Galois theory, elliptic curves, and root numbers
Galois theory, elliptic curves, and root numbers
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伽罗瓦理论、椭圆曲线和根数
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发表时间：
1996
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通讯作者：
David E. Rohrlich
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作者：
David E. Rohrlich
伽罗瓦理论的逆问题是问一个任意有限群G是否可以对Q的某个伽罗瓦扩张K实现为Gal（K/Q）。当这样一个实现已经给了一个特定的G然后一个自然的续集是找到算术实现的不可约表示的G。一种可能性是要求在Q上椭圆曲线的Mordell-Weil群中实现：给定Gal（K/Q）的不可约复表示τ，是否存在Q上的椭圆曲线E，使得τ出现在Gal（K/Q）的自然表示中C Z E（K）上？本文不试图直接探讨这一问题。相反，我们采用格林伯格的观点，在他的评论nonabelian岩泽理论[5]，并考虑一个相关的问题，根号码。设ρE表示Gal（K/Q）在C ∈ Z E（K）和τ上的表示，ρE ∈ τ在ρE中的重数，记L（E，τ，s）为与E和τ相关联的张量积L-函数。Birch-Swinnerton-Dyer和Deligne-Gross的著作暗示，
The inverse problem of Galois theory asks whether an arbitrary finite group G can be realized as Gal(K/Q) for some Galois extension K of Q. When such a realization has been given for a particular G then a natural sequel is to find arithmetical realizations of the irreducible representations of G. One possibility is to ask for realizations in the Mordell-Weil groups of elliptic curves over Q: Given an irreducible complex representation τ of Gal(K/Q), does there exist an elliptic curve E over Q such that τ occurs in the natural representation of Gal(K/Q) on C ⊗Z E(K)? The present paper does not attempt to investigate this question directly. Instead we adopt Greenberg’s point of view in his remarks on nonabelian Iwasawa theory [5] and consider a related question about root numbers. Let ρE denote the representation of Gal(K/Q) on C ⊗Z E(K) and 〈τ, ρE〉 the multiplicity of τ in ρE , and write L(E, τ, s) for the tensor product L-function associated to E and τ . The conjectures of Birch-Swinnerton-Dyer and Deligne-Gross imply that