Selmer groups for elliptic curves in \mathbb{Z}_l^d-extensions of function fields of characteristic p

Characteristic ideals and Selmer groups

“…s )), by the analogue of diagram (2.1)) of order bounded by |H 1 (Gal(F (p) /F t ), A[p ∞ ](F n F (p) ))|, which is finite by [4,Lemma 3.4]. Hence the inverse limit of those kernels (with respect to multiplication by powers of p) is 0 and the restriction map between Tate modules is injective.…”

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

“…Let F be a global function field of trascendence degree one over its constant field F q , where q is a power of a fixed prime p ∈ Z, and K a Galois extension of F unramified outside a finite set of primes S and such that G = Gal(K /F) is an infinite -adic Lie group with = p. Let A/F be an abelian variety: the structure of S := Sel A (K ) ∨ (the Pontrjagin dual of the Selmer group of A over K ) as a (G)-module has been extensively studied, for example, in [2], [3] and [23] (see also the short survey in [1,Section 2] and the references there) for the abelian case, and in [19], [25] and [5] for the noncommutative one (these results cover also the case = p). In most cases S has been proved to be a finitely generated (sometimes torsion) (G)-module and here we shall deal with the presence of nontrivial pseudo-null submodules in S. For the number field setting and K = F( A[ ∞ ]), this issue was studied by Ochi and Venjakob ([18,Theorem 5.1]) when A is an elliptic curve, and by Ochi for a general abelian variety in [17] (see also [10] and [11] for analogous results and/or alternative proofs).…”

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

“…In order to generalize this statement to our extensions K /F, we shall study the structure of Sel A (K ) _ (the Pontrjagin dual of the Selmer group) as a module over both 3(G) and 3(H ) via the classical tool provided by Mazur's Control Theorem (see [19] and, for recent generalizations to Z d -extensions of function fields, [3][4][5] and [34]). We will prove that Sel A (K ) _ is a finitely generated 3(G)-module and, in a similar way, we obtain that Sel A (K ) _ is a finitely generated 3(H )-module as well, provided that G contains a suitable closed subgroup H (and, for`= p, under certain hypotheses on the splitting of primes in the Z`-extension K H /F).…”