  In this paper, we study a (p-adic) geometric analogue for abelian varieties
over a function field of characteristic p of the cyclotomic Iwasawa theory and
the non-commutative Iwasawa theory for abelian varieties over a number field
initiated by Mazur and Coates respectively. We will prove some analogue of the
principal results obtained in the case over a number field and we study new
phenomena which did not happen in the case of number field case. We propose
also a conjecture which might be considered as a counterpart of the principal
conjecture in the case over a number field. \par This is a preprint which is
distributed since 2005 which is still in the process of submision. Following a
recent modification of some technical mistakes in the previous version of the
paper as well as an amelioration of the presentation of the paper, we decide
wider distribution via the archive.
