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Counting abelian varieties over finite fields via Frobenius densities

Jeffrey D. Achter, S. Ali Altuğ, Luis Garcia and Julia Gordon

Appendix: Wen-Wei Li and Thomas Rüd

Vol. 17 (2023), No. 7, 1239–1280
DOI: 10.2140/ant.2023.17.1239

Let [X,λ] be a principally polarized abelian variety over a finite field with commutative endomorphism ring; further suppose that either X is ordinary or the field is prime. Motivated by an equidistribution heuristic, we introduce a factor νv([X,λ]) for each place v of , and show that the product of these factors essentially computes the size of the isogeny class of [X,λ].

The derivation of this mass formula depends on a formula of Kottwitz and on analysis of measures on the group of symplectic similitudes and, in particular, does not rely on a calculation of class numbers.

abelian variety, isogeny class, orbital integral
Mathematical Subject Classification 2010
Primary: 11G10
Secondary: 14G15, 20G25, 22E35
Received: 31 July 2019
Revised: 2 May 2022
Accepted: 10 June 2022
Published: 30 May 2023
Jeffrey D. Achter
Department of Mathematics
Colorado State University
Fort Collins, CO
United States
S. Ali Altuğ
New York, NY
United States
Luis Garcia
Department of Mathematics
University College London
United Kingdom
Julia Gordon
Department of Mathematics
University of British Columbia
Vancouver, BC
Wen-Wei Li
Beijing International Center for Mathematical Research
Peking University
Thomas Rüd
Massachusetts Institute of Technology
Cambridge, MA
United States

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