Equivariant Tamagawa Numbers/Deformation Theory

这是一个算术代数几何的课题，由两部分组成。第一部分研究L在数域上的变元(动机)函数的取值问题。在以前的工作中，研究者和他的合作者已经将BlochKato，Fontaine，Perrin-Riou的猜想推广到具有(可能是非对易的)系数的动机。同时，他们在这些猜想和经典Galois模理论之间建立了精确的联系，从而将后者领域中的各种猜想和定理推广到任意动机。随着猜想的图景牢牢地摆在适当的位置，现在的主要任务是证明更多的案件。目前的技术似乎可以达到数域上的Tate动机、CM椭圆曲线(与CM按非最大阶)和模形式的伴随(与积分Hecke代数的作用)。最后两起案件最近被调查员的学生调查过。同样在力所能及的范围内，似乎证明了等变的特殊值猜想与L函数的函数方程的相容。项目的第二部分是变形理论中相当具体的问题(格式、向量丛或原定群的表示)。利用余切复形理论，我们可以定义高阶Kodaira Spencer映射，并且研究者建议研究这些映射的内射性。在1阶中，这是已知的，并导致变形环的光滑性的一个判据。最令人感兴趣的情况是2阶，在这种情况下，类似的内射性将导致变形环是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都是整数长度和90度角的三角形以来，它一直是数学遗产的一部分。根据毕达哥拉斯的定理，这给出了一个代数方程的整数解。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中的概念的更深层次的联系而得到的，而不是本导论中提到的例子。L函数的特殊值论就是一个很好的例子。通过对模素数方程的解进行计数来定义，结果表明，这些函数的值有时可以用微分形式的积分或某些格子的体积来表示。没有人确切地知道为什么这种关系应该保持下去。人们可以证明的例子似乎都依赖于一些幸福的巧合，尽管它们遵循的是一种模式，导致人们猜测(“猜测”)总体上发生了什么。这很像实验科学中的情况。这位研究人员和他的合作者将这一图景推广到了方程系统具有一些额外对称性的情况，现在他们试图收集更多的证据来证明(或者实际上是证伪的！)他们的猜想。就应用而言，数学家可能会同意L函数是其领域中的一个关键概念。另一方面，数论作为一个整体不再需要对其适用性进行辩护，因为互联网上的许多密码学和编码都是基于它的结果的。

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.