A case of simultaneous non-vanishing

TL;DR

This paper proves the existence of certain modular forms with non-zero special values of associated L-functions for large even weights and primes, advancing understanding in analytic number theory.

Contribution

It demonstrates the existence of newforms with non-vanishing central L-values for large weights and primes, a novel result in the theory of automorphic forms.

Findings

Existence of newforms with non-zero L(1/2,f) and L(1/2,Sym^2 f) for large even weights and primes.

Establishment of non-vanishing results in the context of automorphic L-functions.

Contribution to the analytic theory of automorphic forms and L-functions.

Abstract

We show that for $k>1000$ an even number and a sufficiently large prime $q$, there exists a newform $f$ of weight $k$ and level $q$ such that $$ L(1/2,f)L(1/2,\text{Sym}^2 f)\neq 0. $$