The Taylor-Wiles construction and multiplicity one
The Taylor-Wiles construction and multiplicity one
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泰勒-怀尔斯构造和重数一
DOI：
10.1007/s002220050144
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发表时间：
1997
影响因子：
3.1
通讯作者：
Fred Diamond
中科院分区：
文献类型：
--
作者：
Fred Diamond
Wiles对Q上半稳定椭圆曲线的模性的证明依赖于Taylor和Wiles[16]的构造，证明了某些Hecke代数是完全交。这些Hecke代数是通过考虑Hecke算子在模形式为“极小级”的空间上的作用，或等价地在模曲线的同调群或雅可比上的作用来定义的。粗略地说，泰勒和怀尔斯通过“修补”不同层次的形式产生的代数来继续工作。在它们的构造中使用的深层结果之一是，模曲线的同调在某些极大理想的局部化时成为Hecke代数上的自由模(阶为2)。这个结果被称为“重数一”结果，是Mazur[11]的一个定理的推广。它的证明依赖于Deligne-Rapoport和Katz的Q-展开原理，以及mod‘Betti和de Rham上同调的比较(见[17]的第2.1节)。多重性一被认为是泰勒-威尔斯结构的一个重要组成部分，也是威尔斯证明的其他部分。本文的目的是解释如何改变[16]和[17]的论点，使多重性结果成为副产品而不是成分。这一改进背后的关键概念变化如下：我们证明了模曲线的同调是这个变形环上的自由模，而不是证明(局部化后)Hecke代数可以与mod‘Galois表示的泛变形环相同。为了实现这一点，我们改变了Taylor-Wiles结构，1)对模和代数进行“补丁”，以及2)应用Auslander-Buchsbaum
Wiles’ proof [17] of the modularity of semistable elliptic curves over Q relies on a construction of Taylor and Wiles [16] showing that certain Hecke algebras are complete intersections. These Hecke algebras are defined by considering the action of Hecke operators on spaces of modular forms of “minimal level”, or equivalently, on homology groups or Jacobians of modular curves. Taylor and Wiles proceed, roughly speaking, by “patching” algebras arising from forms of different levels. One of the deep results used in their construction was the fact that the homology of the modular curve becomes a free module (of rank two) over the Hecke algebra upon localization at certain maximal ideals. This result, known as a “multiplicity one” result, is a generalization of a theorem of Mazur [11]. Its proof relied on the q-expansion principle of Deligne-Rapoport and Katz, and the comparison of mod ` Betti and de Rham cohomologies (see Sect. 2.1 of [17]). Multiplicity one was thought to be a crucial ingredient of the Taylor-Wiles construction as well as other parts of Wiles’ proof. The purpose of this paper is to explain how to alter the arguments of [16] and [17] so that multiplicity one results are a byproduct rather than an ingredient. The key conceptual change underlying this improvement is the following: Rather than prove that (after localization) the Hecke algebra can be identified with the universal deformation ring of a mod ` Galois representation, we prove that the homology of the modular curve is a free module over this deformation ring. To carry this out, we change the Taylor-Wiles construction1 by 1) “patching” the modules as well as the algebras, and 2) applying the Auslander-Buchsbaum