Endomorphism algebras of geometrically split abelian surfaces over ℚ

Fité, Francesc; Guitart, Xavier

doi:10.2140/ant.2020.14.1399

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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

DOI: 10.2140/ant.2020.14.1399

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 $C_{2} \times C_{2}$ 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

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

Vol. 14, No. 6, 2020