Fité Naya, FrancescGuitart Morales, Xavier2023-02-142023-02-142020-07-301937-0652https://hdl.handle.net/2445/193621We 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.23 p.application/pdfeng(c) Fité Naya, Francesc et al., 2020Teoria de nombresVarietats de ShimuraVarietats abelianesGeometria algebraicaNumber theoryShimura varietiesAbelian varietiesAlgebraic geometryinfo:eu-repo/semantics/article7083922023-02-14info:eu-repo/semantics/openAccess