FUNCTIONAL EQUATION OF CHARACTERISTIC ELEMENTS OF ABELIAN VARIETIES OVER FUNCTION FIELDS (l ≠ p)

In this paper we apply methods from the number field case of Perrin-Riou [20] and Zabradi [32] in the function field setup. In Z(l)- and GL(2)-cases (l not equal p), we prove algebraic functional equations of the Pontryagin dual of Selmer group which give further evidence of the main conjectures of Iwasawa theory. We also prove some parity conjectures in commutative and non-commutative cases. As a consequence, we also get results on the growth behavior of Selmer groups in commutative and non-commutative extension of function fields.