Let us consider the symmetric square transfer of the automorphic representation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>π</mi> </math></EquationSource> </InlineEquation> associated to a modular form <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$f \in S_k(N, \epsilon )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <msub> <mi>S</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>,</mo> <mi>ϵ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. In this article, we study the variation of the epsilon factor of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\textrm{sym}^2(\pi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mtext>sym</mtext> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> under twisting in terms of the local Weil-Deligne representation at each odd prime p. As an application, we detect the possible types of the symmetric square transfer of the local representation at p. We also consider the case <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$p=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> which is more subtle. Furthermore, as the conductor of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\textrm{sym}^2(\pi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mtext>sym</mtext> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is involved in the variation number, we compute it in terms of N.

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

期刊论文
发表日期：

2025年11月26日

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Tathagata Mandal,
Sudipa Mondal

摘要

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