Isogeny Classes of Hilbert–Blumenthal Abelian Varieties over Finite Fields

Abstract

This paper gives an explicit formula for the size of the isogeny class of a Hilbert–Blumenthal abelian variety over a finite field. More precisely, let OL be the ring of integers in a totally real field dimension g over Q, let N0 and N be relatively prime square-free integers, and let k be a finite field of characteristic relatively prime to both N0N and disc(L, Q). Finally, let (X/k, ι, α) be a g-dimensional abelian variety over k equipped with an action by OL and a Γ0(N0, N)-level structure. Using work of Kottwitz, we express the number of (X′/k, ι′, α′) which are isogenous to (X, ι, α) as a product of local orbital integrals on GL(2); then, using work of Arthur and Clozel and the affine Bruhat decomposition we evaluate all the relevant orbital integrals, thereby finding the cardinality of the isogeny class.