Fine Selmer groups, Heegner points and anticyclotomic ℤp-extensions

Abstract

Let E/Q be an elliptic curve, p a prime where E has ordinary reduction and K∞/K the anticyclotomic Zp-extension of a quadratic imaginary field K satisfying the Heegner hypothesis. We give sufficient conditions on E and p in order to ensure that Sel p ∞ (E/K∞) is a cofree Λ-module of rank one. We also show that these conditions imply that rank(E(Kn)) = p n and that X(E/Kn)[p ∞ ] = {0} for all n ≥ 0.

Anticyclotomic $\largeμ$-invariants of residually reducible Galois Representations

“…Several authors have studied classical Selmer groups of elliptic curves (more generally, abelian varieties or modular forms) in anticyclotomic Z p -extensions of imaginary quadratic fields (see for example [33,2,30]). In line with the conjecture of Coates and Sujatha, for an elliptic curve defined over the imaginary quadratic field K, its fine Selmer group over the anticyclotomic Z p -extension is expected to be Λ-cotorsion with µ fine (E/K ac ) = 0 (see [24,Conjecture B]). Most results in literature focus on the case when the residual representation is irreducible.…”

“…both E 1 and E 2 satisfy the Heegner hypothesis. It is known that µ fine (E 1 /K ac ) = 0 (see [24,Table in §4]). It follows from Theorem 5.2 that µ fine (E 2 /K ac ) = 0, as well.…”

On fine Selmer groups and signed Selmer groups of elliptic modular forms

“…Various conjectures on the structure of these groups have been formulated and they are still wild open to this date. See also [11,16,17,18,19] for some recent results on the topic. In the context of modular forms, the fine Selmer groups have been studied in [1,7,8,6].…”

On the Lambda-cotorsion subgroup of the Selmer group

“…Let p be a fixed odd prime. If K is a number field, we let K cyc be the cyclotomic Z p -extension of K and if K is an imaginary quadratic field, we let K anti be the anticyclotomic Z p -extension of K. Coates and Sujatha ( [6] conjecture A) and the author ( [16] conjecture B) have conjectured when…”

“…We give one more example. Let K = Q( √ −7) and K ∞ /K the anticyclotomic Z 29 -extension of K. By using Wuthrich's work, the author has shown ( [16] sec 4) that R p ∞ (E/K ∞ ) * is Λ-torsion with µ = 0. Hence by theorem 1.1 Sel p ∞ (E/K ∞ ) * has µ = 0.…”