Polarized products of elliptic curves with complex multiplication and field of moduli $\mathbb{Q}$

TL;DR

This paper establishes a categorical equivalence linking polarized abelian varieties with CM by a maximal order in an imaginary quadratic field to hermitian lattices, and uses this to classify certain genus curves with rational moduli.

Contribution

It introduces a new categorical equivalence and applies it to classify genus 2 and 3 curves with Jacobians as products of CM elliptic curves over ield.

Findings

Classified genus 2 and 3 curves with ield of moduli ield.

Established an equivalence between polarized CM abelian varieties and hermitian lattices.

Provided a framework for enumerating such curves using lattice theory.

Abstract

Let $R$ be the maximal order in a quadratic imaginary field $K$. We give an equivalence of categories between the category of polarized abelian varieties isomorphic to a product of elliptic curves over $\mathbb{C}$ with complex multiplication (CM) by $R$ and the category of integral hermitian $R$-lattices. Then we apply this equivalence to enumerate all the genus $2$ and $3$ curves with field of moduli $\mathbb{Q}$ and with Jacobian isomorphic to a product of elliptic curves with CM by $R$.