On the Bloch-kato Conjecture for Hecke L-functions

Let K be an imaginary quadratic eld and let O K be the ring of integers of K. Let E be an elliptic curve deened over Q with complex multiplication by O. Let be the Grr ossencharacter attached to the curve E over K by the theory of complex multiplication and let be the conjugate character. For k > j > 0, k ?j is a Grr ossencharacter. We will study p? part of the conjecture of Bloch-Kato for the special value of the Hecke L-function L(?k j ; s) at s = 0 when the prime p splits in K and p > k +1. The Bloch-Kato conjecture suggests a very deep connection between the special values of a motivic L-function and the arithmetic properties of the motive. It's a vast generalization of the conjecture of Birch and Swinnerton-Dyer on elliptic curves. In our case the motive M = M k;j comes from the Grr ossencharacter k ?j and its motivic L-function coincides with the Hecke L-function L(?k j ; s). After introducing some notations and giving preliminary results in Section 1, the construction of the motive is carried out in Section 2 in a way similar to the one given in Ha]. In Section 3 we show that the p-part of the Bloch-Kato conjecture is equivalent to a formula for the special values of L(?k j ; s) proved in Gu]. Thus we obtain the following theorem Theorem 1 Let k > j > 0. Let p > k + 1 be a prime where E has good, ordinary reduction and let Tam(M) be the Tamagawa number of the motive M = M k;j. If L(?k j ; 0) 6 = 0,then the p? part of the Bloch-Kato conjecture for Tam(M) is true, that is p? partTam(M)] = #(M p =L p) GQ #X(M)(p 1) where X(M) is the Tate-Shafarevich group of M, M p = Hom(M p ; Q p (1)) is the dual of the p-adic G Q-representation induced from the Grr ossencharacter k ?j and L p is a canonical Z p-lattice in M p. Remarks: 1. As to be shown later in sec.1 L(?k j ; 0) 6 = 0 if k > j + 1. When k = j + 1 the study of L(?k j ; s) at s = 0 is traditionally more diicult, j = 0 being the case described …