Nonvanishing of L-Functions Associated to Cusp Forms of Half-Integral Weight

Abstract

In this article, we prove non-vanishing results for L-functions associated to holomorphic cusp forms of half-integral weight on average (over an orthogonal basis of Hecke eigenforms). This extends a result of W. Kohnen [4] to forms of half-integral weight.

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

“…In [5], 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.…”

“…The kernel function of the map f → L * (f, s)in the case of half integral weight on 0 (4) is given in [5] by…”

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

“…In the case of any integral weight Hecke eigenforms of level 1, Kohnen proved a non‐vanishing result at an arbitrary point which lies in the critical strip but not on the critical line using certain kernel functions. Following the method of Kohnen, Ramakrishnan and Shankhadhar proved an analogous result for cusp forms of half‐integral weight of arbitrary level. 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.…”

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

“…the other conditions of Lemma 4-12 being satisfied trivially. For every ε ∈ R >0 , we can write the left-hand side of (5)(6)(7)(8)(9)(10)(11)(12)(13)(14) as the sum…”

“…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.…”