Abstract

In this paper, we make a study of the Iwasawa theory of an elliptic curve at a supersingular prime p along an arbitrary ℤ_p-extension of a number field K in the case when p splits completely in K. Generalizing work of Kobayashi [S. Kobayashi, Iwasawa theory for elliptic curves at supersingular primes, Invent. Math. 152 (2003), no. 1, 1–36.] and Perrin-Riou [B. Perrin-Riou, Arithmétique des courbes elliptiques á réduction supersingulière en p, Experiment. Math. 12 (2003), no. 2, 155–186.], 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 ℤ_p-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 ℤ_p-extension of ℚ_p.