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

Abstract

In this paper, we make a study of the Iwasawa theory of an elliptic curve at a supersingular prime p along an arbitrary Zp-extension of a number field K in the case when p splits completely in K. Generalizing work of Kobayashi [8] and Perrin-Riou [16], we define restricted Selmer groups and λ ± , µ ± -invariants; we then derive asymptotic formulas describing the growth of the Selmer group in terms of these invariants. To be able to work with non-cyclotomic Zp-extensions, a new local result is proven that gives a complete description of the formal group of an elliptic curve at a supersingular prime along any ramified Zp-extension of Qp.

The algebraic functional equation of an elliptic curve at supersingular primes

“…Let K ∞ be the cyclotomic Z p -extension of K. We define Sel − p (E/K ∞ ) following [5], [2], and [4]. We will explain this construction in the following sections.…”

“…The construction of plus norm subgroups in [4] does not seem to always work unlike minus norm subgroups. However, when K = Q(µ p ) or p splits completely over K/Q, we have the plus norm subgroups as constructed in [5] and [2], and we can prove the algebraic functional equation for the plus Selmer groups without modifying our technique.…”

The Plus/Minus Selmer Groups for Supersingular Primes

“…But, under the given condition, Proposition 2.3 was proven in [5]. But, under the given condition, Proposition 2.3 was proven in [5].…”

An algebraic version of a theorem of Kurihara

“…The trace relations between the c n then yield relations between the P n by diagrams (5) and (6). We have n+1/n (P n+1 ) = a p P n − n−1/n (P n−1 ),…”