The main conjecture for CM elliptic curves at supersingular primes.

*(English)*Zbl 1082.11035The 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.

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.

Reviewer: R. Sujatha (Bombay)