# Equations for abelian subvarieties

## TL;DR

This paper develops explicit equations for abelian subvarieties arising from group actions on abelian varieties, linking representation theory with algebraic geometry, and provides simplified formulas in special cases with illustrative examples.

## Contribution

It introduces a method to derive explicit equations for abelian subvarieties associated with subgroup actions, connecting representation theory and algebraic geometry.

## Key findings

- Explicit equations for abelian subvarieties are derived.
- Simplified equations are provided in special cases.
- Examples illustrate the application of the formulas.

## Abstract

Given a finite group $G$ and an abelian variety $A$ acted on by $G$, to any subgroup $H$ of $G$, we associate an abelian subvariety $A_H$ on which the associated Hecke algebra $\mathcal{H}_H$ for $H$ in $G$ acts. Any irreducible rational representation $\widetilde W$ of $\mathcal{H}_H$ induces an abelian subvariety of $A_H$ in a natural way. In this paper we give equations for this abelian subvariety. In a special case these equations become much easier. We work out some examples.