Non-vanishing of automorphic L-functions of prime power level

Abstract

Abstract. Iwaniec and Sarnak showed that at the minimum 25% of L-values associated to holomorphic newforms of fixed even integral weight and level N → ∞ do not vanish at the critical point when N is square-free and φ(N ) ∼ N . In this paper we extend the given result to the case of prime power level N = p ν , ν ≥ 2.

A uniform asymptotic formula for the second moment of primitive L-functions on the critical line

“…From the explicit formula we derive the asymptotics that generalizes theorem 1 of [4] proved for N prime, k = 1, t = 0 and l < N. Removing the restriction l < N is of crucial importance here. As shown in [1] this is required to prove the best known lower bound on the proportion of non-vanishing L-values when N = p ν , ν is fixed and p → ∞.…”

The first moment of cusp form $L$-functions in weight aspect on average

“…This formula was proved in [1, Sections 4-5] for prime power level N = p v , p prime, v ≥ 2. When the level N is equal to 1, the function under the integral in [1,Eq. 4.16] has a pole in view of [1,Eq.…”

“…When the level N is equal to 1, the function under the integral in [1,Eq. 4.16] has a pole in view of [1,Eq. 4.15].…”

“…Consider the family H 2k (1) of primitive forms of level 1 and weight 2k ≥ 12. Every f ∈ H 2k (1) has a Fourier expansion of the form (1.1) f (z) = n≥1 λ f (n)n (2k−1)/2 e(nz).…”

Moments of Central $L$-values for Maass Forms over Imaginary Quadratic Fields

“…There are abundant non-vanishing results for holomorphic modular forms over Q in [Duk,IS,KM1,KM2,Van,KMV,Dja,Rou1,Rou2,BF1,LT,Luo,BF2,Liu2,Job] and also over a totally real field in [Tro].…”