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P-adic L-functions and explicit reciprocity laws

P 进 L 函数和显式互易定律

基本信息

批准号：
EP/S020977/1
负责人：
David Loeffler
金额：
$ 41.76万
依托单位：
University of Warwick
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2019
资助国家：
英国
起止时间：
2019 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FS020977%2F1
关键词：
adic functions explicit reciprocity laws

项目摘要

An important strand of research in number theory concerns the relation between algebraic properties of arithmetical objects, such as elliptic curves, and the values of analytic functions associated to them (L-functions). The best-known example of this is the Birch and Swinnerton-Dyer conjecture (one of the Clay Millennium Prize problems), which predicts that the size of the set of rational points on an elliptic curve is determined by the behaviour of its L-function at a specific point -- in particular, rational points exist if and only if the value of the L-function at this point is zero. However, this is only the first instance of a much more general theme.In recent years there has been some very exciting progress in understanding the links between L-functions and arithmetic, using an algebraic tool called an "Euler system". These are powerful tools, but difficult to construct. In my previous work with Lei and Zerbes in 2014, I discovered a new Euler system arising from products of modular forms, and this new construction has played a central role in many recent works on the BSD conjecture and related problems. The focus of my research program at present is to try to find a systematic approach to constructing new Euler systems, using methods from a variety of mathematical fields including representation theory and algebraic geometry; this research is funded by a grant from the Royal Society. My team have already found several new examples of Euler systems, including one related to Siegel modular forms which could potentially have very interesting consequences for the arithmetic of genus 2 algebraic curves.However, there is a significant gap in our understanding of these objects, which is that in many cases we cannot prove that the new objects are not zero. In the earlier constructions of Euler systems, this input was provided by theorems called "explicit reciprocity laws", which relate the Euler system to the values of an L-function. The goal of the proposed research is to prove explicit reciprocity laws for some of the newly-discovered Euler systems. Until recently this problem seemed to be entirely inaccessible; but recent breakthroughs in the theory of p-adic automorphic forms, arising from work of Vincent Pilloni, suggest a strategy for attacking the problem.

数论中一个重要的研究方向是算术对象（如椭圆曲线）的代数性质和与之相关的解析函数（L-函数）的值之间的关系。这方面最著名的例子是伯奇和斯温纳顿-戴尔猜想（克莱千年奖问题之一），它预测椭圆曲线上有理点集的大小由其L函数在特定点的行为决定-特别是，有理点存在当且仅当L函数在该点的值为零。然而，这只是一个更普遍的主题的第一个例子。近年来，在理解L函数和算术之间的联系方面有了一些非常令人兴奋的进展，使用一种称为“欧拉系统”的代数工具。这些都是强大的工具，但很难构建。在2014年我与Lei和Zerbes的工作中，我发现了一个新的由模形式的乘积产生的欧拉系统，这个新的构造在最近关于BSD猜想和相关问题的许多工作中发挥了核心作用。目前我的研究计划的重点是试图找到一个系统的方法来构建新的欧拉系统，使用的方法从各种数学领域，包括表示论和代数几何;这项研究是由赠款资助的皇家学会。我的团队已经发现了几个新的欧拉系统的例子，包括一个与Siegel模形式相关的例子，它可能对亏格2代数曲线的算术产生非常有趣的影响。然而，我们对这些对象的理解有很大的差距，那就是在许多情况下我们不能证明新的对象不为零。在欧拉系统的早期构造中，这种输入由称为“显式互易定律”的定理提供，该定理将欧拉系统与L函数的值联系起来。所提出的研究的目标是证明一些新发现的欧拉系统明确的互易律。直到最近，这个问题似乎是完全无法解决的;但最近的突破，理论的p-adic自守形式，所产生的工作文森特Pilloni，提出了一个战略攻击的问题。