Iwasawa Theory of elliptic curves at supersingular primes over higher rank Iwasawa extensions

Abstract

Suppose K is an imaginary quadratic field over which the prime p is inert, $K_{\infty }$ is its Iwasawa extension of rank 2, and E is an elliptic curve defined over K with good supersingular reduction at the prime above p. Unlike the case where p splits completely over K as in the author's previous work, no good Iwasawa Theory has been established in this case. We construct series of local points of E over $K_n$ satisfying certain norm relations by Fontaine's theory of group schemes, establish the algebraic side of Iwasawa Theory in this case, compatible with the author's theory on the analytic side, and propose a conjecture relating the algebraic side of the theory and the analytic side of it. (And, to do that, we also show that the author's previous work on the analytic side, which the author did only for the primes that split over K, also applies to the inert primes.)