New York Journal of Mathematics
Volume 25 (2019), 123-144


Jonathan Gerhard and Cassandra Williams

Local heuristics and an exact formula for abelian varieties of odd prime dimension over finite fields

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Published: January 5, 2019.
Keywords: abelian varieties, finite fields, matrix groups.
Subject: 14K02.

Consider a q-Weil polynomial f of degree 2g. Using an equidistribution assumption that is too strong to be true, we define and compute a product of local relative densities of matrices in GSp2g(Fl) with characteristic polynomial f mod l when g is an odd prime. This infinite product is closely related to a ratio of class numbers. When g=3 we conjecture that the product gives the size of an isogeny class of principally polarized abelian threefolds.


We thank Everett Howe for sharing with us his work related to isogeny classes of principally polarized abelian varieties, and Jeff Achter for very helpful conversations. We also thank the referee for useful suggestions and insightful questions.

Author information

Jonathan Gerhard:
James Madison University
Harrisonburg, VA 22807, USA


Cassandra Williams:
James Madison University
Harrisonburg, VA 22807, USA