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 . 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 C2 × C2 for which there exists an abelian surface A defined over 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 -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.

Keywords
products of CM elliptic curves, Coleman's conjecture, endomorphism algebras, singular abelian surfaces
Mathematical Subject Classification 2010
Primary: 11G18
Secondary: 11G15, 14K22
Milestones
Received: 8 February 2019
Revised: 31 October 2019
Accepted: 26 February 2020
Published: 30 July 2020
Authors
Francesc Fité
Massachusetts Institute of Technology
Cambridge, MA
United States
Xavier Guitart
Universitat de Barcelona
Barcelona
Catalonia
Spain