论文标题
一致性模块和较高编码中的Wiles-Lenstra-Diamond数值标准
Congruence modules and the Wiles-Lenstra-Diamond numerical criterion in higher codimensions
论文作者
论文摘要
我们定义了与过滤$ \ Mathcal O $ -Algebra形态相关的一致模块$ψ_A（m）$ $ $ \ mathfrak {p} $，$λ$的内核和有限生成的$ a $ module。我们建立了$ m $的数值标准，以超过$ $ $的正等级的免费直接汇总。就$ψ_A（m）$的长度和$ \ mathfrak {p}/\ mathfrak {p}^2 $的扭转部分而言。它概括了Wiles，Lenstra和Diamond的结果，这些结果涉及$ \ Mathfrak {P} $的编码为零时的情况。 数字理论应用程序包括在某些标准猜想的有条件的积极缺陷的情况下进行积分（非最小值）$ r = \ mathbb t $定理。这里$ r $是一个变形环参数为某些galois表示形式，$ \ mathbb t $是Hecke代数。一个示例是对假想二次次数的二维$ \ ell $ -adic galois表示的模块化举重。这些证明将我们的交换代数结果与胆汁和泰勒的修补方法引起的概括结合了概括 - wiles和wiles和水平提出了回到肋骨的参数。该结果提供了新的证据，以支持古典兰德兰对应关系的有趣和刚刚起步的扭转类似物。 我们还证明了无条件的积分$ r = \ mathbb t $ hecke代数$ \ mathbb t $作用于$ \ mathbb q $的Shimura曲线的共同体。这导致了扭力jacquet-山林对应关系，比较了作用于shimura曲线和模块化曲线的重量共同体的积分代数。在这种情况下，共同体学具有丰富的扭转，因此我们的对应关系不能通过经典的jacquet--langlands对应来推导。
We define a congruence module $Ψ_A(M)$ associated to a surjective $\mathcal O$-algebra morphism $λ\colon A \to \mathcal{O}$, with $\mathcal{O}$ a discrete valuation ring, $A$ a complete noetherian local $\mathcal{O}$-algebra regular at $\mathfrak{p}$, the kernel of $λ$, and $M$ a finitely generated $A$-module. We establish a numerical criterion for $M$ to have a free direct summand over $A$ of positive rank. It is in terms of the lengths of $Ψ_A(M)$ and the torsion part of $\mathfrak{p}/\mathfrak{p}^2$. It generalizes results of Wiles, Lenstra, and Diamond, that deal with the case when the codimension of $\mathfrak{p}$ is zero. Number theoretic applications include integral (non-minimal) $R=\mathbb T$ theorems in situations of positive defect conditional on certain standard conjectures. Here $R$ is a deformation ring parametrizing certain Galois representations and $\mathbb T$ is a Hecke algebra. An example is a modularity lifting for 2-dimensional $\ell$-adic Galois representations over an imaginary quadratic field. The proofs combine our commutative algebra results with a generalization due to Calegari and Geraghty of the patching method of Wiles and Taylor--Wiles and level raising arguments that go back to Ribet. The results provide new evidence in favor of the intriguing, and as yet fledgling, torsion analog of the classical Langlands correspondence. We also prove unconditional integral $R=\mathbb T$ results for Hecke algebras $\mathbb T$ acting on weight one cohomology of Shimura curves over $\mathbb Q$. This leads to a torsion Jacquet--Langlands correspondence comparing integral Hecke algebras acting on weight one cohomology of Shimura curves and modular curves. In this case the cohomology has abundant torsion and so our correspondence cannot be deduced by means of the classical Jacquet--Langlands correspondence.
