 
Summary: Isogeny classes of HilbertBlumenthal abelian varieties over finite
fields
Jeffrey D. Achter
Department of Mathematics, Columbia University, New York, NY 10027
Email: achter@math.columbia.edu
and
Clifton L. R. Cunningham
Department of Mathematics, University of Calgary, Alberta T2N 1N4
Email: cunning@math.ucalgary.ca
This paper gives an explicit formula for the size of the isogeny class of a
HilbertBlumenthal 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 squarefree integers, and let k be a finite field of characteristic
relatively prime to both N0N and disc(L, Q). Finally, let (X/k, , ) be a
gdimensional abelian variety over k equipped with an action by OL and a
0(N0, N)level structure. Using work of Kottwitz, we express the number
of (X /k, , ) which are isogenous to (X, , ) as a product of local orbital
integrals on GL(2); then, using work of ArthurClozel and the affine Bruhat
decomposition we evaluate all the relevant orbital integrals, thereby finding
the cardinality of the isogeny class.
