On the Lower Bound of the Number of Abelian Varieties Over 𝔽p. (1st July 2020)

• Record Type: Journal Article
• Authors: Lee, Jungin
• Is Part Of: International mathematics research notices. Volume 2022:Number 6(2022)
• Page Start: 4290
• Page End: 4317
• Publication Date: 2020-07-01
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnaa153
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: In this paper, we prove that the number $B(p, g)$ of isomorphism classes of abelian varieties over a prime field $\mathbb{F}_p$ of dimension $g$ has a lower bound $p^{\frac{1}{2}g^2(1+o(1))}$ as $g \rightarrow \infty$ . This is the 1st nontrivial result on the lower bound of $B(p, g)$ . We also improve the upper bound $2^{34g^2}p^{\frac{69}{4} g^2 (1+o(1))}$ of $B(p, g)$ given by Lipnowski and Tsimerman [ 7 ] to $p^{\frac{45}{4} g^2(1+o(1))}$ .