TAILIEUCHUNG - Đề tài " The main conjecture for CM elliptic curves at supersingular primes "

At a prime of ordinary reduction, the Iwasawa “main conjecture” for elliptic curves relates a Selmer group to a p-adic L-function. In the supersingular case, the statement of the main conjecture is more complicated as neither the Selmer group nor the p-adic L-function is well-behaved. Recently Kobayashi discovered an equivalent formulation of the main conjecture at supersingular primes that is similar in structure to the ordinary case. Namely, Kobayashi’s conjecture relates modified Selmer groups, which he defined, with modified padic L-functions defined by the first author. In this paper we prove Kobayashi’s conjecture for elliptic curves with complex multiplication. . | Annals of Mathematics The main conjecture for CM elliptic curves at supersingular primes By Robert Pollack and Karl Rubin Annals of Mathematics 159 2004 447-464 The main conjecture for CM elliptic curves at supersingular primes By Robert Pollack and Karl Rubin Abstract At a prime of ordinary reduction the Iwasawa main conjecture for elliptic curves relates a Selmer group to a p-adic L-function. In the supersingular case the statement of the main conjecture is more complicated as neither the Selmer group nor the p-adic L-function is well-behaved. Recently Kobayashi discovered an equivalent formulation of the main conjecture at supersingular primes that is similar in structure to the ordinary case. Namely Kobayashi s conjecture relates modified Selmer groups which he defined with modified p-adic L-functions defined by the first author. In this paper we prove Kobayashi s conjecture for elliptic curves with complex multiplication. Introduction Iwasawa theory was introduced into the study of the arithmetic of elliptic curves by Mazur in the 1970 s. Given an elliptic curve E over Q and a prime p there are two parts to such a program an Iwasawa-Selmer module containing information about the arithmetic of E over subfields of the cyclotomic Zp-extension Q of Q and a p-adic L-function attached to E belonging to a suitable Iwasawa algebra. The goal or main conjecture is to relate these two objects by proving that the p-adic L-function controls in precise terms is a characteristic power series of the Pontrjagin dual of the Iwasawa-Selmer module. The main conjecture has important consequences for the Birch and Swinnerton-Dyer conjecture for E . The first author was supported by an NSF Postdoctoral Fellowship. The second author was supported by NSF grant DMS-0140378. 2000 Mathematics Subject Classification. Primary 11G05 11R23 Secondary 11G40. 448 ROBERT POLLACK AND KARL RUBIN For primes p where E has ordinary reduction Mazur introduced and studied the Iwasawa-Selmer module Ma .

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