Equivariant Tamagawa Numbers/Deformation Theory

这是一个算术、代数、几何学的项目，分为两部分。第一部分研究数域上簇（模）的L-函数的值。在以前的工作中，Reverator和他的合作者已经将BlochKato，方丹，Perrin-Riou的理论扩展到具有（可能是非交换的）系数的动机。与此同时，他们建立了这些定理与经典伽罗瓦模理论之间的精确联系，从而将后者领域的各种定理和定理推广到任意动机。随着自然图景的稳固，现在的主要任务是证明更多的案例。目前的技术似乎可以达到的是Tate动机的数域，CM椭圆曲线（CM的非最大阶）和伴随的模形式（与行动的积分Hecke代数）。最后两个是由研究者的学生们研究的。同样的within reach似乎是一个证明的兼容性的同变specialvalue代数与功能方程的L-function.The第二部分的项目是一个相当具体的问题deformationtheory（的计划，向量丛或表示的profitegroups）。利用余切复形的理论可以定义高阶科代拉Spencer映射，并研究了这类映射的内射性。在1次中，这是已知的，并且导致变形环的光滑度的苛刻。最令人感兴趣的情况是2度，其中类似的内射性将导致变形环成为局部完全相交的简单判据。这是数论中的一个项目，自从巴比伦人发现可以有所有边长为整数且一个角为90度的三角形以来，它就一直是数学遗产的一部分。现代数论仍然在寻找代数方程的整数解，但这种搜索是通过与几何和拓扑学思想的更深层次的联系而得到的，而不是在这个介绍性的例子中所提到的. L-函数的特殊值理论就是一个很好的例子。虽然通过计算方程模素数的解来定义，但事实证明，这些函数的值有时可以用微分形式的积分或某些格的体积来表示。没有人确切地知道为什么这样的关系应该保持。人们可以证明的例子似乎都依赖于一些令人高兴的巧合，尽管它们遵循一种模式，导致人们猜测（"猜想"）一般发生了什么。这很像实验科学中的情况。这位研究者和他的合作者已经将这幅图推广到了方程组具有一些额外对称性的情况，他们现在试图收集更多的证据（或者确实是伪造的!）他们的肖像。就应用而言，数论学家可能会同意L-函数是他们领域的一个关键概念。另一方面，数论作为一个整体不再需要为它的适用性辩护，因为互联网上的许多密码学和编码都是基于它的结果。

This is a project in Arithmetic Algebraic Geometry comprisingtwo parts. The first part deals with the study of values of L-functionsof varieties (motives) over number fields. In previous work the investigatorand his collaborator have extended the conjectures of BlochKato, Fontaine, Perrin-Riou to motives with (possibly noncommutative)coefficients. At the same time they have established a precise linkbetween these conjectures and classical Galois module theory, therebygeneralizing various conjectures and theorems in this latter area toarbitrary motives. With the conjectural picture firmly in place, themain task now is to prove more cases. What seems within reach ofcurrent techniques are Tate motives over number fields, CM ellipticcurves (with CM by non-maximal orders) and the adjoint of a modularform (with action of the integral Hecke algebra). The last two arecurrently looked into by students of the investigator. Equally withinreach seems to be a proof of the compatibility of the equivariant specialvalue conjectures with the functional equation of the L-function.The second part of the project is a rather concrete question in deformationtheory (of schemes, vector bundles or representations of profinitegroups). Using the theory of the cotangent complex one can definehigher Kodaira Spencer maps and the investigator proposes to studythe injectivity of these maps. In degree 1 this is known and leads to acriterion for smoothness of the deformation ring. The case of mostinterest is degree 2 where a similar injectivity would lead to a simplecriterion for the deformation ring to be a local complete intersection.This is a project in number theory which has been part of the mathematicalheritage ever since the Babylonians discovered that there can be triangleswith all sides of integer length and one angle of ninety degrees. By thetheorem of Pythagoras this gives integer solutions of an algebraic equation.Modern number theory still looks for integer solutions of algebraic equationsbut this search is informed and enriched by much deeper connections with ideasfrom geometry and topology than the ones alluded to in this introductoryexample. The theory of special values of L-functions is a case in point. While definedby counting solutions of equations modulo prime number it turns out that valuesof these functions can sometimes be expressed in terms of integrals ofdifferential forms, or volumes of certain lattices. Nobody knows exactly why such a relationshipshould hold. The examples one can prove all seem to rely on some happycoincidences, although they follow a pattern that leads one to guess ("conjecture") whathappens in general. This is much like the situation in an experimental science. Theinvestigator and his collaborator have generalized this picture to situations where thesystem of equations has some additional symmetries and they now try to collectfurther evidence for (or indeed falsify!) their conjectures. As far as applications areconcerned, number theorists would probably agree that L-functions are a key concept in theirfield. On the other hand, number theory as a whole no longer needs to be defensive about itsapplicability, with much of cryptography and coding, on the internet andotherwise, being based on its results.