On the Iwasawa theory of CM fields for supersingular primes

Büyükboduk, Kâzım

doi:10.1090/tran/7029

On the Iwasawa theory of CM fields for supersingular primes
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Trans. Amer. Math. Soc. 370 (2018), 927-966 Request permission

Abstract:

The goal of this article is two-fold: First, to prove a (two-variable) main conjecture for a CM field $F$ without assuming the $p$-ordinary hypothesis of Katz, making use of what we call the Rubin-Stark $\mathcal {L}$-restricted Kolyvagin systems, which is constructed out of the conjectural Rubin-Stark Euler system of rank $g$. (We are also able to obtain weaker unconditional results in this direction.) The second objective is to prove the Park-Shahabi plus/minus main conjecture for a CM elliptic curve $E$ defined over a general totally real field again using (a twist of the) Rubin-Stark Kolyvagin system. This latter result has consequences towards the Birch and Swinnerton-Dyer conjecture for $E$.

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