Modular forms of half - integral weight on ( ) Γ 4 0 with few nonvanishing coe ﬃ cients modulo ℓ

: Let k be a nonnegative integer. Let K be a number ﬁ eld and K (cid:2) be the ring of integers of K . Let 5 ℓ ≥ be a prime and v be a prime ideal of K (cid:2) over ℓ . Let f be a modular form of weight k 1 2 + on Γ 0 ( 4 ) such that its Fourier coe ﬃ cients are in K (cid:2) . In this article, we study su ﬃ cient conditions that if f has the form with square - free integers s i , then f is congruent to a linear combination of iterated derivatives of a single theta function modulo v . ,


2
+ on Γ 0 (4) such that its Fourier coefficients are in K . In this article, we study sufficient conditions that if f has the form

Introduction
The Fourier coefficients of modular forms of half-integral weight are related to various objects in number theory and combinatorics such as the algebraic parts of the central critical values of modular L-functions, orders of Tate-Shafarevich groups of elliptic curves, the number of partitions of a positive integer, and so on. With a lot of application to these objects, Bruinier [1], Bruinier and Ono [2], Ono and Skinner [3], Ahlgren and Boylan [4,5], and the others studied congruence properties modulo a power of a prime for Fourier coefficients of modular forms of half-integral weight. Many of them considered modular forms of half-integral weight whose the Fourier coefficients are supported on only finitely many square classes modulo a prime ℓ. Let f be a modular form of half-integral weight on N Γ 4 1 ( ). Vignéras [6] proved that if the q-expansion of f has the form  with a positive integer t and square-free integers s i , then f is a linear combination of single variable theta functions (a different proof of this result was given by Bruinier [1]). Many of the aforementioned results can be considered as positive characteristic extensions of Vignéras' result on classification of modular forms of half-integral weight such that their nonvanishing Fourier coefficients lie in only finitely many square classes. Especially, Ahlgren et al. [7] obtained an explicit mod ℓ analog of the result of Vignéras for modular forms of half-integral weight on Γ 4 0 ( ) satisfying the Kohnen-plus condition. Let K be a number field and K be the ring of integers of K . Let M Γ 4 ; with square-free integers s i , then k is even and f z a n q v 1 m o d . with square-free integers s i , then f z a n q a n q v 1 m o d .
f n k n f n k n The following theorem proves that the portion of P ε in the set of primes is one.
For a positive integer X, there is an absolute constant C such that P and For a nonnegative real number r, we define an operator Θ r on q [[ ]] by a n q n a n q r Θ if , 0 elsewhere. For convenience, we let Θ Θ 1 ≔ . As in Theorem 1.1, the previous results on modular forms of half-integral weight having the form (1.2) such as [1,2,4,5,7] and so on imply that in many cases, if f has the form (1.2), then f Θ( ) is congruent to a linear combination of iterated derivatives of a single theta function modulo v. These lead us to the following conjecture on modular forms f of half-integral weight having the form (1.2). Conjecture 1.2. Let K be a number field and K be the ring of integers of K . Let 5 ℓ ≥ be a prime and v be a prime ideal of K over ℓ. Assume that f S Γ 4 ; has the form f z sn a sn q v Θ mod n f sn with a square-free integer s, then f z a n q v Assume that ℓ is a prime and m is a nonnegative integer. Let r m ( ) ℓ be the least positive integer such that with square-free integers s i , then the following statements are true.
To give numerical evidence for Conjecture 1.
is a -basis of the space of modular forms of weight k , obtained by removing n 1 2 + th rows for all nonnegative integers n with n , ) be the null space of B k m , . With this notation, we give the following conjecture.   The remainder of this article is organized as follows. In Section 2, we review some properties of f having the form (1.3) and the filtration for modular forms. In Section 3, we prove Theorems 1.1, 1.3, 1.5, and 1.6.

Preliminaries
In this section, we review some notions and properties of the filtration for modular forms, and then we introduce some properties about modular forms of half-integral weight on Γ 4 0 ( ) such that their Fourier coefficients are supported on finitely many square classes modulo a prime ℓ. For further details, see [8].
Throughout the rest of this article, we fix the following notation. For a congruence subgroup Γ and be the space of modular forms (resp. cusp forms) of weight w on Γ.
w 0 ( ( ) )) be the space of modular forms (resp. cusp forms) of weight w on N Γ 0 ( ) with character χ. Let k be a nonnegative integer and 5 ℓ ≥ be a prime. Let K be a number field and K be the ring of integers of K . Let v be a prime ideal of K over ℓ. Let M N Γ 4 ; ) be the space of modular forms (resp. cusp forms) of weight k 1 2 + on N Γ 4 0 ( ) such that their Fourier coefficients are in K and S Γ 4 ; satisfying the Kohnen-plus condition. Now, we review the basic notions and properties about the Shimura correspondence. Assume that f is a cusp form of weight k where T n w , denotes the nth Hecke operator on the space of modular forms of weight w. For each prime ℓ, operators U ℓ and V ℓ on formal power series are defined by a n q U a n q n n n n 0 0 and a n q V a n q .

Filtration for modular forms of half integral weight modulo a prime ℓ
The theory of filtration for modular forms of integral weight was developed by Serre [9], Swinnerton-Dyer [10], Katz [11], and Gross [12]. From this, the theory of filtration for modular forms of half-integral weight on Γ 4 0 ( ) was studied. In this section, we review some properties of filtration for modular forms of half-integral weight on Γ 4 0 ( ). For the details, we refer to [13, Section 2]. We say that a n q , then let ω f ( ) = −∞. We summarize the properties of ω f ( ) in the following lemma.
. Then, the following statements are true.
(3) There is a nonnegative integer k′ such that Proof. The proofs of (1) and (2) 2 Modular forms of half-integral weight such that their Fourier coefficients are supported on finitely many square classes modulo ℓ In this section, we introduce some properties of modular forms of half-integral weight on Γ 4 0 ( ) such that their Fourier coefficients are supported on finitely many square classes modulo v.
Ahlgren and Boylan [4] obtained the necessary conditions for the weight of f M Γ 4 k 0 1 2 ( ( )) ∈ + such that their Fourier coefficients are supported on finitely many square classes modulo v by using the theory of Galois representations. This was reproved in [15] by using only the theory of filtration for modular forms of integral weight. The Choi and Kilbourn [16] improved the necessary conditions for the weight by using only the theory of filtration for modular forms of integral weight. We review the results [4,16] in the following theorem.
with square-free integers s i . Let k and i k be nonnegative integers, which satisfy k i k Then, the following statements are true.
(1) If n i ℓ∤ for some i, then Bruinier and Ono [2, Theorem 3.1] proved the following theorem by using an argument in [1].
Ahlgren et al. [7] proved that if f S Γ 4 ; and the Fourier coefficients of f are supported on finitely many square classes modulo v, then f has the form f z a n q a n q v mod . with square-free integers s i . Then, the following statements are true.
(1) If k 2| and 1 mod 4 ( ) ℓ ≡ , then f z a n q a n q v mod .
f z a n q v mod .
Shimura lift of f i . Since the Shimura correspondence commutes with the Hecke operators, for each prime p with p s p where s s i i ℓ ′ = . By the result of Carayol [17], the conductor of ρ i divides N i . By  with square-free integers s i . Then, f z a a n q a n q v 0 m o d.
Proof. Without loss of generality, we assume that there is a positive integer n 1 such that a s n 0 ). Let a be the exponent of ℓ in s n
Proof of Theorem 1.3. We fix a prime 5 ℓ ≥ . We prove Theorem 1.3 by induction on k. When k 1 < ℓ − , Theorem 1.3 is true by Lemma 2.6. Thus, we assume that Theorem 1.3 is true when k k 0 < with a fixed positive integer k 0 , where k 0 is a positive integer larger than Let k 2 be the largest integer satisfying k k k k k 1 2 1 1 2 and 1 2 1 2 mod 1 .   . Combining the assumption that Conjecture 1.2 is true, we have By the induction hypothesis, g 2 is congruent to a linear combination of k2 . Since we deduce that f is congruent to a linear combination of To complete the proof, it is sufficient to show that α k β k β k α k , , a n d , 1 , .
First, we assume that k 0 Further, assume that k 0 2 ≠ and k 2 , and then  and . Let p ℓ be the smallest positive prime p such that p 1 mod ( ) ≡ ℓ. If k p 2 1 2 + < ℓ , then k is even and We follow the proof of [7,Theorem 5.2]. By Proposition 2.4, we obtain that k is even. By Theorem 2.3, for each odd prime p with p 0, 1 mod Hence, for any positive odd integer m which is not divisible by any prime p with for n p 2 < ℓ . Since  . Then, f z a T z a n q v The following proposition is a refinement of Theorem 1.1. Let P be a set of primes ℓ such that for every f S Γ 4 ; Then, there is an absolute constant C such that Proof. Let p ℓ be the smallest positive prime p with p 1 mod ( ) ≡ ℓ. By using Theorem 3.