tailieunhanh - Đề tài " Finding large Selmer rank via an arithmetic theory of local constants "

We obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields. Suppose K/k is a quadratic extension of number fields, E is an elliptic curve defined over k, and p is an odd prime. Let K− denote the maximal abelian p-extension of K that is unramified at all primes where E has bad reduction and that is Galois over k with dihedral Galois group (., the generator c of Gal(K/k) acts on Gal(K− /K) by inversion). We prove (under mild hypotheses on p) that if the Zp -rank of the pro-p Selmer group Sp. | Annals of Mathematics Finding large Selmer rank via an arithmetic theory of local constants By Barry Mazur and Karl Rubin Annals of Mathematics 166 2007 579 612 Finding large Selmer rank via an arithmetic theory of local constants By BARRy Mazur and Karl Rubin Abstract We obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields. Suppose K k is a quadratic extension of number fields E is an elliptic curve defined over k and p is an odd prime. Let K- denote the maximal abelian p-extension of K that is unramified at all primes where E has bad reduction and that is Galois over k with dihedral Galois group . the generator c of Gal K k acts on Gal K- K by inversion . We prove under mild hypotheses on p that if the Zp-rank of the pro-p Selmer group Sp E K is odd then rankzpSp E F F K for every finite extension F of K in K-. Introduction Let K k be a quadratic extension of number fields let c be the nontrivial automorphism of K k and let E be an elliptic curve defined over k. Let F K be an abelian extension such that F is Galois over k with dihedral Galois group . a lift of the involution c operates by conjugation on Gal F K as inversion x x-1 and let X Gal F K Qx be a character. Even in cases where one cannot prove that the L-function L E K x s has an analytic continuation and functional equation one still has a conjectural functional equation with a sign e E K x nv e E KV Xv 1 expressed as a product over places v of K of local -factors. If e E K x 1 then a generalized Parity Conjecture predicts that the rank of the x-part E F x of the Gal F K -representation space E F Q is odd and hence positive. If F K is odd and F K is unramified at all primes where E has bad reduction then e E K x is independent of X and so the Parity Conjecture predicts that if the rank of E K is odd then the rank of E F is at least F K . The authors are supported by NSF grants DMS-0403374 and DMS-0457481 respectively. 580 BARRY MAZUR AND KARL RUBIN .