#### Vol. 14, No. 6, 2020

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

### Francesc Fité and Xavier Guitart

Vol. 14 (2020), No. 6, 1399–1421
##### Abstract

We determine the set of geometric endomorphism algebras of geometrically split abelian surfaces defined over $ℚ$. 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 ${C}_{2}×{C}_{2}$ for which there exists an abelian surface $A$ defined over $ℚ$ 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 $ℚ$-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 $ℚ$-curve.

##### Keywords
products of CM elliptic curves, Coleman's conjecture, endomorphism algebras, singular abelian surfaces
##### Mathematical Subject Classification 2010
Primary: 11G18
Secondary: 11G15, 14K22