Mathematical Research Letters

Volume 27 (2020)

Number 3

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

Pages: 903 – 944



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


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.

This research is supported in part by the National Natural Science Foundation of China (#11601087), by the Program of Shanghai Academic/Technology Research Leader (#16XD1400400) and by the General Program of National Natural Science Foundation of China (#11771086).

Received 1 July 2018

Accepted 28 December 2018

Published 20 August 2020