The anticyclotomic main conjecture for elliptic curves at supersingular primes.

*(English)*Zbl 1146.11057Let \(E\) 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 \({\mathcal C}^-|L_p^-(E, K)\).

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 \({\mathcal C}^-|L_p^-(E, K)\).

Reviewer: Andrzej Dąbrowski (Szczecin)