Rankin-Selberg integrals for local symmetric square factors on $GL\mathrm{(2)}$

TL;DR

This paper proves the equality of local symmetric square L-functions for GL(2) representations with Artin L-functions via integral representations, and demonstrates the stability of associated gamma-factors under ramified twists.

Contribution

It establishes the equality between integral and Artin L-functions for local symmetric squares on GL(2), confirming a key aspect of the local Langlands correspondence.

Findings

Proves the equality of local symmetric square L-functions and Artin L-functions.

Shows stability of symmetric gamma-factors under highly ramified twists.

Provides a new integral representation approach for local L-functions.

Abstract

Let $\pi$ be an irreducible admissible (complex) representation of $GL(2)$ over a non-archimedean characteristic zero local field with odd residual characteristic. In this paper we prove the equality between the local symmetric square $L$-function associated to $\pi$ arising from integral representations and the corresponding Artin $L$-function for its Langlands parameter through the local Langlands correspondence. With this in hand, we show the stability of local symmetric $\gamma$-factors attached to $\pi$ under highly ramified twists.