Iwasawa Theory of Quadratic Twists of X0（49）

作     者：Junhwa CHOI John COATES 

作者机构：Department of MathematicsPOSTECH Emmanuel College Cambridge University 

出 版 物：《Acta Mathematica Sinica,English Series》 (数学学报（英文版）)

年 卷 期：2018年第34卷第1期

页      面：19-28页

核心收录：

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

主　　题：Birch-Swinnerton-Dyer conjecture elliptic curves Iwasawa theory 

摘      要：The field K ---- Q（x/-AT） is the only imaginary quadratic field with class number 1, in which the prime 2 splits, and we fix one of the primes p of K lying above 2. The modular elliptic curve X0 （49） has complex multiplication by the maximal order O of K, and we let E be the twist of Xo （49） by the quadratic extension K（v/M）/K, where M is any square free element of O with M -- i mod 4 and （M, 7） = 1. In the present note, we use surprisingly simple algebraic arguments to prove a sharp estimate for the rank of the MordeII-Weil group modulo torsion of E over the field F∞=k（Ep∞ ）, where Epic denotes the group of p∞-division points on E. Moreover, writing B for the twist of X0（49） by K（C/ET）/K, our Iwasawa-theoretic arguments also show that the weak form of the conjecture of Birch and Swinnerton-Dyer implies the non-vanishing at s =1 of the complex L-series of B over every finite layer of the unique Z2-extension of K unramified outside p. We hope to give a proof of this last non-vanishing assertion in a subsequent paper.