Isogeny classes of abelian varieties over finite fields

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...

Tate’s Isogeny Theorem for Abelian Varieties 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.