Reductions of abelian surfaces over global function fields

Universal Norms on Abelian Varieties over Global Function Fields

We examine the Mazur-Tate canonical height pairing defined between an abelian variety over a global field and its dual. By expressing local factors of this pairing in terms of nonarchimedean theta functions, we show in the case of global function fields that certain of these pairings are annihilated by universal norms coming from Carlitz cyclotomic extensions. Furthermore, for elliptic curves w...

Abelian Surfaces over Finite Fields as Jacobians

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.

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...