Bulletin of the Belgian Mathematical Society - Simon Stevin

On abelian surfaces with potential quaternionic multiplication

Luis V. Dieulefait and Victor Rotger

Full-text: Open access

Abstract

An abelian surface $A$ over a field $K$ has potential quaternionic multiplication if the ring $\End _{\bar K}(A)$ of geometric endomorphisms of $A$ is an order in an indefinite rational division quaternion algebra. In this brief note, we study the possible structures of the ring of endomorphisms of these surfaces and we provide explicit examples of Jacobians of curves of genus two which show that our result is sharp.

Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 12, Number 4 (2005), 617-624.

Dates
First available in Project Euclid: 5 December 2005

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1133793348

Digital Object Identifier
doi:10.36045/bbms/1133793348

Mathematical Reviews number (MathSciNet)
MR2206004

Zentralblatt MATH identifier
1146.11033

Subjects
Primary: 11G18: Arithmetic aspects of modular and Shimura varieties [See also 14G35] 14G35: Modular and Shimura varieties [See also 11F41, 11F46, 11G18]

Keywords
Abelian surface Galois representation quaternion algebra modularity

Citation

Dieulefait, Luis V.; Rotger, Victor. On abelian surfaces with potential quaternionic multiplication. Bull. Belg. Math. Soc. Simon Stevin 12 (2005), no. 4, 617--624. doi:10.36045/bbms/1133793348. https://projecteuclid.org/euclid.bbms/1133793348


Export citation