Nonvanishing and Central Critical Values of Twisted $L$-functions of Cusp Forms on Average

TL;DR

This paper constructs a kernel function to study the nonvanishing of averaged twisted L-values of cusp forms and proves an averaged Waldspurger's theorem relating central L-values to Fourier coefficients.

Contribution

It introduces a kernel function approach to analyze nonvanishing of averaged twisted L-values and establishes an averaged Waldspurger's theorem for cusp forms.

Findings

Average twisted L-values do not vanish on certain line segments in the critical strip for large weight or level.

Established an averaged Waldspurger's theorem linking central L-values to Fourier coefficients.

Provided a new method to study nonvanishing and special value formulas for L-functions of cusp forms.

Abstract

Let $f$ be a holomorphic cusp form of integral weight $k \geq 3$ for $\Gamma_{0}(N)$ with nebentypus character $\psi$. Generalising work of Kohnen and Raghuram we construct a kernel function for the $L$-function $L(f,\chi,s)$ of $f$ twisted by a primitive Dirichlet character $\chi$ and use it to show that the average $\sum_{f \in S_{k}(N,\psi)}\frac{L(f,\chi,s)}{<f,f>}\bar{a_{f}(1)}$ over an orthogonal basis of $S_{k}(N,\psi)$ does not vanish on certain line segments inside the critical strip if the weight $k$ or the level $N$ is big enough. As another application of the kernel function we prove an averaged version of Waldspurger's theorem relating the central critical value of the $D$-th twist ($D < 0$ a fundamental discriminant) of the $L$-function of a cusp form $f$ of even weight $2k$ to the square of the $|D|$-th Fourier coefficient of a form of half-integral weight $k+1/2$ associated to $f$ under the Shimura correspondence.