# Grothendieck groups of categories of abelian varieties

## TL;DR

This paper calculates the Grothendieck groups for categories of abelian varieties over algebraically closed fields, providing explicit results for isotypic varieties and elliptic curves across different characteristics.

## Contribution

It explicitly determines the Grothendieck groups of categories of abelian varieties and their isotypic subcategories, extending to elliptic curves in all characteristics.

## Key findings

- Grothendieck group of abelian varieties over algebraically closed fields computed.
- Grothendieck group of A-isotypic abelian varieties determined in characteristic 0.
- Grothendieck group for elliptic curves established in any characteristic.

## Abstract

We compute the Grothendieck group of the category of abelian varieties over an algebraically closed field $k$. We also compute the Grothendieck group of the category of $A$-isotypic abelian varieties, for any simple abelian variety $A$, assuming $k$ has characteristic 0, and for any elliptic curve $A$ in any characteristic.