Endomorphism algebras of geometrically split abelian surfaces over $\mathbb{Q}$

TL;DR

This paper classifies the geometric endomorphism algebras of geometrically split abelian surfaces over nd, identifying 92 possible algebras through a detailed analysis involving quadratic imaginary fields and urves with complex multiplication.

Contribution

It provides a complete classification of endomorphism algebras for these abelian surfaces, linking the problem to quadratic imaginary fields with specific class groups and urves with CM.

Findings

The set of possible endomorphism algebras has cardinality 92.

Identifies conditions on quadratic imaginary fields for the existence of such surfaces.

Employs Nakamura's method to compute endomorphism algebras of restriction of scalars.

Abstract

We determine the set of geometric endomorphism algebras of geometrically split abelian surfaces defined over $\mathbb{Q}$. In particular we find that this set has cardinality 92. The essential part of the classification consists in determining the set of quadratic imaginary fields $M$ with class group $\mathrm{C}_2 \times \mathrm{C}_2$ for which there exists an abelian surface $A$ defined over $\mathbb{Q}$ which is geometrically isogenous to the square of an elliptic curve with CM by $M$. We first study the interplay between the field of definition of the geometric endomorphisms of $A$ and the field $M$. This reduces the problem to the situation in which $E$ is a $\mathbb{Q}$-curve in the sense of Gross. We can then conclude our analysis by employing Nakamura's method to compute the endomorphism algebra of the restriction of scalars of a Gross $\mathbb{Q}$-curve.