Algebraic and Arithmetic Geometry Seminar: "Abelian varieties of prescribed order over finite fields"

Abstract: Given a prime power q and n >> 1, we prove that every integer in a large subinterval of the Hasse--Weil interval is the order of a geometrically simple ordinary principally polarized abelian variety of dimension n over F_q. As a consequence, we generalize a result of Howe and Kedlaya for F_2 to show that for each prime power q, every sufficiently large positive integer is realizable. Our methods are effective: We prove that if q<= 5, then every positive integer is realizable, and for arbitrary q, every positive integer >>q^{3 \sqrt{q} \log q} is realizable. This is joint work with Raymond van Bommel, Edgar Costa, Bjorn Poonen and Alexander Smith.

Host: Matt Kerr