Abelian varieties of prescribed order over finite fields

Abstract: We give several new constructions of Weil polynomials to show that given a prime power q and n >> 1, every integer in a large subinterval of the Hasse-Weil interval is realized as #A(F_q) for some n-dimensional abelian variety A over F_q. Moreover, we can make A geometrically simple, ordinary, and principally polarized. On the one hand, our work generalizes a theorem of Howe and Kedlaya for F_2. On the other hand, it improves upon theorems of DiPippo and Howe; Aubry, Haloui, and Lachaud; and Kadets. This talk will focus on one construction that leads to explicit (and nearly best possible) bounds, in terms of q, on the largest integer that is not A(F_q) for any A. This is joint work with Raymond van Bommel, Edgar Costa, Wanlin Li, and Alexander Smith.

number theory

Audience: researchers in the topic

Around Frobenius distributions and related topics II