Jacobians of Curves over Finite Fields

On Curves over Finite Fields with Jacobians of Small Exponent

We show that finite fields over which there is a curve of a given genus g ≥ 1 with its Jacobian having a small exponent, are very rare. This extends a recent result of W. Duke in the case g = 1. We also show when g = 1 or g = 2 that our bounds are best possible.

Curves and Jacobians over function fields

Abelian Surfaces over Finite Fields as Jacobians

Elliptic Curves over Finite Fields

In this chapter, we study elliptic curves defined over finite fields. Our discussion will include the Weil conjectures for elliptic curves, criteria for supersingularity and a description of the possible groups arising as E(Fq). We shall use basic algebraic geometry of elliptic curves. Specifically, we shall need the notion and properties of isogenies of elliptic curves and of the Weil pairing....

Computing endomorphism rings of Jacobians of genus 2 curves over finite fields

We present algorithms which, given a genus 2 curve C defined over a finite field and a quartic CM field K, determine whether the endomorphism ring of the Jacobian J of C is the full ring of integers in K. In particular, we present probabilistic algorithms for computing the field of definition of, and the action of Frobenius on, the subgroups J [l] for prime powers l. We use these algorithms to ...