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

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.

We have added the relevant references [BS17], [Cre92], and [KS23], absent in the published version in Algebra & Number Theory

References in corpus (2)

• Finiteness theorems for K3 surfaces and abelian varieties of CM type
• Principally polarized squares of elliptic curves with field of moduli equal to Q