Isogeny classes of Hilbert-Blumenthal abelian varieties over finite fields

JOURNAL OF NUMBER THEORY

卷:92

Article

Achter, JD ; Cunningham, CLR

关键词: base change;    orbital integrals;    abelian varieties;    finite fields;   

DOI  :  10.1006/jnth.2001.2716

来源: Elsevier

【 摘 要 】

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 O-L be the ring of integers in a totally real field dimension g over Q, let N-0 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, 0). Finally, let (X/k, iota, alpha) be a g-dimensional abelian variety over k equipped with an action by O-L and a Gamma(0)(N-0, N)-level structure. Using work of Kottwitz, we express the number of (X'/k, iota', alpha') which are isogenous to (X, iota, alpha) 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. (C) 2002 Elsevier Scienoe (USA).

【 授权许可】

Free