Statistics of abelian varieties over finite fields
Statistics of abelian varieties over finite fields

Michael Lipnowski, Duke University
Fine Hall 214
Joint work with Jacob Tsimerman. Let B(g,p) denote the number of isomorphism classes of gdimensional 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.