BENOIT TRIHAN FABIEN

Let A be an abelian variety over a global function field K of characteristic p. We study the µ-invariant appearing in the Iwasawa theory of A over the unramified ℤp-extension of K. Ulmer suggests that this invariant is equal to what he calls the dimension of the Tate–Shafarevich group of A and that it is indeed the dimension of some canonically defined group scheme. Our first result is to verify his suggestions. He also gives a formula for the dimension of the Tate–Shafarevich group (which is now the µ-invariant) in terms of other quantities including the Faltings height of A and Frobenius slopes of the numerator of the Hasse–Weil L-function of A/K assuming the conjectural Birch–Swinnerton-Dyer formula. Our next result is to prove this µ-invariant formula unconditionally for Jacobians and for semistable abelian varieties. Finally, we show that the “µ = 0” locus of the moduli of isomorphism classes of minimal elliptic surfaces endowed with a section and with fixed large enough Euler characteristic is a dense open subset.

We state and prove certain cases of the equivariant Tamagawa number conjecture of a semistable Abelian variety over an everywhere unramified finite Galois extension of a global field of characteristic p > 0 under a semisimplicity hypothesis.

We establish the Iwasawa main onje ture for semistable abelian varieties over a function field of charateristi p under certain restrictive assumptions. Namely we consider p-torsion free p-adic Lie extensions of the base field which contain the constant Zp-extension and are everywhere unramifield. Under the usual μ = 0 hypothesis, we give a proof which mainly relies on the interpretation of the Selmer complex in terms of p-adic cohomology [TV] together with the trace formulas of [EL1].

We prove a functional equation for two projective systems of finite abelian p-groups, {an} and {abn}, endowed with an action of ℤdp such that an can be identified with the Pontryagin dual of bn for all n. Let K be a global field. Let L be a ℤdp-extension of K (d ≥ 1), unramified outside a finite set of places. Let A be an abelian variety over K. We prove an algebraic functional equation for the Pontryagin dual of the Selmer group of A.

We study a geometric analogue of the Iwasawa Main Conjecture for constant ordinary abelian varieties over Z(p)(d)-extensions of function fields ramifying at a finite set of places.