Iwasawa Theory of Modular Elliptic Curves of Analytic Rank at most 1
Iwasawa Theory of Modular Elliptic Curves of Analytic Rank at most 1
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解析阶最多 1 的模椭圆曲线岩泽理论
DOI：
10.1112/jlms/50.2.243
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发表时间：
1994
影响因子：
1.2
通讯作者：
Gary McConnell
中科院分区：
文献类型：
--
作者：
J. Coates;Gary McConnell
Let E be an elliptic curve over Q. We assume that E is modular. Let L {E, s) be the Hasse-Weil L-series of E, so that the analytic continuation and functional equation for L (E, s) are known. Let rE be the multiplicity of the zero of L (E, s) at s= 1 (rB is usually called the analytic rank of E). Kolyvagin [5] has proven the deep theorem that if rE^ 1 then the group E (Q) of rational points of E has rank rE, and the Tate-Shafarevich group UL (E) of E over Q is finite. The aim of the present paper is to point out some simple consequences of Kolyvagin's theorem for the arithmetic of E over the cyclotomic/^-extension of Q (that is, the unique subfield of the field generated over O by all/>-power roots of unity, whose Galois group over Q is topologically isomorphic to the additive group of the ring Zp of p-adic integers) for all odd primes p. To our knowledge, these consequences have not been pointed out before, and it seems to us interesting that they are valid irrespective of whether p is a bad, ordinary or supersingular prime for E. As above, let p denote any odd prime number, and let Ep<* be the group of all p-power division points on E. Throughout, S will denote any finite set of nonarchimedean primes of Q, which is always assumed to contain/? and the primes of bad reduction of E. Let Qs be the maximal extension of Q which is unramified outside S and the place at infinity, and write Gs= G (QS/Q) for the Galois group of Qs over< Q>. As is well known, Q (Ep «>) cr Qs, and so we can regard Ep<* as a Gs-module. Let FK denote the cyclotomic Zp-extension of Q. Since F^ is unramified outside/?, we have
