Non-vanishing of Rankin-Selberg Convolutions for Hilbert Modular Form

TL;DR

This paper proves that for large weights, a significant number of Hilbert primitive forms have non-vanishing central Rankin-Selberg L-values when paired with a fixed form, advancing understanding of automorphic L-functions.

Contribution

It establishes a lower bound on the number of Hilbert primitive forms with non-zero central L-values for large weights, demonstrating non-vanishing in a new setting.

Findings

At least k/(log k)^c forms have non-zero L-values for large k

Non-vanishing occurs with a quantitative lower bound

Results extend non-vanishing results to Hilbert modular forms

Abstract

In this paper, we study the non-vanishing of the central values of the Rankin-Selberg $L$-function of two ad\`elic Hilbert primitive forms ${\bf f}$ and ${\bf g}$, both of which have varying weight parameter $k$. We prove that, for sufficiently large $k$, there are at least $\frac{k}{(\log k)^{c}}$ ad\`elic Hilbert primitive forms ${\bf f}$ of weight $k$ for which $L(\frac12, {\bf f}\otimes{\bf g})$ are nonzero.