Mathematical Research Letters

Volume 27 (2020)

Number 3

Virtual Abelian varieties of $\mathrm{GL}_2$-type

Pages: 903 – 944

DOI: https://dx.doi.org/10.4310/MRL.2020.v27.n3.a14

Author

Chenyan Wu (Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, China; and School of Mathematics and Statistics, University of Melbourne, Victoria, Australia)

Abstract

This paper studies a class of Abelian varieties that are of $\mathrm{GL}_2$-type and with isogenous classes defined over a number field $k$. We treat the cases when their endomorphism algebras are either (1) a totally real field $K$ or (2) a totally indefinite quaternion algebra over a totally real field $K$. Among the isogenous class of such an Abelian variety, we identify one whose Galois conjugates can be described in terms of actions of Atkin–Lehner operators and the class group of $K$. Thus we deduce that such Abelian varieties are parametrised by finite quotients of certain PEL Shimura varieties. These new families of moduli spaces are further analysed when they are of dimension $2$. We provide explicit numerical bounds for when they are surfaces of general type. In addition, for two particular examples, we show that they are both rational surfaces by computing the coordinates of inequivalent elliptic points and studying the intersections of Hirzebruch cycles with exceptional divisors.

Received 1 July 2018

Accepted 28 December 2018

Published 20 August 2020