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# Nonzero coefficients of half-integral weight modular forms mod \(\ell \)

- Joël Bellaïche
^{1}, - Ben Green
^{2}and - Kannan Soundararajan
^{3}Email author

**5**:6

https://doi.org/10.1007/s40687-018-0123-7

© SpringerNature 2018

**Received: **10 May 2017

**Accepted: **25 October 2017

**Published: **31 January 2018

## Abstract

We obtain new lower bounds for the number of Fourier coefficients of a weakly holomorphic modular form of half-integral weight not divisible by some prime \(\ell \). Among the applications of this we show that there are \(\gg \sqrt{X}/\log \log X\) integers \(n \le X\) for which the partition function *p*(*n*) is not divisible by \(\ell \), and that there are \(\gg \sqrt{X}/\log \log X\) values of \(n \le X\) for which *c*(*n*), the *n*th Fourier coefficient of the *j*-invariant, is odd.

## 1 Introduction

Let *K* be a number field and \({ \mathcal O}\) its ring of integers. Let \(\ell \) be a rational prime, and let \(\lambda \) be a maximal ideal of \({ \mathcal O}\) above \(\ell \). We denote by \( \mathbb F\) the residue field \({ \mathcal O}/\lambda \), a finite extension of \( \mathbb F_\ell \). The reader will lose little by supposing that \(K = { \mathbb Q } ,\) \({ \mathcal O}= { \mathbb Z }\), \(\lambda = (\ell )\) and \( \mathbb F= \mathbb F_\ell \); our main applications use only this case.

### Theorem 1

*N*a positive integer. Let \(f=\sum _{n=n_0}^\infty a_n q^n\) be a weakly holomorphic modular form

^{1}of weight

*k*and level \(\Gamma _1(N)\). Suppose that the coefficients \(a_n\) lie in the ring \({ \mathcal O}\). If \(\ell \ge 3\), we assume that \(f \not \equiv 0 \pmod {\lambda }\), and for \(\ell =2\) we assume that \(f \pmod {\lambda }\) is not a constant. Then

Theorem 1 improves, by a factor of about \(\log X\), the earlier work of Ahlgren and Boylan [2]. Here are some sample applications of Theorem 1.

### Example 1

*f*is a weakly holomorphic modular form of weight \(-1/2\) and level \(\Gamma _0(576)\). The Fourier expansion of

*f*is

*p*(

*n*) is the

*partition function*(cf. [17, Theorem 1.60, Corollary 1.62 and Theorem 5.3] for a proof of these well-known facts). Applying the theorem to

*f*, we conclude that

### Example 2

Theorem 1 applies in particular when *f* is a *holomorphic* cusp form of half-integral weight *k* (which must then be positive). In this case, it improves on the main theorem of [8] (itself an improvement of [18]) which proves the slightly weaker estimate \(\# \{ n \le X, a_n \not \equiv 0 \pmod {\ell }\} \gg {\sqrt{X}}/{\log X}\) under the supplementary assumption that the coefficients \(a_n \pmod {\lambda }\) are not supported in a finite union of sequences of the form \((c n^2)_{n \in { \mathbb N }}\). We remark that in [8, 18] the Shimura correspondence between holomorphic half-integral weight modular forms and integral weight modular forms plays a crucial role, whereas our proof of Theorem 1 does not involve the Shimura correspondence.

### Example 3

*integral*weight, since those forms are congruent modulo \(\lambda \) to forms of

*half-integral*weight (see Lemma 3). In particular, for the modular invariant \(j(q)=\sum _{n=-1}^\infty c(n) q^n\), which is of weight 0 and level \({\text{ SL }}_2({ \mathbb Z })\), we obtain that

Theorem 1 unfortunately does not give any improvement for the number of class numbers of imaginary quadratic fields not divisible by a prime \(\ell \). Although these class numbers arise as the Fourier coefficients of \(\theta ^3\), in order to produce distinct fields we need a variant of Theorem 1 that produces square-free *n* with \(a_n \not \equiv 0 \pmod {\lambda }\), which is not something that our methods allow.

For completeness, we remark that [6] establishes, for \(\ell \ge 3\), an asymptotic for the number of nonzero coefficients (mod \(\ell \)) of holomorphic modular forms and that [5] establishes such an asymptotic for \(\ell =2\) and holomorphic forms of level 1. The situation for *weakly* holomorphic forms of integral weight modulo a prime \(\ell >2\) remains mysterious, and (for example) we do not have lower bounds for the number of \(c(n) \ne 0 \pmod \ell \) for primes \(3 \le \ell \le 11\). (It is known [19, Théorème 5.2(a)] that \(\sum _{n\ge 0} c(\ell n) q^{n}\) is an holomorphic modular form mod \(\ell \) of positive integral weight, and this form is constant if and only if \(\ell \le 11\) [19, Exercise (6.16)]. Thus in the case \(\ell \ge 13\), from [6] it follows that there are \(\gg X/(\log X)^{3/4}\) values of \(n\le X\) with *c*(*n*) not divisible by \(\ell \).)

The proof of Theorem 1 uses the standard idea of multiplying *f* by a suitable lacunary holomorphic cusp form *g* of half-integral weight, to obtain an holomorphic cusp form \(h=fg\) of integral weight. The panoply of results stemming from the existence of Galois representations associated with integral weight holomorphic eigenforms may then be used to study the coefficients of *h*. Finding a suitable form *g* is easy when \(\ell >2\), and somewhat less so in the case \(\ell =2\). We describe this deduction in Sect. 2 for the sake of completeness, but we should note that closely related arguments appear already in the work of Ahlgren and Boylan [2]. It will follow from the work of Sect. 2 (and this is also implicit in [2]) that the sumset of the set of squares and the set of nonzero Fourier coefficients of \(f \pmod {\ell }\) contains all numbers of the form *up* for a fixed integer *u* and *p* lying in a positive density subset of the primes. Our improvement over previous work comes from a more careful analysis of this problem in analytic number theory/additive combinatorics.

### Theorem 2

*X*be large. For any subset \({\mathcal {A}}\subset \{1,\ldots ,X\}\) the number of primes

*p*such that \(pu \le X\) and

*pu*may be written as \(a + m^2\) for some \(a \in {\mathcal {A}}\) and some integer

*m*is

Our interest in Theorem 2 is in the situation where a positive proportion of the primes *p* are known to be of the form \(a+m^2\), when it follows that \(|{\mathcal {A}}|\) must have \(\gg \sqrt{X}/\log \log X\) elements. This statement is optimal, as we shall show in Sect. 4 by constructing an example of a set \({\mathcal {A}}\) with \(|{\mathcal {A}}| \asymp {\sqrt{X}}/{\log \log X}\) and with a positive proportion of primes below *X* being of the form \(a+m^2\).

*f*of half-integral weight however (specifically for \(f(q)=\eta _1^{-1}(q)=\sum _n p(n) q^{24 n-1}\), and perhaps for all forms that are not congruent mod \(\lambda \) to a one-variable theta series), it is expected that \(f \pmod {\lambda }\) is not

*lacunary*, which is to say that \( \# \{ n \le X, \ a_n \not \equiv 0 \pmod {\lambda }\} \gg X\).

We thank Scott Ahlgren for kindly drawing our attention to [2] and the referees for a careful reading.

## 2 Deduction of Theorem 1 from Theorem 2

### 2.1 A preliminary lemma

Let \(M_k(\Gamma _1(N),{ \mathcal O})\) be the \({ \mathcal O}\)-module of holomorphic modular forms of integral weight \(k\ge 0\), level \(\Gamma _1(N)\) and coefficients in \({ \mathcal O}\). Let \({ \mathcal O}_\lambda \) be the completion of \({ \mathcal O}\) at the place defined by the ideal \(\lambda \), and set \(M_k(\Gamma _1(N),{ \mathcal O}_\lambda ) = M_k(\Gamma _1(N),{ \mathcal O}) \otimes _{ \mathcal O}{ \mathcal O}_\lambda \). Let *A* be the closure of the \({ \mathcal O}_\lambda \)-subalgebra of \({\text {End}}_{{ \mathcal O}_\lambda }(M_k(\Gamma _1(N),{ \mathcal O}_\lambda ))\) generated by the Hecke operators \(T_n\) for *n* running among integers relatively primes to \(N \ell \).

*n*-th coefficient of a modular form

*h*by \(a_n(h)\). We recall that if \(h =\sum _{n=0}^\infty a_n q^n\) is a holomorphic modular form of integral weight

*k*for \(\Gamma _1(N)\), and if

*p*is a prime not dividing

*N*, the

*m*-th coefficient of the form \(T_p h\) is

Denote by \(G_{{ \mathbb Q } ,N\ell }\) the Galois group of the maximal extension of \({ \mathbb Q } \) unramified outside \(N\ell \), and by \(\mathrm{{Frob}}_p\), for *p* a prime not dividing \(N\ell \), the Frobenius element of *p* in \(G_{{ \mathbb Q } ,N\ell }\), well defined up to conjugation.

### Lemma 1

There exists a unique continuous map \(t: G_{{ \mathbb Q } ,N\ell } \rightarrow A\) which is central and satisfies \(t(\mathrm{{Frob}}_p) = T_p\) for every prime *p* not dividing \(N\ell \). This map also satisfies \(t(1)=2\).

### Proof

This follows from a well-known argument of Wiles based on the existence of Galois representations attached to eigenforms due to Deligne; see, for example, [4, Thm 1.8.5] for a detailed proof. \(\square \)

### 2.2 The case \(\ell >2\)

We begin with a lemma.

### Lemma 2

Assume that \(\ell \) is odd. Let \(k \in { \mathbb N }\) and \(h(q) = \sum _{n=0}^\infty a_n q^n\in M_k(\Gamma _1(N),{ \mathcal O})\). Assume that \(u \ge 1\) is an integer such that \(a_u \not \equiv 0 \pmod {\lambda }\). Then there is a positive density set of primes \({\mathcal {P}}\) such that \(a_{up} \not \equiv 0 \pmod {\lambda }\) for every \(p\in {\mathcal P}\).

### Proof

*t*as in Lemma 1, the map from \(G_{{ \mathbb Q } ,N\ell }\) to \( \mathbb F\) sending

*g*to \(a_u( t(g) h) \pmod {\lambda }\) is a continuous map. Thus there exists an open neighborhood

*U*of 1 in \(G_{{ \mathbb Q } ,N\ell }\) such that \(g \mapsto a_u( t(g) h) \pmod {\lambda }\) is constant on

*U*. Let \({\mathcal {P}}\) be the set of primes not dividing \(N\ell u\) such that \(\mathrm{{Frob}}_p \in U\). By Chebotarev, \({\mathcal {P}}\) has positive density. Further for \(p \in {\mathcal {P}}\) using (3) and Lemma 1 we have

*m*be an even integer such that \(\ell ^m\) is larger than the order of any pole of

*f*. Since

*f*has half-integer weight

*k*, the holomorphic cuspidal modular form \(h = f \eta _1^{\ell ^m}\) has integral weight \(k+\ell ^m/2\). Since \(f\pmod \lambda \) and \(\eta _1 \pmod {\lambda }\) are nonzero, the power series \(h \pmod {\lambda } \in \mathbb F[[q]]\) is also nonzero, and indeed \(h \pmod {\lambda }\) is not a constant (because

*h*is cuspidal, and a cuspidal constant form must be 0).

*n*is such that \(a_n(h) \not \equiv 0 \pmod \lambda \), then

*n*must be of the form \(a + v^2\) for some \(a\in {\mathcal {A}}\) and some integer

*v*. Now, by Lemma 2, the set of

*n*such that \(a_n(h) \not \equiv 0 \pmod {\lambda }\) contains a set of the form \(u {\mathcal {P}}\), for some fixed natural number

*u*and a set of primes \({\mathcal {P}}\) of positive density. For large

*X*, it follows from Theorem 2 that the number of primes

*p*with \(up \le X\) and

*up*of the form \(a+ v^2\) with \(a\in {\mathcal {A}}\) is \(\ll |{\mathcal {A}}\cap [1,X]| \sqrt{X} (\log \log X)/\log X + |{\mathcal {A}}\cap [1,X]|^{\frac{1}{2}} X^{\frac{3}{4}}/\log X\). It follows that \(|{\mathcal {A}}\cap [1,X]| \gg {\sqrt{X}}/{\log \log X}\), proving Theorem 1.

### 2.3 The case \(\ell =2\)

This case needs a little more care, and we begin by recalling a well-known result that (for \(\ell =2\)) modular forms of integer weights are congruent to modular forms of half-integer weight.

### Lemma 3

Assume that \(\ell =2\). For every weakly holomorphic modular form *f* of weight *k* and level \(\Gamma _1(N)\) with coefficients in \({ \mathcal O}\), there exists a weakly holomorphic modular form \(f'\) of weight \(k+1/2\), some level \(\Gamma _1(N')\), with coefficients in \({ \mathcal O}\), such that \(f \equiv f' \pmod {\lambda }\).

### Proof

Recall that (see, for example, [17, Prop 1.4]) the theta series \(\theta _0(q)=\sum _{n=-\infty }^\infty q^{n^2} = 1 + \sum _{n=1}^\infty 2 q^{n^2}\) is a holomorphic modular form of weight 1 / 2, level \(\Gamma _0(4)\), coefficients in \({ \mathbb Z }\). Now take \(f'=f \theta _0\). \(\square \)

### Lemma 4

Let \(n_0\) be a nonzero integer, and let *N* be a positive integer. There are only finitely many natural numbers *s* such that \(2^s + n_0\) equals \(uy^2\) for some square-free divisor *u* of 2*N* and some integer *y*.

### Proof

Write \(s= 3 s_0 + r\) with \(r=0\), 1, or 2, and set \(x= 2^{s_0}\). The equation \(2^s+n_0 = uy^2\) becomes \(2^r x^3 +n_0 = uy^2\), which for a given *r* and *u* may be viewed as an elliptic curve (since \(n_0 \ne 0\)). By Siegel’s theorem, there are only finitely many integer points (*x*, *y*) on this elliptic curve. Since there are only three possible values for *r*, and finitely many possibilities for *u* (being a square-free divisor of 2*N*), the lemma follows. \(\square \)

### Lemma 5

Let \(k \in { \mathbb N }\) and \(h(q) = \sum _{n=0}^\infty a_n q^n\in M_k(\Gamma _1(N),{ \mathcal O})\). Assume that there exists an integer \(n \ge 1\) and a prime \(p_0\) not dividing 2*N* such that \({\text {ord}}_{p_0}{n}\) is odd and \(a_{n} \not \equiv 0 \pmod {\lambda }\). Then there exists an integer \(u \ge 1\) and a set of primes \({\mathcal {P}}\) of positive density such that \(a_{up} \not \equiv 0 \pmod {\lambda }\) for every \(p \in {\mathcal {P}}\).

### Proof

*if*

*p*

*is a prime not dividing*2

*N*

*and if*\(T_p h \pmod {\lambda }\)

*is a constant, then*\(a_n(h)\equiv 0 \pmod {\lambda }\)

*if*\({\text {ord}}_p(n)\)

*is odd*. We prove that claim, for all

*h*such that \(T_p h \pmod {\lambda }\) is constant, by induction over the odd number \({\text {ord}}_{p}(n)\). If \({\text {ord}}_{p}(n)=1\), applying (3) to the form

*h*and the integer \(m=n/p\) and reducing mod \(\lambda \) gives (using that \(p \equiv -1 \equiv 1 \pmod {\lambda }\)):

*n*with \({\text {ord}}_p(n)\) odd, we get similarly

*t*as in Lemma 1, \(a_u(t(\mathrm{{Frob}}_{p_0}) h) \not \equiv 0 \pmod {\lambda }\). By continuity of

*t*, there exists an open set

*U*in \(G_{{ \mathbb Q } ,2N}\) such that for

*p*a prime not dividing 2

*Nu*, if \(\mathrm{{Frob}}_p \in U\), then

*p*is a set of primes of positive density by Chebotarev. \(\square \)

We are now ready to prove Theorem 1 in the case \(l=2\) using Theorem 2. Let *f* be a weakly holomorphic modular form of half-integral weight with coefficients in \({ \mathcal O}_\lambda \). By Lemma 3, we may assume that *f* has integral weight instead.

We consider the form \(h:=f \eta _1^{2^m}\) for a suitable *m* that will be specified below. We observe that if \(m \ge 1\), \(\eta _1^{2^m}\) has integral weight, and so does *h*. Moreover, since \(\eta _1\) is cuspidal, *h* is also cuspidal holomorphic for *m* large enough.

Write \(f \equiv \sum _{n = n_0}^\infty a_n q^n \pmod {\lambda }\) with \(a_{n_0} \not \equiv 0\). The first term of \(h \pmod {\lambda }\) is \(a_{n_0} q^{2^m+n_0}\). When \(n_0 \ne 0\), Lemma 4 ensures that we can choose *m* even, large enough in the sense of the preceding paragraph, and such that there is a prime \(p_0\) not dividing 2*N* such that \({\text {ord}}_{p_0}(2^m+n_0)\) is odd. When \(n_0=0\), let \(a_{n_1} q^{n_1}\) with \(n_1>0\), \(a_{n_1} \not \equiv 0\) the term of smallest positive degree in \(f \pmod {\lambda }\) (such an \(n_1\) exists because we assume that \(f \pmod {\lambda }\) is not a constant). If *m* is such that \(2^m > n_1\), then the form *h* has a term \(a_{n_1} q^{2^m+n_1}\). Again by Lemma 4, we can find *m* large enough and such that there is a prime \(p_0\) not dividing 2*N* such that \({\text {ord}}_{p_0}(2^m+n_1)\) is odd.

So in both cases (\(n_0 \ne 0\) and \(n_0 =0\)), we have shown the existence of an integer *m* such that \(h =f \eta _1^{2^m}\) is a cuspidal holomorphic modular form of integral weight and such that, by Lemma 5, there is \(u\ge 1\) and a set of primes \({\mathcal {P}}\) of positive density, with \(a_{u p}(h) \not \equiv 0 \pmod {\lambda }\) for every \(p \in {\mathcal {P}}\). The rest of the proof is now exactly as in the case \(\ell >2\).

## 3 Proof of Theorem 2

Given a positive integer *a*, we let \(\chi _{-4a}=(\frac{-4a}{\cdot })\) denote the Kronecker symbol, which is a Dirichlet character \(\pmod {4a}\). Note that \(-4a\) is a discriminant, but it need not be a fundamental discriminant. We denote the associated (negative) fundamental discriminant by \({\widetilde{a}}\), so that \(-4a = {\widetilde{a}} a_2^2\) for a suitable natural number \(a_2\).

### Lemma 6

*u*be a fixed natural number, and let

*X*be large. For every integer \(1\le a \le X\),

### Proof

*m*below \(\sqrt{X}\) such that \(a+ m^2\) is of the form

*ur*for a prime number

*r*. We may restrict attention to \(r> X^{\frac{1}{4}}\), since the smaller primes

*r*contribute negligibly to the number of

*m*. For each prime \(p \not \mid 2u\) and \(p\le X^{\frac{1}{4}}\), we see that \(m^2\) cannot be \(\equiv -a \pmod {p}\), which means that \(1+ \chi _{-4a}(p)\) residue classes \(\pmod p\) are forbidden for

*m*. Any standard upper-bound sieve (for example, Brun’s sieve or Selberg’s sieve; or see Theorem 2.2 of [13]) then shows that the number of possible \(m \le \sqrt{X}\) is

Call a fundamental discriminant *d* *good* if the corresponding Dirichlet *L*-function \(L(s,\chi _d)\) has no zeros in the region \(\{ \sigma > 99/100, \ |t| \le |d| \}\), and call the discriminant *d* *bad* otherwise.

### Lemma 7

### Proof

*O*(1), and partial summation shows that the second term is also

*O*(1). Therefore

*p*with \(up \le X\) and

*p*of the form \(a + m^2\) with \(a \in {{\mathcal {A}}}\) coming from a

*good*associated fundamental discriminant \({\widetilde{a}}\) is bounded by

*bad*fundamental discriminants \({\widetilde{a}}\). Note that, with \(-4a ={\widetilde{a}} a_2^2\),

*p*arising from bad fundamental discriminants is

*d*with \(Y \le |d| \le 2Y\). Therefore, the sum over bad \({\widetilde{a}}\) in (7) converges, and we conclude that the quantity in (7) is \(\ll |{\mathcal {A}}|^{\frac{1}{2}} X^{\frac{3}{4}}/\log X\). This completes the proof of Theorem 2.

## 4 Optimality of Theorem 2

In this section, we show the existence of a subset \({\mathcal {A}}\) of [1, *X*] with \(|{\mathcal {A}}| \asymp \sqrt{X}/\log \log X\), and such that a positive proportion of the primes below *X* may be written as \(a+ m^2\) with \(a\in {\mathcal {A}}\) and \(m \in {\mathbb {Z}}\). Since this is only an example to show the optimality of Theorem 2, we shall be content with sketching the proof.

Set \(Z= \exp ( (\log X)^{\frac{1}{10}})\). Note that \(\log \log Z \asymp \log \log X\). Let \({\mathcal {D}}\) be a set of about \(\sqrt{Z}/\log \log X\) odd square-free numbers *d* with \(Z\le d \le 2Z\) and such that \(L(1,\chi _{-4d}) \asymp 1/\log \log X\). Then our set \({\mathcal {A}}\) will consist of all numbers of the form \(d k^2\) with \(d \in {\mathcal {D}}\) and \(k\le \sqrt{X/2Z}\). By construction, the set \({\mathcal {A}}\) has \(\asymp \sqrt{X}/\log \log X\) elements.

*L*-functions, we may see that for any \(d\in {\mathcal {D}}\) the number of primes up to

*X*/ 2 of the form \(dk^2+b^2\) with \(b,k \in { \mathbb N }\) is

*p*as \(a+b^2\) with \(a \in {\mathcal {A}}\) and \(b \in { \mathbb N }\), it follows that

*X*/ 2 that may be represented as \(d_1 k^2 + b^2\) and also as \(d_2 r^2 + s^2\) is at most

In Theorem 2, we were interested in lower bounds for the size of a set \(\mathcal {A} \subset \{1,\ldots , X\}\) such that \(\mathcal {A} + \mathcal {B} \supset \mathcal {C}\), where \(\mathcal {B}, \mathcal {C} \subset \{1,\ldots , X\}\) are given sets. (In this case, \(\mathcal {B}\) is the set of squares, and \(\mathcal {C}\) is a set consisting of a positive proportion of the primes.) One might say that \(\mathcal {A}\) is an *additive complement of* \(\mathcal {B}\) *relative to* \(\mathcal {C}\). In the case \(\mathcal {C} = \{1,\ldots , X\}\), one recovers the usual notion of additive complement. To our knowledge, the relative case has not been studied in any generality. A large number of questions suggest themselves.

### Acknowledgements

Joël Bellaïche was supported by NSF Grant DMS 1405993. Ben Green was supported by a Simons Investigator grant from the Simons Foundation. Kannan Soundararajan was partially supported by NSF Grant DMS 1500237, and a Simons Investigator grant from the Simons Foundation. Part of the work was carried out when the second and third authors were in residence at MSRI, Berkeley, during the Spring semester of 2017, supported in part by NSF Grant DMS 1440140.

## Declarations

## Authors’ Affiliations

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