 

Jonathan Gerhard
and Cassandra Williams
Local heuristics and an exact formula for abelian varieties of odd prime dimension over finite fields
view print


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


Abstract
Consider a qWeil 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
GSp_{2g}(F_{l}) 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. 

Acknowledgements
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
gerha2jm@dukes.jmu.edu
Cassandra Williams:
James Madison University
Harrisonburg, VA 22807, USA
willi5cl@jmu.edu

