A short note on sign changes and non-vanishing of Fourier coefficients of half-integral weight cusp forms

We study sign changes and non-vanishing of a certain double sequence of Fourier coefficients of cusp forms of half-integral weight.


Introduction
Starting with the paper [6] many authors have investigated sign change properties of Fourier coefficients of cusp forms, in various directions. In particular, the case of half-integral weight has been the focus of much research. If g is a cusp form of half-integral weight k + 1 2 with real Fourier coefficients c(m) (m ≥ 1) and in addition g is a Hecke eigenform, then there are at least two important themes in this area: on the one hand the study of sign changes of (c(tn 2 )) n≥1 where t is a fixed positive integer, and on the other hand the corresponding question for the sequence (c(t)) t≥1squarefree where t runs over positive squarefree integers only. Of course, similar questions can be studied for forms of weight k + 1 2 in the plus subspace in which case t has to be replaced by |D| where D is a fundamental discriminant with (−1) k D > 0 . For a good (at least partial) survey the reader may look up the literature given in [4].
Note that sign change results trivially imply corresponding non-vanishing results and in general non-vanishing properties of Fourier coefficients a priori are easier to handle. We recall that non-vanishing of products of Fourier coefficients was studied in [3].
In this short note we will investigate sign change and non-vanishing properties of the double sequence (c(4n + r 2 )) n≥1,r∈ where g is a cusp form of weight k + 1 2 with k even and level 4 in the plus subspace S + k+1∕2 (so c(m) = 0 unless m ≡ 0, 1 (mod 4) , see [7]). These Communicated by Jens Funke.
coefficients turn up naturally when one considers the adjoint linear map with respect to the Petersson scalar products of (essentially) the linear map "multiplication with ", where is the standard theta function of weight 1 2 and level 4. Here as throughout q = e 2 iz for z ∈ H , the complex upper half-plane.
Our results will be stated in the next section; the proofs will be given in section 3. They rely on a detailed study of the above mentioned adjoint map, on growth properties of Fourier coefficients of cusp forms of integral weight due to Ram Murty and on a strong bound for the Fourier coefficients of cusp forms of half-integral weight due to Blomer-Harcos. Detailed references will be given below.

Statement of results
If M ⊂ we denote by #M the cardinality of M (thus #M is either a non-negative integer or ∞).
By k we always understand a positive even integer. We let S k be the space of cusp forms of weight k on Γ 1 ∶= SL 2 ( ) . There is a linear map Note that in general L is not Hecke equivariant.
We denote by L * ∶ S + k+1∕2 → S k the linear map adjoint to L with respect to the Petersson scalar products. Note that since L is injective, L * is surjective.
Let g ∈ S + k+1∕2 be fixed, with Fourier coefficients c(m) (m ≥ 1) . For each n ∈ we then put and if in addition the c(m) are real k+1∕2 with real Fourier coefficients c(m) (m ≥ 1) and suppose that L * g is a normalized Hecke eigenform. Then there are sequences (n ) ≥1 and (m ) ≥1 in such that for any < 1 16 one has lim →∞ Remark It is easy to see that for any normalized Hecke eigenform F ∈ S k there exists g ∈ S + k+1∕2 with real Fourier coefficients such that F = L * g. If we drop the assumption that L * g is an eigenform, we still can get non-vanishing results for the Fourier coefficients. Let us put V ∶= imL and denote by V ⊥ the orthogonal Theorem 2 Let g ∈ S + k+1∕2 with real Fourier coefficients c(m) (m ≥ 1) and suppose that g is not contained in V ⊥ . Then there exists a sequence (n ) ≥1 in such that for any < 1 16 one has lim →∞ n n = ∞ . In particular one has lim →∞ n = ∞.
Remark Applying the above result with g replaced by g − g 0 where g 0 ∈ V ⊥ has Fourier coefficients c 0 (m) , we obtain a corresponding statement with " c(4n + r 2 ) ≠ 0 " replaced by " c(4n + r 2 ) ≠ c 0 (4n + r 2 ) " in the definition of n . A corresponding assertion mutatis mutandis (and in the case where the c 0 (m) are real) of course is valid also in the context of Theorem 1.

Proof of results
We start with briefly indicating the explicit construction of the map L * adjoint to L following [9, sect. 5], and [8], mutatis mutandis.
Let g ∈ S + k+1∕2 . The n-th Fourier coefficient of L * g is given by by the usual Petersson formula, where P k,n denotes the n-th Poincaré series in S k . By definition where z = x + iy, dV = dxdy y 2 is the invariant measure, F is a fundamental domain for Γ 0 (4) ⊂ Γ 1 and G(z) ∶= √ y g(z) (z) behaves like a modular form of weight k under Γ 0 (4) . Recall that Γ 0 (4) consists of those matrices in Γ 1 whose left lower component is divisible by 4. The integral in the last line above can be computed by the usual unfolding argument.
Altogether one finds that where C k is a real positive constant depending only on k and The convergence of the sum is clear by the usual Hecke estimate for the coefficients c(m) (observe that we may assume that k ≥ 4 , otherwise S + k+1∕2 = {0} ). This gives an explicit description of the map L * .
Since the P k,n (n ≥ 1) generate S k , we also see that V ⊥ = kerL * consists of those g with the property that (g, n) = 0 for all n ≥ 1.
For the proof of our results we also need Ω-results for the Fourier coefficients a(n) (n ≥ 1) of cusp forms f ∈ S k . Recall that for arithmetic functions v, w with w(n) ultimately strictly positive, one defines We shall now prove the first assertion of Theorem 1. We put F ∶= L * g and denote by A(n) (n ≥ 1) the Fourier coefficients of F. According to (4) (applied with Ω + ) we can choose a sequence (n ) ≥1 in such that for all and We claim that a(n) = Ω ± n (k−1)∕2 exp(c ± log n log log n ) , Suppose that this is not true, for a given . Then we can find a sequence n 1 < n 2 < … and K > 0 such that for all ≥ 1.
It follows from (1) and (2) that where r in ∑ + r runs over those r ∈ with c(4n + r 2 ) > 0 and r in ∑ − r runs over those r with c(4n + r 2 ) ≤ 0 . Note that the sum ∑ + r is non-empty by (1) and (5) and for each fixed is finite by (7). By [1] the Fourier coefficients c(m) of g can be estimated by where one can take = 1 16 . This estimate is slightly better than the Weil bound with = 0 . It is important to us that the bound (9) holds for all m ≥ 1 . Bounds better than the Weil bound for m squarefree were obtained in [2,5,10].
Inserting (9) into (8) we obtain where in the last line we have used (7). Choosing = − = 1 16 − we therefore find that Letting going to ∞ we obtain a contradiction to (6). This proves the assertion of Theorem 1 regarding + n . To obtain the assertion with − n one proceeds in the same way, mutatis mutandis, using (4) with Ω − . Finally to prove Theorem 2, one again proceeds in the same way, using (3). Note that the assumption that g ∉ V ⊥ is used to guarantee that L * g ≠ 0.
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