties discussed above for -expansions and CM values, with replaced with . Additionally, these forms satisfy a natural correspondence with the space given by matching forms with the same principal parts at cusps.
Assume the notation above, with any infinite place of , and any finite prime of not dividing . There exists a subspace satisfying a one-to-one correspondence with . This correspondence maps each function to a corresponding , satisfying the following properties.
The principal parts of the -expansions of and at all cusps are equal.
If and denote the -th coefficients of and respectively, then there exist and , so that
More generally, there exist and so that
where is as above.
The above correspondence is equivariant with respect to the Hecke algebra and Atkin–Lehner involutions.
By transitivity, the correspondence given in Theorem 1.3 extends to any two places and of finite or infinite which do not divide . Building on the earlier notation, we write for functions which correspond in this manner.
In Section 5 we will outline various structural results about the space . We will show that this space can be generated by the action of the Hecke algebra and Atkin–Lehner involutions acting on a single element. We show that the image under the modular derivative is a distinguished subspace of , orthogonal to the weight cusp forms under a natural -adic analog of the Petersson inner product. This space is distinguished by the -adic slopes of the forms (see (5.1)). When the slope is not negative, these align with the over convergent -adic modular forms of Candelori and Castella’s theory.
For primes dividing , we can define a similar space which nearly satisfies Theorem 1.1, and subspaces which satisfy Theorem 1.3 parts (1),(2), and (4). However, aside from weakly holomorphic modular functions, the functions in these spaces are not well defined on the supersingular locus. Away from the supersingular locus, the results are -adic modular forms. The construction also converges on the supersingular locus, but the results seem to be incomplete. The limits branch depending on the cusp at which the expansion is taken, and the construction might be termed at best mock modular. We will primarily focus on the case does not divide , with a few exceptions in Theorem 1.4 below.
The correspondence described in Theorem 1.3 and Remark 1 between places of raises the natural question if there is an adelic theory connecting these forms. This question requires bounds on the denominators that can arise.
Let be a family of functions which are equivalent under , and such that the principal part of each at each cusp is defined over the ring of integers of Denote the -expansion of at a cusp by Then there are integers , , and explicitly defined in section 6, all independent of the family so that the following are true.
Suppose is a finite place of . Given any cusp and a positive integer with square free, we have
In particular, for fixed and cusp the vector is an adele.
Suppose , with not supersingular at if divides . Then is -integral.
The bound on the denominators for evaluations is not sharp when is weakly holomorphic, and may not be sharp in general. Improvements in this bound could be used to improve bounds for denominators appearing in algebraic coefficients of weight harmonic Maass forms.
Let , , and , and let be the unique newform for with rational coefficients,
There is a unique weight harmonic Maass form for with the -expansion at given by
which is invariant under the Fricke involution. Then where
and The form has the -expansion
and satisfies where . Here we have represented each -adic number in base format so that, for instance,
Now let For and chosen as in Proposition 1.2, the traces are the coefficients of a weight vector valued harmonic Maass form , as seen in Theorem 8.1. However for simplicity in this example we will consider a projection of this vector valued form to a scalar-valued form obtained by summing the vector components and multiplying by . Then , and lies in the Kohnen plus space. The newform
maps to under the Shimura correspondence. We have , where
The remainder of this paper will be organized as follows. In Section 2 we review basic results about modular functions used throughout this paper. In Section 3 we review the theory of harmonic Maass forms. Section 4 contains the construction of the -adic harmonic Maass forms and the proofs of Theorem 1.1 and Theorem 1.3 parts (1),(2), and (4). Section 5 contains additional results about the structure of the spaces of -adic harmonic Mass forms that we will find useful later. In Section 6 we prove the integrality results for the -series and values given in Theorem 1.4. In Section 7 we will review the theory of half-integer weight vector-valued modular forms and Hecke operators. In Section 8 we will review the lifts connecting integral weight and half-integral weight forms and prove Proposition 1.2. In Section 9 we extend the lifts studied in the previous section to the -adic harmonic Maass forms. Part (3) of Theorem 1.3 will follow as a corollary to Theorem 9.1.
This research was was supported by the National Science Foundation grant DMS-1502390 and by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant agreement n. 335220 - AQSER. It was conducted during postdoctoral work at Princeton University and at the University of Cologne. The author thanks these institutions for their support, with special thanks to his postdoctoral advisors Professor ShouWu Zhang (Princeton) and Professor Kathrin Bringmann (Cologne).
The author is also grateful to Claudia Alfes and Luca Candelori for helpful discussions, and to Jonas Kaszian, Michael Mertens, Grant Molnar, and Michael Woodbury for their comments on earlier versions of this paper.
2. Modular functions
Throughout this paper, we will treat modular forms interchangeably as functions on elliptic curves, line bundles over the modular curve, as formal -series, and in the complex case as functions in the complex variable in the upper half plane. We will treat harmonic Maass forms similarly.
Given a model of an elliptic curve , let and be periods which generate the associated lattice, ordered so that . If is modular of weight and level , then we have that
If has level then different choices of generators of the lattice may give different evaluations. A choice of level structure is a choice among the -equivalence classes of periods which generate the lattice.
As usual any rational matrix with positive determinant acts on modular forms over by
Regardless of the field of definition of the forms under consideration, the matrix group acts as an algebra of linear operators on modular forms, where the image of acts trivially on level modular forms. This algebra contains both the Hecke algebra and the group of Atkin–Lehner involutions.
Equivalently, we may consider the evaluation of modular forms algebraically. Let be the set of Weierstraas models of elliptic curves over with a specified level structure. If , the evaluations of the Eisenstein series and can be read from the Weierstrass model. This suffices to evaluate any level meromorphic modular form. If is a modular function of level then it is related to the -function by a polynomial for some field , defined by
Here the matrices act by permuting the level structure of the input. Because the action by any matrix in simply permutes the cosets, the coefficient functions must all be level , and hence rational functions in . Moreover, must be a perfect power of an irreducible polynomial. The level structure of then specifies an evaluation of among the roots of
The geometry of the modular curve gives a more uniform characterization of the level structure. The modular curve is a smooth affine curve over , which satisfies and [32, Theorem 13.1].
We may fix a model
So that the projection gives the standard projection to each and each satisfies a monic polynomial Modular functions are on the curve are can be given as rational functions in the coordinates, and weakly holomorphic forms are polynomials,
Each coordinate gives the value of an associated modular function Up to a linear change of variable we may take
If is a field with ring of integers , we define the integral modular functions
It will be useful to fix a complete integral model of , so that
More generally, if is any fractional ideal of then define the submodule by
The modular curve has certain important regions. If is a prime of , then the -integral locus is the region The -integral locus splits into two distinguished subregions: the supersingular locus
and the complement, the -ordinary locus.
2.1. -series and the Tate curve
The -expansion of a weakly holomorphic modular form corresponds to the evaluation of on a model of the Tate curve. The various level models of the Tate curve correspond to the action of a matrix in on a level modular form, and is related to the -expansions of at the various cusps.
The inequivalent cusps of with square-free can be indexed by the divisors of , with cusp indexed by . The cusp then has index , while the cusp has index . The Atkin–Lehner involutions permute these cusps. Here, can be represented by any integer matrix with determinant . Then swaps the cusp of index with that of index .
We denote the standard series of a modular form at the cusp by . This corresponds to the model of the Tate curve which satisfies
where is the image of under the Fricke involution In the complex case, this model corresponds to the usual Fourier expansion at . More generally, each -model of the Tate curve corresponds to the action of some right-coset representative . The resulting action on the -expansion of a modular form can be found by factoring
for some Atkin–Lehner involution with , and . Since is square-free in our case, and are co-prime, the factorization is well defined. The action of an upper triangular matrix on a -series is simply with a fixed primitive -th root of . Therefore
The -expansion principle allows us to use the various models of the Tate curve to evaluate a modular form when for some place of . Given such a curve and any model of the Tate curve, there are parameters and in with so that for every form we have
Evaluating at the Tate curve easily shows that the the integral forms are exactly those level modular functions whose coefficients at all cusps are in .
2.2. The Hecke algebra
For our construction we will need an extension of the Hecke algebra
generated by the standard Hecke operators for , the Atkin -operators for divisible only by primes dividing , and the Atkin–Lehner involutions for . For this is a non-commutative algebra. While operators with coprime index commute, the and operators for primes have non-trivial commutativity relations which can be worked out in terms of the action of matrices.
The operator satisfies
The operators and both satisfy the polynomial relation
The action of on a -expansion is that of
where as usual sends
We will find it useful to define the operators
where , is the greatest divisor of with and A short exercise then shows that the weight and weight operators satisfy the same multiplicative relation
In particular we have an isomorphism
The normalizations for the non-positive weight operators also preserve integrality of -expansions. If , this follows easily from the formula in terms of the and operators,
where or depending on whether or respectively. If , then the action on -expansions of can be worked out using (2.5). We find
The normalizations also allow simpler commutativity relations with the modular differential operators and defined in the next section.
3. Harmonic Maass forms
In this section we define harmonic Maass forms and lay out certain key properties that will be used later. We begin by recalling the definition of harmonic Maass forms of weight Here we set with and real, and . The weight hyperbolic Laplacian is defined by
Let for some , and let . Then a real analytic function is a harmonic Maass form of weight for if:
The function is invariant under the slash operator so that
for every matrix
The function is harmonic so that
The function has a meromorphic principal part at each cusp. That is, if is the expansion of at , then there is some polynomial and constant so that as
We denote the vector space of weight harmonic Maass forms for by . The differential equation given by implies that harmonic Maass forms have Fourier expansions which split into two components: one part which is a holomorphic -series, and one part which is non-holomorphic.
Lemma 3.1 ([9, Proposition 3.2]).
Let be a harmonic Maass form of weight for as defined above. Then we have that
where is the holomorphic part of or mock modular form, given by
and is the non-holomorphic part given by
3.1. Differential operators and the Petersson inner product
Differential operators yield some important relations between spaces of harmonic Maass forms and weakly holomorphic modular forms of dual weight. Let be an even and define the operators
These maps yield the exact sequences
Here, the space is a distinguished subspace of consisting of those forms with vanishing constant term at all cusps and which are orthogonal to the cusp forms with respect to the regularized Petersson inner product described below.
The operator preserves inegrality of coefficients, and so extends to a map
As noted before, the operators commute with these differential operators. If and , then
The same relations hold for the Atkin-Lehner involutions .
The Petersson inner product is defined by the regularized integral
Borcherds’ regularization of the inner product (see ) allows the inner product to make sense if we have growth towards the cusps. The normalization by the group index ensures that the inner product is independent of the level.
Bruinier–Funke define a pairing connected to the inner product, defined by
This pairing, and therefore the resulting inner product, can be computed in terms of the coefficients of harmonic Maass forms.
Theorem 3.2 (Bruinier–Funke).
Let and with -expansions at cusps given by and Then
The pairing is a sum of the constant terms at cusps of the non-holomorphic weight modular form . The formula presented here differs slightly from Bruinier and Funke’s original statement which is given in terms of vector valued forms. The formula for the pairing is more easily recognized as a sum over cosets,