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The main conjecture for CM elliptic curves at supersingular primes. (English) Zbl 1082.11035
The main conjectures in Iwasawa theory predict a relationship between an (analytically defined) $$p$$-adic $$L$$-function and an algebraically defined characteristic element for the Selmer group of an elliptic curve. When $$E$$ is an elliptic curve over a number field and $$p$$ an odd prime such that $$E$$ has good ordinary reduction at $$p$$, significant results around the main conjectures are known. However, when $$E$$ has supersingular reduction at $$p$$, it was not even clear what form a precise formulation of the Main Conjecture should take. Let $$E$$ be an elliptic curve over $${\mathbb Q}$$ and let $${\mathbb Q}_{\infty}$$ be the cyclotomic $${\mathbb Z}_p$$-extension of $${\mathbb Q}$$.
Following work of I. Kobayashi [Invent. Math. 152, 1–36 (2003; Zbl 1047.11105)] and R. Pollack [Duke Math. J. 118, 523–558 (2003; Zbl 1074.11061)], it emerged that one could define two modules Sel$$_p^+(E/{\mathbb Q}_{\infty})$$ and Sel$$_p^-(E/{\mathbb Q}_{\infty})$$, whose direct sum is the usual classical Selmer module over the Iwasawa algebra $${\mathbb Z}_p[[T]]$$, and further that the classical $$p$$-adic $$L$$-function corresponds in a precise way to two elements denoted $${\mathcal L}_E^+$$ and $${\mathcal L}_E^-$$, of the Iwasawa algebra. The main conjecture is then formulated to assert that the characteristic ideal of the + and $$-$$ part of the Selmer groups equals the ideal generated respectively by $${\mathcal L}^+_E$$ and $${\mathcal L}_E^-$$. In this paper the authors prove this conjecture when $$E$$ is an elliptic curve over $${\mathbb Q}$$ with complex multiplication.

##### MSC:
 11G05 Elliptic curves over global fields 11R23 Iwasawa theory 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
##### Keywords:
Iwasawa theory; main conjecture; supersingular primes
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