Notes on an analogue of the Fontaine-Mazur conjecture

Abelian varieties over finite fields with a specified characteristic polynomial modulo \ell

“…For each g, there are finitely many isogeny classes of abelian varieties with dimension g. Let d r,g be the fraction of isogeny classes of dimension g over F p r for which divides P (1). Then This result and the other major result of this paper could also be obtained using the techniques of [1]. The proofs given here are perhaps more elementary, and also give some access to the number of isogeny classes and not merely the proportion satisfying each condition.…”

“…In the paper [1], Jeffrey Achter and the author address the question of how many function fields are associated with a given P (T ) modulo , and thus how many fall under the purview of Theorem 1 and Theorem 2. In this paper we will address the different but related question of how many isogeny classes of abelian varieties have a given characteristic polynomial P (T ) modulo .…”

The square sieve and a Lang–Trotter question for generic abelian varieties

“…In the proof of our main Theorems, we will employ a bound on the size of particular subsets of GSp 2g Z/lZ. The bound appears (essentially) as stated below in [AH03] and originates in [Cha97].…”

The probability that the number of points on the Jacobian of a genus 2 curve is prime

“…Moreover, they seem more flexible towards generalizing Conjecture 1 to Jacobians of curves of higher genus, which have also been proposed for use in cryptography. The required analogues of Lenstra's theorem are provided by a recursive formula due to Achter and Holden [3,Lemma 3.2], which we turn into a closed expression in Section 5.…”