Mazur–Rubin initiated the study of Selmer companion elliptic curves. Two elliptic curves over a number field K are said to be n-Selmer companion if for every quadratic twist their n-Selmer groups over K are isomorphic. Mazur–Rubin gave several sufficient conditions for two curves to be <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1215_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$p^r$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>p</mi> <mi>r</mi> </msup> </math></EquationSource> </InlineEquation>-Selmer companion for a positive power of a prime p. Jha–Majumdar–Shekhar extended this idea to the case of modular forms of weight <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1215_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$k \geqslant 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>⩾</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. In this article, we extend earlier works to give sufficient conditions which ensure that if two modular representations are <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1215_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$p^r$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>p</mi> <mi>r</mi> </msup> </math></EquationSource> </InlineEquation>-Selmer companion then their symmetric square representations are also <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1215_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$p^r$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>p</mi> <mi>r</mi> </msup> </math></EquationSource> </InlineEquation>-Selmer companion.

Selmer companion symmetric square representations of modular forms

期刊论文
出版模式：

开放获取
发表日期：

2025年9月16日

查看全文

Jishnu Ray,
Selvam V

摘要

Mazur–Rubin initiated the study of Selmer companion elliptic curves. Two elliptic curves over a number field K are said to be n-Selmer companion if for every quadratic twist their n-Selmer groups over K are isomorphic. Mazur–Rubin gave several sufficient conditions for two curves to be $p^r$ $p^{r}$ -Selmer companion for a positive power of a prime p. Jha–Majumdar–Shekhar extended this idea to the case of modular forms of weight $k \geqslant 2$ $k ⩾ 2$ . In this article, we extend earlier works to give sufficient conditions which ensure that if two modular representations are $p^r$ $p^{r}$ -Selmer companion then their symmetric square representations are also $p^r$ $p^{r}$ -Selmer companion.