Non-vanishing of the symmetric square $L$-function at the central point

TL;DR

This paper proves that a positive proportion of holomorphic Hecke eigenforms have non-vanishing symmetric square L-functions at the central point, using mollifier techniques, aligning with results for similar symplectic families.

Contribution

It demonstrates non-vanishing at the central point for a positive proportion of forms in a new family using mollifiers, extending known results to symmetric square L-functions.

Findings

Positive proportion of forms with non-vanishing L-functions

Same proportion as other symplectic families

Uses mollifier method effectively

Abstract

Using the mollifier method, we show that for a positive proportion of holomorphic Hecke eigenforms of level one and weight bounded by a large enough constant, the associated symmetric square $L$-function does not vanish at the central point of its critical strip. We note that our proportion is the same as that found by other authors for other families of $L$-functions also having symplectic symmetry type.