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 we find conditions for the triviality of these universal norms.