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arXiv · 1511.02212

How Large is $A_g(\mathbb{F}_q)$?

Michael Lipnowski·Jacob Tsimerman
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Abstract

Let $B(g,p)$ denote the number of isomorphism classes of $g$-dimensional abelian varieties over the finite field of size $p.$ Let $A(g,p)$ denote the number of isomorphism classes of principally polarized $g$ dimensional abelian varieties over the finite field of size $p.$ We derive upper bounds for $B(g,p)$ and lower bounds for $A(g,p)$ for $p$ fixed and $g$ increasing. The extremely large gap between the lower bound for $A(g,p)$ and the upper bound $B(g,p)$ implies some statistically counterintuitive behavior for abelian varieties of large dimension over a fixed finite field.

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Michael Lipnowski, Jacob Tsimerman. 2015-11-06. How Large is $A_g(\mathbb{F}_q)$?. https://doi.org/10.1215/00127094-2018-0029

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