Vol. 14, No. 6, 2020

Download this article
Download this article For screen
For printing
Recent Issues

Volume 14
Issue 6, 1331–1667
Issue 5, 1055–1329
Issue 4, 815–1054
Issue 3, 545–813
Issue 2, 275–544
Issue 1, 1–274

Volume 13, 10 issues

Volume 12, 10 issues

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Editorial Board
Editors' Interests
Submission Guidelines
Submission Form
Editorial Login
Ethics Statement
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Author Index
To Appear
Other MSP Journals
Endomorphism algebras of geometrically split abelian surfaces over $\mathbb{Q}$

Francesc Fité and Xavier Guitart

Vol. 14 (2020), No. 6, 1399–1421

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.

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