Galois action on $\bar{\mathbb Q}$-isogeny classes of abelian $L$-surfaces with quaternionic multiplication
Volume 181 / 2017
Acta Arithmetica 181 (2017), 369-392
MSC: Primary 11G18; Secondary 11G10.
DOI: 10.4064/aa170504-20-9
Published online: 8 December 2017
Abstract
We construct a projective Galois representation attached to an abelian $L$-surface with quaternionic multiplication, describing the Galois action on its Tate module. We prove that such a representation characterizes the Galois action on the isogeny class of the abelian $L$-surface, seen as a set of points of a certain Shimura curve.
