Control theorems for $\ell$-adic Lie extensions of global function fields

Abstract

Let F be a global function field of characteristic p>0, K/F an`-adic Lie extension unramified outside a finite set of places S and A/F an abelian variety. We study Sel A (K ) _ (the Pontrjagin dual of the Selmer group) and (under some mild hypotheses) prove that it is a finitely generated Z`[[Gal(K /F)]]module via generalizations of Mazur's Control Theorem. If Gal(K /F) has no elements of order`and contains a closed normal subgroup H such that Gal(K /F)/H ' Z`, we are able to give sufficient conditions for Sel A (K ) _ to be finitely generated as Z`[[H ]]-module and, consequently, a torsion Z`[[Gal(K /F)]]-module. We deal with both cases`6 = p and`= p.

Euler characteristic and Akashi series for Selmer groups over global function fields

“…Moreover, to shorten notations, in this section we put S for the p-Selmer group of F d , i.e., what we previously denoted by Sel(F d ) and S for its Pontrjagin dual S ∨ . The fact that S ∈ M G1 (G) has been proved in many different cases for example in [2], [5], [17] and [20].…”

“…Then equation (3) shows that H 1 (G, Sel(K)) ≃ Coker(ψ G K ) . We consider the classical descent diagram, which has been already used to study the structure of Selmer groups as modules over some Iwasawa algebra (see, e.g., [5] and the references there) (5) Sel(F )…”

Stickelberger series and Main Conjecture for function fields

“…This approach has been applied also to Iwasawa theory for elliptic curves and abelian varieties in [6], building on structure theorems for Selmer groups (see e.g. [29], [30] and [7]) and the only avaliable Main Conjecture in this setting, i.e. the one for constant abelian varieties in [21].…”

On Euler characteristics of Selmer groups for abelian varieties over global function fields

“…For infinite extensions we define the Selmer groups by taking direct limits on the finite subextensions. In particular, Sel A (K) ℓ is a Λ(G)-module whose structure has been studied in [5].…”