On Selmer groups of abelian varieties over ℓ-adic Lie extensions of global function fields

Abstract

Let F be a global function field of characteristic p > 0 and A/F an abelian variety. Let K /F be an -adic Lie extension ( = p) unramified outside a finite set of primes S and such that Gal(K /F) has no elements of order . We shall prove that, under certain conditions, Sel A (K ) ∨ has no nontrivial pseudo-null submodule.

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)…”

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

“…(where the * denotes the ℓ-adic completion, see, e.g., [4,Proposition 4.4]). Hence, since A t (F ) is finite by hypothesis,…”

“…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]. If L/F is a finite extension the group Sel A (L) ℓ is a cofinitely generated Z ℓ -module (see, e.g.…”

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

“…When this is the case, since H has infinite index in G, Sel A (K ) _ is also a torsion 3(G)module. For some related results regarding the presence of pseudo-null submodules in Sel A (K ) _ (for`6 = p), see [7].…”