Non-commutative Iwasawa theory of abelian varieties over global function fields

Let A be an abelian variety defined over a global function field F. We investigate the structure of the p-primary Selmer group Sel (A/F_) for any prime number p distinct from the characteristic of F, over p-adic Lie extensions F_ of F which contain the cyclotomic Zₚ-extension F^cyc. In particular, we prove that the Pontryagin dual of the Selmer group Sel (A/Fᶜyc) is a torsion Zₚ[Gal (Fᶜyc/F) ]-module with trivial -invariant, and we establish the MH (G) -conjecture of Coates-Fukaya-Kato-Sujatha-Venjakob for A/F_. In view of the validity of the MH (G) -conjecture, it therefore makes sense to speak of the characteristic element (in the sense of Coates et al. ) attached to the Pontryagin dual of Sel (A/F_). We relate the order of vanishing of the characteristic elements, evaluated at Artin representations, to the corank of the Selmer group of the corresponding twist of A over the base field F. Combining this with the deep results of Tate, Milne and Kato-Trihan, we show that the order of vanishing of the characteristic elements is equal to the order of vanishing of the L-function of A/F at s=1 under appropriate assumptions. Finally, we relate the generalised Euler characteristic of Sel (A/F_) to the Euler characteristic of Sel (A/F^cyc). This is a natural analogue of Zerbes' result in the number field context and generalises previous results of Sechi and Valentino in the function field context.