MODULAR GALOIS REPRESENTATIONS OF "NEBEN" TYPE
MODULAR GALOIS REPRESENTATIONS OF "NEBEN" TYPE
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“NEBEN”类型的模伽罗瓦表示
DOI：
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
H. Hida
中科院分区：
文献类型：
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作者：
H. Hida
1995年，A.Wiles在[W]定理0.2中证明了，在一些较温和的条件下(并依赖于[TW]中Hecke代数的一个环论结果)，如果一个p-进Galois表示与椭圆尖点形式的模p同余，则该表示本身与一个尖点形式相关。在那之后，我一直在考虑刻画剩余二面体Galois表示，因为二维诱导Galois表示总是模的。在此背景下，本文件是[H98]和[HM97]中所做调查的继续。我们在[H98](可从www.math.ucla.edu/hida下载)中证明，对于具有实二次Neben型的权k2的模Galois表示，给出剩余二面体表示的(奇)素数p总是二次扩张的正基本单位的Nεk11的因子，推广了1972年Shimura关于权2的一个结果。同时，我提出了一个描述二面体表示的泛p-普通变形环的猜想，这个猜想现在已经由Cho-Vatsal[CV](可在www.math.ubc.ca/vatsal下载)基本上解决了。然而，形变环的知识并不能给我们一种构造剩余二面体伽罗瓦表示的直接方法。在这里，我们想给出一个简单的(非常初级的)构造方法，它实际上给出了所有剩余的二面体表示。作为我们结果的一个明显的应用，我们可以自动地显示许多非有理椭圆曲线的模性，-HBAV或秩二动机，尽管我们在本文中只详细讨论了一些椭圆曲线的例子(见第5节)。在我即将出版的书[H00]中，将包括更多关于这种方法来解决动机的模块化问题的背景材料。也是我的一个学生：阿米·费希曼现在正试图利用这个想法来使[HM97]的结果更加系统化。此后，本文假定素数p为奇数。
In 1995 in [W] Theorem 0.2, A. Wiles proved, under some mild conditions (and relying on a ring theoretic result on Hecke algebras in [TW]), that if a p–adic Galois representation is congruent modulo p to that of an elliptic cusp form, then the representation itself is associated to a cusp form. After that, I have been thinking of characterizing residually dihedral Galois representations, because a 2-dim induced Galois representation is always modular. In this context, this paper is a continuation of the investigation done in [H98] and [HM97]. We have shown in [H98] (downloadable at www.math.ucla.edu/ hida) that for modular Galois representations of weight k 2 with real quadratic Neben type, the (odd) prime p giving residually dihedral representation is always a factor of N εk 1 1 for the positive fundamental unit ε of the quadratic extension, generalizing a result of Shimura in 1972 for weight 2. A simple application of this fact to the base-change problem was discussed in [HM97]. At the same time, I made a conjecture describing the universal p–ordinary deformation ring of a dihedral representation, which has been basically solved now by Cho-Vatsal [CV] (downloadable at www.math.ubc.ca/ vatsal). However the knowledge of the deformation ring does not give us an immediate method of constructing residually dihedral Galois representations. Here we would like to present a simple and (very elementary) constructive method, which actually gives all residually dihedral representations. As an obvious application of our result, we can show automatically the modularity of many non –rational elliptic –curves, –HBAV or rank two –motives, although we only discuss some examples of elliptic –curves in details in this paper (see Section 5). Some more background materials of this type of approach to the modularity problems of motives will be included in my forthcoming book [H00]. Also one of my students: Ami Fischman is now trying to use the idea here to make the result in [HM97] more systematic. Hereafter in this paper the prime p is assumed to be odd.