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Endomorphism algebras of geometrically split abelian surfaces over $Q$
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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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FITÉ NAYA, Francesc and GUITART MORALES, Xavier. Endomorphism algebras of geometrically split abelian surfaces over $Q$. Algebra & Number Theory. 2020. Vol. 14, num. 6, pags. 1399-1421. ISSN 1937-0652. [consulted: 11 of August of 2026]. Available at: https://hdl.handle.net/2445/193621