An algebraic version of a theorem of Kurihara

Abstract

Let E/Q be an elliptic curve and let p be an odd supersingular prime for E. In this article, we study the simplest case of Iwasawa theory for elliptic curves, namely when E(Q) is finite, X(E/Q) has no p-torsion and the Tamagawa factors for E are all prime to p. Under these hypotheses, we prove that E(Q n ) is finite and make precise statements about the size and structure of the p-power part of X(E/Q n ). Here Q n is the n-th step in the cyclotomic Z p -extension of Q.

Iwasawa theory of elliptic curves at supersingular primes over ℤ p -extensions of number fields

“…Then S n /S n−1 is finite if and only if ker(π n )/Q n (ker(Tr n n−1 )) is finite by Corollary 4.12. We have that ker(π n ) ∼ = ω n−1 X/ω n X and from (18)…”

Bounding the Iwasawa invariants of Selmer groups

“…We will use a modification of this result in order to prove that the fine Selmer groups of abelian varieties over an arbitrary Z p -extension yield a Fukuda module under suitable assumptions. Note: in the case of supersingular reduction, there does not exist a control theorem for usual Selmer groups, since the cokernels of the maps pr n ∶ (X (K∞) ) n → X (K∞) n do not have bounded orders (see [25,Section 2]). The task of finding an appropriate control theorem for fine Selmer groups of elliptic curves in the supersingular case has been considered also by Kurihara (cf.…”

Iwasawa theory for elliptic curves at supersingular primes: A pair of main conjectures

“…Although Kobayashi assumed a p = 0, Pollack has pointed out in [Po1] that this could be done for a p 0 as well. Both [Kobayashi] and [Po1] also assume p is odd, but Honda's theorems hold for p = 2 as well [Ho], so we make no special assumption on a p or p in this paper until Section 7. We denote the trace map from F ss (m n+1 ) to F ss (m n ) by Tr n+1/n .…”