Equivariant Tamagawa Numbers/Deformation Theory

等变玉川数/变形理论

• 批准号: 0088930
• 负责人: Matthias Flach
• 金额: $ 10.24万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0088930&HistoricalAwards=false
• 关键词: Equivariant; Tamagawa; Numbers; Deformation; Theory

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.

这是一个算术代数几何的课题，由两部分组成。第一部分研究了数域上l -函数的变量（动机）值。在之前的工作中，研究者和他的合作者已经将BlochKato， Fontaine， Perrin-Riou的猜想扩展到具有（可能非交换）系数的动机。同时，他们在这些猜想和经典伽罗瓦模理论之间建立了精确的联系，从而将后者领域的各种猜想和定理推广到任意动机。有了这张推测的图片，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数域上的Tate动机，CM椭圆曲线（非极大阶CM）和模形式的伴随（积分Hecke代数的作用）。最后两个目前正在调查的学生的调查员。同样触手可及似乎证明了等变特值猜想与l函数的泛函方程的相容性。项目的第二部分是一个相当具体的问题，在变形理论（方案，矢量束或营利群的表示）。利用余切配合物的理论可以定义更高的Kodaira Spencer图，并提出对这些图的注入性进行研究。在度1中，这是已知的，并导致变形环的平滑准则。最感兴趣的情况是2度，其中类似的注入率将导致变形环是局部完全相交的简单准则。这是数论领域的一个项目，自从巴比伦人发现可以存在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。根据毕达哥拉斯定理，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思想的更深层次的联系而丰富的，而不是在这个介绍性的例子中暗示的那些。l函数的特殊值理论就是一个很好的例子。虽然是通过计算方程的模素数的解来定义的，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下去。人们可以证明的例子似乎都依赖于一些令人愉快的巧合，尽管它们遵循一种模式，使人们猜测（“猜想”）通常会发生什么。这很像实验科学中的情况。研究者和他的合作者已经将这种情况推广到方程系统有一些额外的对称性的情况，他们现在试图收集进一步的证据来证明（或者确实证伪！）他们的猜想。就应用而言，数论学家可能会同意l函数是他们领域的关键概念。另一方面，数论作为一个整体，不再需要为其适用性辩护，因为互联网上的许多密码学和编码都是基于它的结果。