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The anticyclotomic main conjecture for elliptic curves at supersingular primes. (English) Zbl 1146.11057
Let $$E$$ be an elliptic curve over $$\mathbb{Q}$$ of conductor $$N_0$$. Let $$K$$ be an imaginary quadratic field of discriminant prime to $$N_0$$. Chose a rational prime $$p$$ and let $$K_\infty$$ denote the anticyclotomic $$\mathbb{Z}_p$$-extension of $$K$$. The anticyclotomic main conjecture of Iwasawa theory in the ordinary case was studied by M. Bertolini and H. Darmon in [Ann. Math. (2) 162, No. 1, 1–64 (2005; Zbl 1093.11037)]. Let $${\mathcal C}$$ be the characteristic power series of the Pontriyagin dual of the Selmer group $$\text{Sel}(K_\infty, E_{p^\infty})$$ (we put $${\mathcal C}= 0$$ if $$\text{Se1}(K_\infty, E_{p^\infty})^\wedge$$ is not torsion over the Iwasawa algebra $$\Lambda$$) and let $$L_p(E, K)$$ denote the $$p$$-adic $$L$$-function. They proved (under certain technical hypotheses) that $${\mathcal C}|L_p(E, K)$$ in $$\Lambda$$.
The authors of this paper formulate and prove analogous results in the case where $$p$$ is a prime of supersingular reduction. The foundational study of supersingular main conjecture carried out by Perrin-Riou, Pollack, Kurihara, Kobayashi, Iovita and others, are required to handle this case in which many of the simplifying features of the ordinary setting break down. In this case we have two $$p$$-adic $$L$$-functions $$L_p^\pm(E, K)$$ and two restricted Selmer groups. The main conjecture in this case is formulated as follows: the characteristic power series $${\mathcal C}^+$$ and $${\mathcal C}^-$$ generate the same ideal of the Iwasawa algebra $$\Lambda$$ as the $$p$$-adic $$L$$-functions $$L_p^+(E, K)$$ and $$L_p^-(E, K)$$ respectively. The main result of this article (Theorem 1.4) says that, under certain technical hypothesis, $${\mathcal C}^+|L^+_p(E,K)$$ and $${\mathcal C}^-|L_p^-(E, K)$$.

##### MSC:
 11R23 Iwasawa theory 11G05 Elliptic curves over global fields
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