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arXiv · 1807.10010

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

Francesc Fité·Xavier Guitart
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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.

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BibTeXRIS

Francesc Fité, Xavier Guitart. 2018-07-26. Endomorphism algebras of geometrically split abelian surfaces over $\mathbb{Q}$. https://doi.org/10.2140/ant.2020.14.1399

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