Non-vanishing of L-functions associated to cusp forms of half-integral weight in the plus space

Abstract

In this paper, we show a non-vanishing result for L-functions associated to cuspidal Hecke eigenforms of half integral weight in plus space. BackgroundThe first named author had shown in [4] that given any real number t 0 and > 0, then for k large enough, the average of the normalized L-functions L * (f, s) with f varying over a basis of Hecke eigenforms of weight k on SL 2 (Z) does not vanish on the line segment, the authors extend the result for cuspidal Hecke eigenforms of half integer weight. In what follows, we prove a non-vanishing result for sums of L-functions associated to cuspidal Hecke eigenforms of half-integral weight k + 1/2 where k ∈ 2Z on level 4 in the plus space. For this, we determine the Fourier coefficients of the projected kernel function.

On the non-vanishing of L-functions associated to cusp forms of half-integral weight

“…By (4-10) and (4-11), (S2) implies (4)(5)(6)(7)(8). Thus, by Theorem 3-2 Γ\SL 2 (R) ∼ P Λ\Γ,χ F f dµ Γ\SL 2 (R) ∼ > 0.…”

“…By (f1), the terms of the series P Λ\Γ,χ f are well-defined. Next, the function F f satisfies the assumptions of Lemma 3-1: it satisfies (F1) by (2-10) and (f1), and it satisfies (F2) by (2)(3)(4)(5)(6)(7)(8)(9)(10)(11) and (f2). By Lemma 3-1.…”

“…where c m,s ∈ C × depends only on m and s. Kohnen's method inspired Muić's work in [9] and was adapted in [11] to the case of cusp forms with non-trivial level and character, and in [13] and [4] to the case of cusp forms of half-integral weight.…”

“…The function F ( • , s) satisfies the condition (f1) for Λ = Γ ∞ since it is h-periodic and (5-1) holds, and by(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) it satisfies (f2). The condition (4-6) is satisfied by (5-1) and(2)(3)(4)(5)(6).Next, we defineS := [0, h[× 1 N , ∞ .Let us prove that S has the property (S1) of Theorem 4-5. Suppose that z ∈ S and γ = a b c d , η γ ∈ Γ are such that γ.z ∈ S. We need to prove that γ.z = z.…”

Zeros of L-functions attached to modular forms of half-integral weight

“…Recently, Choie and Kohnen have investigated the non‐vanishing properties of ‐functions attached to certain half‐integral weight cusp forms on on the real line. Also very recently, Kohnen and Raji have proved a non‐vanishing result in certain half planes for ‐functions attached to modular forms of half‐integral weight in the Kohnen plus space. In this article we concentrate on the zeros of ‐functions attached to half‐integral weight cusp forms on on the critical line.…”