Olga Balkanova, Brown University, USA

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

Consider the family of L-functions associated with holomorphic newforms of fixed even integral weight and level N → ∞. When N is square-free and φ(N) ∼ N, Iwaniec and Sarnak showed that at least 25% of L-values do not vanish at the critical point. This problem for the prime-power level N = p^v, v ≥ 2 was investigated by Rouymi. He proved that at least (p-1)/6p of all L-functions in the family are non-zero when v → ∞ and p is fixed. In this talk, we show how to replace (p-1)/6p by (p-1)/4p. We also prove that the proportion of non-vanishing L-values is at least 25% in the opposite case: N = p^v, v is fixed and p → ∞ over primes.
This is a joint work with Dmitry Frolenkov.