A note on cusp forms as p-adic limits

A note on cusp forms as -adic limits

Scott Ahlgren Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA sahlgren@illinois.edu  and  Detchat Samart Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA dsamart@illinois.edu
July 20, 2019
Abstract.

Several authors have recently proved results which express cusp forms as -adic limits of weakly holomorphic modular forms under repeated application of Atkin’s -operator. The proofs involve techniques from the theory of weak harmonic Maass forms, and in particular a result of Guerzhoy, Kent, and Ono on the -adic coupling of mock modular forms and their shadows. Here we obtain strengthened versions of these results using techniques from the theory of holomorphic modular forms.

Key words and phrases:
Modular forms, Cusp forms as -adic limits
2010 Mathematics Subject Classification:
11F33, 11F11, 11F03
The first author was supported by a grant from the Simons Foundation (#208525 to Scott Ahlgren).

1. Introduction

In a recent paper [4], El-Guindy and Ono study a cusp form and a modular function related to the elliptic curve . Following their notation, define

(1.1)
(1.2)
(1.3)

The main result of [4] states that if is a prime for which , then as a -adic limit, we have

(1.4)

The proof involves the theory of harmonic Maass forms, and in particular a result of Guerzhoy, Kent, and Ono [5] on the -adic coupling of mock modular forms and their shadows. Similar results were proved in [5] and [2].

Our goal is to prove strengthened versions of these results. We use a direct method; it does not involve harmonic Maass forms but rather an investigation of the action of the Hecke operators on a family of weakly holomorphic modular forms. A similar approach was recently employed in the study of the congruences of Honda and Kaneko [1]. For the modular forms described above, we prove the following, of which (1.4) is an immediate corollary. Note in addition that the case of (1.5) gives . Let denote the -adic valuation on .

Theorem 1.1.

Let be prime. Then for all integers we have

(1.5)
(1.6)

In Theorems 4.1 and 5.1 below we obtain similar improvements of results given in [5] and [2]. It is clear that the present approach would give similar results for a number of other spaces of modular forms.

2. Background

If is an integer, is a function of the upper half-plane, and , we define

If , , and is a Dirichlet character modulo , let be the space consisting of functions which satisfy for all and which are holomorphic on the upper half plane and at the cusps. Let be the space of forms which are meromorphic at the cusps, and let denote the subspace of forms which are holomorphic at all cusps of other than . We drop the character from this notation when it is trivial. Each can be identified with its -expansion; with we have for some coefficients .

For each positive integer , the and -operators are defined on -expansions by

Let be the usual Hecke operator on . If is prime, then for and we have

(2.1)

Define

Lemma 2.1.

If , then we have

(2.2)

If then

(2.3)
Proof.

For the first statement, it suffices to show that for each prime we have

We have

(2.4)

Let be a cusp of inequivalent to and choose with . Given set . By a standard argument (see e.g. [6, §6.2]) we find that

where the first matrix on the right is in . It follows that each term from the sum on in (2.4) is holomorphic at cusps other than . To see that the last summand is also holomorphic at these cusps, let . Then

where the first matrix on the right is in .

Let be the Maass raising operator in weight , so that we have the basic relation

Bol’s identity (see for example [3, Lemma 2.1]) states that for we have

It follows that

and that

The claim (2.3) follows from these two facts. ∎

If and , let denote the subset of consisting of forms whose coefficients are -integral rational numbers. If , define the filtration

We require two facts, which can be found for example in [7, §1]. First, if and , then . Also, we have

(2.5)

3. Proof of Theorem 1.1

Recall the definitions (1.1)–(1.3), and note that and in the notation of the next proposition.

Proposition 3.1.

We have the following.

  1. For every odd integer there exists a unique of the form

  2. Suppose that is an odd prime and that . Then

Proof.

For each integer , let

Using standard criteria (see, e.g. [8, Thm. 1.64, Thm. 1.65]) we find that . The forms can then be constructed as linear combinations of forms with . Uniqueness follows since the space is one-dimensional. This gives the first assertion.

From (2.1) we have

Observe that

and that

Assertion (2) follows from assertion (1) together with Lemma 2.1. ∎

Before proving Theorem 1.1 we require two lemmas.

Lemma 3.2.

For each prime and each integer we have

Proof.

Lemma 2.3 and Corollary 2.4 of [4] show that for each , there is a modular function of the form

(3.1)

(we have corrected a sign error in the proof of the corollary). From Lemma 2.1 we have

On the other hand, Proposition 3.1 gives

Therefore

(3.2)

or equivalently

(3.3)

Applying to both sides of (3.3) and arguing inductively, we obtain the following for each :

(3.4)

For any we have . Therefore for each each we have

(3.5)

The lemma follows by comparing coefficients of in (3.5). ∎

The authors of [4] speculated that for every prime . We prove that this is the case.

Lemma 3.3.

For each prime we have .

Proof.

Assume to the contrary that From (3.2) and Proposition 3.1 it follows that

from which it follows that for some integral coefficients we have

Let

Then has the form

Since , we find that has the form

so that

(3.6)

Using (2.5) we obtain

Since and we must have , so that . Thus there exists such that

However, by examining a basis for the eight-dimensional space we find that there is no such form . This provides the desired contradiction. ∎

Proof of Theorem 1.1.

Assertion (1.5) follows from Lemmas 3.2 and 3.3. To prove (1.6), we use Proposition 3.1 and (2.1) to find that

(3.7)

Using (2.1) we obtain

Since for , we see from Proposition 3.1 that . It follows that

Assertion (1.6) now follows from (3.7) and (1.5). ∎

4. An example in weight and level

In [5], the authors study the -adic coupling of mock modular forms and their shadows. As an application of their general result, they prove two -adic limit formulas involving the hypergeometric functions and evaluated at certain modular functions. We will use the following notation:

After rewriting using (3.3) and (3.4) of [5], we find that each of the two formulas in Theorem 1.3 of [5] is equivalent to the assertion that for every prime with we have

(4.1)

Here we prove a strengthened version of this result.

Theorem 4.1.

Let be a prime. Then for each integer we have

(4.2)
(4.3)

The proof follows the argument in Section 3, so we give fewer details here.

Proposition 4.2.

We have the following.

  1. For every integer with there exists a unique of the form

  2. Let be prime and let be a nonnegative integer. Then we have

Proof.

For each integer let

Then . We construct each form by taking a linear combination of with Uniqueness follows since is spanned by the form .

We deduce assertion (2) as in the last section using (2.2), (2.1), and assertion (1).

Lemma 4.3.

If is prime, then

Proof.

Define

It is seen from the expression of as an infinite product that if . Similarly, if

then for all Therefore, for each positive integer there exist such that

with and if (these coincide with the forms in [5, Prop. 3.1]). Since the constant term in the weight two modular form must be zero, we find as in the last section that . In particular, for any prime we have

By Lemma 2.1, we have

Hence it follows from Proposition 4.2 that

(4.4)

so that

(4.5)

Applying iteratively leads to

(4.6)

for any non-negative integer . Since , we have from (4.6) that

(4.7)

Comparing coefficients of in (4.7) gives the result. ∎

The authors of [5] verified that for every prime less than . Here we prove

Lemma 4.4.

For every odd prime , we have .

Proof.

Suppose by way of contradiction that is an odd prime with Then (4.4) gives

which implies that for some coefficients we have

Since has no zeros on the upper half plane (and does not vanish at any cusp), we have . Moreover,

Therefore so that . Since and , we must have , but this is impossible since contains no non-constant elements. ∎

Proof of Theorem 4.1.

Assertion (4.2) follows from Lemma 4.3, Lemma 4.4, and the fact that . Next, we use Proposition 4.2 and (2.1) to write

(4.8)

Since for any , Proposition 4.2 and (2.1) give

The result follows from (4.8) and (4.2). ∎

5. An example in weight and level

In [2] the authors establish an analogous representation of a weight cusp form as a -adic limit. Let denote the non-trivial Dirichlet character modulo , and define

The two formulas stated in the main theorem of [2] involve the hypergeometric function after rewriting they are equivalent to the following statement: for every prime with we have

Here we prove

Theorem 5.1.

For every prime and every integer we have

(5.1)
(5.2)

We give only a sketch of the proof.

Proposition 5.2.

We have the following.

  1. For every odd integer there exists a unique of the form

  2. Let be an odd prime and let be a nonnegative integer. Then we have

Proof.

For each integer define

We construct the form with the desired properties by taking an appropriate linear combination of , and uniqueness follows since is one-dimensional. Assertion (2) is proved as before. ∎

Lemma 5.3.

If is prime and then

Proof.

For each , let be the form given in [2, Lem. 3.3]. We have . For we have

where has Let be prime. As above we find that

It follows from Proposition 2.1 that

(5.3)

and we deduce using Proposition 5.2 that

Iteratively applying results in

so we have

(5.4)

Comparing coefficients gives the result. ∎

Lemma 5.4.

For every prime we have

Proof.

Suppose by way of contradiction that Then (5.3) and Lemma 5.2 show that whence

Let Then has the form

so that

Analyzing the filtration yields and . However, we find by examining a basis that there is no form with . This provides the desired contradiction. ∎

The proof of Theorem 5.1 follows as before.

References

  • [1] Scott Ahlgren and Nickolas Andersen. Hecke grids and congruences for weakly holomorphic modular forms. In Ramanujan 125, volume 627 of Contemp. Math., pages 1–16. Amer. Math. Soc., Providence, RI, 2014.
  • [2] Matthew Boylan and Sharon Anne Garthwaite. Quadratic AGM and -adic limits arising from modular forms. Bull. Lond. Math. Soc., 42(3):527–537, 2010.
  • [3] Jan H. Bruinier, Ken Ono, and Robert C. Rhoades. Differential operators for harmonic weak Maass forms and the vanishing of Hecke eigenvalues. Math. Ann., 342(3):673–693, 2008.
  • [4] Ahmad El-Guindy and Ken Ono. Gauss’s hypergeometric function and the congruent number elliptic curve. Acta Arith., 144(3):231–239, 2010.
  • [5] Pavel Guerzhoy, Zachary A. Kent, and Ken Ono. -adic coupling of mock modular forms and shadows. Proc. Natl. Acad. Sci. USA, 107(14):6169–6174, 2010.
  • [6] Henryk Iwaniec. Topics in classical automorphic forms, volume 17 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1997.
  • [7] Naomi Jochnowitz. Congruences between systems of eigenvalues of modular forms. Trans. Amer. Math. Soc., 270(1):269–285, 1982.
  • [8] Ken Ono. The web of modularity: arithmetic of the coefficients of modular forms and -series, volume 102 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004.
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