On Siegel Zeros of Symmetric Power L-functions

2023/01/19 by Zhao, Shifan
#11F11 #11F66 #11F67 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2301.07869

Abstract

Let f be a holomorphic cusp form of even weight k for the modular group SL(2,ℤ), which is assumed to be a common eigenfunction for all Hecke operators. For positive integer n, let Symⁿ(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 Symⁿ(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.

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