Abstract
We determine the set of geometric endomorphism algebras of geometrically split abelian surfaces defined over . 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 with class group for which there exists an abelian surface defined over which is geometrically isogenous to the square of an elliptic curve with CM by . We first study the interplay between the field of definition of the geometric endomorphisms of and the field . This reduces the problem to the situation in which is a -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 -curve.
Citation
Francesc Fité. Xavier Guitart. "Endomorphism algebras of geometrically split abelian surfaces over $\mathbb{Q}$." Algebra Number Theory 14 (6) 1399 - 1421, 2020. https://doi.org/10.2140/ant.2020.14.1399
Information