Selmer Groups and AnticyclotomicZ_p-extensions

Abstract

Abstract. Let E/Q be an elliptic curve, p a prime and K∞/K the anticyclotomic Zp-extension of a quadratic imaginary field K satisfying the Heegner hypothesis. In this paper we give a new proof to a theorem of Bertolini which determines the value of the Λ-corank of Sel p ∞ (E/K∞) in the case where E has ordinary reduction at p. In the case where E has supersingular reduction at p we make a conjecture about the structure of the module of Heegner points mod p. Assuming this conjecture we give a new proof to a theorem of Ciperiani which determines the value of the Λ-corank of Sel p ∞ (E/K∞) in the case where E has supersingular reduction at p.

Fine Selmer groups, Heegner points and anticyclotomic ℤp-extensions

“…These observations imply that to prove the proposition, we need to show that for any n if s ∈ Sel p (E/K n ) and res ℓ (s) = 0 for all ℓ ∈ L (U ), then s = 0. This can be shown exactly as in [9] proposition 2.8.…”

“…Consider the Λ-module M := lim − → H 1 (K n,ℓ , E)[p] where the direct limit is taken with respect to the restriction maps. In [9] proposition 2.5 we showed that M is a cofree Λ-module of rank 2 and that M Γn = H 1 (K n,ℓ , E)[p] for any n. Therefore the proposition follows from [10] proposition 2.1. Now for any n let L n = K n (E[p]) and G n = Gal(L n /K n ) which is isomorphic to GL 2 (F p ) by [9] lemma 2.3.…”

“…The claim for n follows. As in [9], we now describe the construction of Kolyvagin classes over ring class fields. For the rest of this section we assume that (E, p) satisfies (⋆).…”

“…However we should note that our proof relies on the fact mentioned above that corank Λ (Sel p (E/K ∞ )) = 1. Therefore this article is a natural continuation of [9].…”

“…Now assume that E has ordinary reduction at p. In [9] we analyzed the images of the maps ψ ℓ for an infinite set of primes ℓ to conclude that rank Λ (lim ← − Sel p (E/K n )) ≤ 1 (see [10] theorem 3.1). Now consider the group lim ← − Sel p (E/K ∞ ) Γn where the inverse limit is taken with respect to the norm maps.…”