Examples of abelian surfaces with everywhere good reduction

Lassina Dembele, Abhinav Kumar

Published 2013-09-16, updated 2015-08-28Version 3

We describe several explicit examples of simple abelian surfaces over real quadratic fields with real multiplication and everywhere good reduction. These examples provide evidence for the Eichler-Shimura conjecture for Hilbert modular forms over a real quadratic field. Several of the examples also support a conjecture of Brumer and Kramer on abelian varieties associated to Siegel modular forms with paramodular level structures.