Jacobians in isogeny classes of abelian surfaces over finite fields

Jacobians in isogeny classes of abelian surfaces over finite fields

We give a complete answer to the question of which polynomials occur as the characteristic polynomials of Frobenius for genus-2 curves over finite fields.

Principally Polarizable Isogeny Classes of Abelian Surfaces over Finite Fields

Let A be an isogeny class of abelian surfaces over Fq with Weil polynomial x4+ax3+bx2+aqx+q2. We show that A does not contain a surface that has a principal polarization if and only if a2 − b = q and b < 0 and all prime divisors of b are congruent to 1 modulo 3.

Abelian Surfaces over Finite Fields as Jacobians

Isogeny classes of Hilbert-Blumenthal abelian varieties over finite fields

This paper gives an explicit formula for the size of the isogeny class of a Hilbert-Blumenthal abelian variety over a finite field. More precisely, let OL be the ring of integers in a totally real field dimension g over Q, let N0 and N be relatively prime square-free integers, and let k be a finite field of characteristic relatively prime to both N0N and disc(L,Q). Finally, let (X/k, ι, α) be a...

Isogeny Classes of Abelian Varieties over Function Fields

Let K be a field, K̄ its separable closure, Gal(K) = Gal(K̄/K) the (absolute) Galois group of K. Let X be an abelian variety over K. If n is a positive integer that is not divisible by char(K) then we write Xn for the kernel of multiplication by n in X(Ks). It is well-known [21] that Xn ia a free Z/nZ-module of rank 2dim(X); it is also a Galois submodule in X(K̄). We write K(Xn) for the field of d...