On the change of epsilon factors for symmetric square transfers under twisting and applications

TL;DR

This work analyzes how the local and global epsilon factors of the symmetric square transfer $\mathrm{sym}^2(\pi)$ attached to a modular form respond to quadratic twisting at a prime $p$, with a focus on $p$-minimal forms. By matching local Langlands parameters and exploiting twisting properties of $\varepsilon$-factors, the authors derive explicit variance formulas $\varepsilon_p$ across all local types (principal series, special, and both non- and dihedral supercuspidal). They classify the possible local types of $\mathrm{sym}^2(\pi_p)$ in terms of the variance data and provide a conductor formula for $\mathrm{sym}^2(\pi)$ in terms of the level $N$, including delicate $p=2$ phenomena. The results yield concrete criteria to detect the local Sym$^2$-types from epsilon-factor variation and connect the conductor of the symmetric square to the global level, extending the parallel theory for symmetric cube transfers. These insights have consequences for understanding local representations in the symmetric square context and for arithmetic applications involving Heegner-type constructions and L-function twists.

Abstract

Let us consider the symmetric square transfer of the automorphic representation $π$ associated to a modular form $f \in S_k(N,ε)$. In this article, we study the variation of the epsilon factor of ${\mathrm{sym}}^2(π)$ under twisting in terms of the local Weil-Deligne representation at each prime $p$. As an application, we detect the possible types of the symmetric square transfer of the local representation at $p$. Furthermore, as the conductor of ${\mathrm{sym}}^2(π)$ is involved in the variation number, we compute it in terms of $N$.

Theorems & Definitions (46)

• Proposition 2.1
• proof
• Proposition 2.2
• Theorem 2.3
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4
• Proposition 4.1
• proof