Realizations of Unisingular Representations by Hyperelliptic Jacobians

TL;DR

This paper demonstrates how certain unisingular representations of finite groups can be realized through hyperelliptic Jacobians over the rationals, and identifies new unisingular representations for symmetric and alternating groups.

Contribution

It shows that specific unisingular representations can be realized via hyperelliptic Jacobians and introduces new unisingular representations for symmetric and alternating groups.

Findings

Unisingular representations can be realized as mod 2 representations of hyperelliptic Jacobians.

New unisingular representations identified for symmetric and alternating groups.

Provides a connection between group representations and algebraic geometry.

Abstract

A representation of a finite group $G$ on a finite dimensional vector space $V$ is called \textbf{unisingular} if every $g\in G$ has 1 as an eigenvalue in its action on $V$. In this paper we show that certain unisingular representations can be realized as mod 2 representations of hyperelliptic Jacobians over $\Q$. We additionally identify new unisingular representations of the symmetric and alternating groups.