Mathematics

Dr. Matteo Longo
Department of Mathematics,Universities of Padua

Let E be an elliptic curve, p a prime number, and K an imaginary quadratic field, of discriminant prime to the conductor of E, in which p is not ramified. One considers the anticyclotomic Z_p-extension of K, which we denote K_\\infty, and is characterized as the unique Z_p-extension of K which is dihedral over Q. In this setting, we formulate an anticyclotomic main conjecture, relating a suitably defined p-adic L-function (which interpolates central critical values of the L-function of E twisted by anticylclotomic characters) and the structure of a Selmer group as module over the Iwasawa algebra of K_\\infty/K. The formulation of the main conjecture depends on a number of arithmetic data, the main of which are the behavior of p in K (i.e., split or inert), the reduction type of E at p (i.e., good ordinary or supersingular) and the behavior of the conductor N of E in K (i.e., the number of prime factors of N which are inert in K). In a joint work, in progress, with M. Bertolini and R. Venerucci, for each of these situations we prove the relative version of the anticyclotomic Iwasawa main conjecture. We use of a uniform approach, which combines previous works of Bertolini-Darmon and works of Skinner-Urban-Wan. I will try to explain some of the ideas underlying the proofs of these results.