On Siegel Zeros of Symmetric Power L-functions

TL;DR

This paper proves the non-existence of Siegel zeros for symmetric power L-functions of certain modular forms, using auxiliary L-functions and functoriality, and provides lower bounds at s=1.

Contribution

It establishes the non-existence of Siegel zeros for symmetric power L-functions of modular forms, leveraging recent automorphy results and constructing auxiliary functions.

Findings

No Siegel zeros for symmetric power L-functions of modular forms.

Provides lower bounds at s=1 for these L-functions.

Utilizes functoriality and auxiliary L-functions in the proof.

Abstract

Let $f$ be a holomorphic cusp form of even weight $k$ for the modular group $SL(2,\mathbb{Z})$, which is assumed to be a common eigenfunction for all Hecke operators. For positive integer $n$, let $\text{Sym}^n(f)$ be the symmetric nth power lifting of $f$ , which was shown by Newton and Thorne to be automorphic and cuspidal. In this paper, we construct certain auxiliary $L$-functions to show that Siegel zeros of $\text{Sym}^n(f)$ do not exist, for each given $n$, utilizing the above functoriality result. As an application, we give a lower bound of those symmetric power $L$-functions at $s=1$ of logarithm power type.