Main Conjectures for Families of Automorphic Forms

我的研究项目旨在解决数论中两个领域的问题，这两个领域目前正经历着令人印象深刻的发展，并且密切相关：Iwasawa理论和自同构形式的p进族。我的项目有许多分支的可能性，这将为硕士生和博士生提供有趣的问题。******数论最重要的目标之一是研究某些多项式方程组的积分解；这是使用代数几何和分析的方法来完成的。众所周知，对于大多数“动机”，人们可以将两个对象联系起来：l函数和Selmer群（它推广了各种有理点的概念）。一系列的推测表明，关于l函数的消失阶和导系数的信息可以转化为关于Selmer群的信息。******对于这些猜想，一个非常有效的方法是通过p进变形。给定一个动机，我们可以研究它在特定的场扩展塔上是如何变化的，比如由单位的p^n个根产生的塔。在许多情况下，可以定义p进l函数和扩展塔上的“大”Selmer群。Iwasawa， Greenberg和Benois的猜想表明，“大”Selmer群是具有Z_p系数的形式级数环上的(co)-扭转，其特征理想是由p进l函数生成的。这通常意味着布洛赫-加藤猜想。******我的研究计划的第一个目标，在与Z. Liu （McGill）的一个联合项目中，每当常数项（即p进l函数）消失时，可以使用Ribet的思想来构建Selmer群中的共环，可以用来约束其大小，从而证明两个理想之间的包含。******这表明p进族对于主要猜想是必要的，但到目前为止，只有当Shimura变种的p-普通轨迹不为空时，才能构造p进族，这排除了许多情况。这就引出了我项目的第二个目标。与R. Brasca （Universite Paris 7）合作，我们正在开发无普通位点的PEL志村品种的Hida理论。其思想是将全称p可分群的乘法部分替换为它的整个过滤连通部分。这种方法主要利用了Dieudonne模块的过滤特性。因此，我们的后续项目是将其推广到志村品种，这些品种不是PEL类型，而是映射到G-zips堆栈，“具有额外结构的Dieudonne模块”，例如正交志村品种。期望将其应用于p进l函数的构造。

My research project aims at solving problems in two areas of number theory which are currently experiencing an impressive development and are closely related: Iwasawa theory and p-adic families of automorphic forms. My project has possibilities for many ramifications which would provide interesting problems for Masters' and PhD students.******One of the most important goals of number theory is the study of integral solutions of certain sets of polynomial equations; this is done using methods from algebraic geometry and analysis. It has been known that to most of "motives" one can associate two objects: the L-function and the Selmer group (which generalizes the concept of rational points of a variety). A series of conjectures state that information on the vanishing order and the leading coefficient of the L-function can be translated to information about the Selmer group. ******A very fruitful approach to these conjectures is via p-adic deformations. Given a motive, one can study how it varies over certain towers of field extensions, such as the tower generated by the p^n-th roots of unity. In many cases, one can define a p-adic L-function and a "big" Selmer group over the tower of extensions. Conjectures by Iwasawa, Greenberg, and Benois state that the "big" Selmer group is (co)-torsion over the ring of formal series with Z_p coefficients, and that its characteristic ideal is generated by the p-adic L-function. This often implies the Bloch--Kato conjecture. ******The first objective of my research program, in a joint project with Z. Liu (McGill whenever the constant term (i.e. the p-adic L-function) vanishes, one can use ideas of Ribet to construct co-cycles in the Selmer group which can be used to bound its size, and hence proving an inclusion between the two ideals. ******This shows that p-adic families are necessary for Main Conjectures but so far families have been constructed only when the p-ordinary locus of the Shimura variety is not empty, which excludes many cases. This leads us to the second objective of my project. With R. Brasca (Universite Paris 7) we are developing Hida theory for PEL Shimura varieties without ordinary locus. The idea is to substitute the multiplicative part of the universal p-divisible group with its whole filtered connected part. This approach uses mainly properties of the Dieudonne modules with its filtration. Hence, our subsequent project is to generalize this to Shimura varieties which are not of PEL type but with a map to the stack of G-zips, "Dieudonne modules with extra structure", such as the orthogonal Shimura varieties. Application to the construction of p-adic L-functions are expected.