Coates J, Howson S
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge, CB2 1SB, United Kingdom.
Proc Natl Acad Sci U S A. 1997 Oct 14;94(21):11115-7. doi: 10.1073/pnas.94.21.11115.
Let E be a modular elliptic curve over [symbol, see text], without complex multiplication; let p be a prime number where E has good ordinary reduction; and let Finfinity be the field obtained by adjoining [symbol, see text] to all p-power division points on E. Write Ginfinity for the Galois group of Finfinity over [symbol, see text]. Assume that the complex L-series of E over [symbol, see text] does not vanish at s = 1. If p >/= 5, we make a precise conjecture about the value of the Ginfinity-Euler characteristic of the Selmer group of E over Finfinity. If one makes a standard conjecture about the behavior of this Selmer group as a module over the Iwasawa algebra, we are able to prove our conjecture. The crucial local calculations in the proof depend on recent joint work of the first author with R. Greenberg.
设(E)是定义在(\mathbb{Q})上的一条模椭圆曲线，且无复乘法；设(p)是使得(E)具有良好普通约化的素数；设(F_{\infty})是通过将(\mathbb{Q})添加到(E)上所有(p)次幂除点所得到的域。记(G_{\infty})为(F_{\infty})在(\mathbb{Q})上的伽罗瓦群。假设(E)在(\mathbb{Q})上的复(L) - 级数在(s = 1)处不为零。如果(p\geq5)，我们对(E)在(F_{\infty})上的塞尔默群的(G_{\infty}) - 欧拉特征值的值提出一个精确的猜想。如果对这个塞尔默群作为岩泽代数上的模的行为做出一个标准猜想，那么我们能够证明我们的猜想。证明中的关键局部计算依赖于第一作者与(R.)格林伯格最近的合作成果。
