A note on cusp forms as adic limits
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 limits2010 Mathematics Subject Classification:
11F33, 11F11, 11F031. Introduction
In a recent paper [4], ElGuindy 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) 
2. Background
If is an integer, is a function of the upper halfplane, 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 .
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
Proposition 3.1.
We have the following.

For every odd integer there exists a unique of the form

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 onedimensional. This gives the first assertion.
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 eightdimensional space we find that there is no such form . This provides the desired contradiction. ∎
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.

For every integer with there exists a unique of the form

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 .
∎
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 nonnegative 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 nonconstant elements. ∎
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 nontrivial 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.

For every odd integer there exists a unique of the form

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 onedimensional. Assertion (2) is proved as before. ∎
Lemma 5.3.
If is prime and then
Proof.
Lemma 5.4.
For every prime we have
Proof.
The proof of Theorem 5.1 follows as before.
References
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