Duke Mathematical Journal

How large is Ag(Fq)?

Michael Lipnowski and Jacob Tsimerman

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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 for B(g,p) implies some statistically counterintuitive behavior for Abelian varieties of large dimension over a fixed finite field.

Article information

Duke Math. J., Volume 167, Number 18 (2018), 3403-3453.

Received: 10 December 2015
Revised: 7 March 2018
First available in Project Euclid: 15 November 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G10: Abelian varieties of dimension > 1 [See also 14Kxx]
Secondary: 11G25: Varieties over finite and local fields [See also 14G15, 14G20]

principally polarized Abelian variety large dimension arithmetic statistics Katz–Sarnak heuristics Cohen–Lenstra heuristics finite field function field


Lipnowski, Michael; Tsimerman, Jacob. How large is $A_{g}(\mathbb{F}_{q})$ ?. Duke Math. J. 167 (2018), no. 18, 3403--3453. doi:10.1215/00127094-2018-0029. https://projecteuclid.org/euclid.dmj/1542250833

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