P-adic L-functions and explicit reciprocity laws

数论中一个重要的研究领域涉及算术对象（如椭圆曲线）的代数性质与与之相关的解析函数（l函数）的值之间的关系。最著名的例子是Birch和Swinnerton-Dyer猜想（克莱千禧年奖问题之一），它预测椭圆曲线上有理点集的大小是由其l函数在特定点的行为决定的——特别是，当且仅当l函数在该点的值为零时存在有理点。然而，这只是一个更普遍的主题的第一个例子。近年来，使用一种称为“欧拉系统”的代数工具，在理解l函数和算术之间的联系方面取得了一些非常令人兴奋的进展。这些都是强大的工具，但很难构建。在我之前与Lei和Zerbes在2014年的工作中，我发现了一个由模形式的乘积产生的新的欧拉系统，并且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。我目前的研究重点是试图找到一种系统的方法来构建新的欧拉系统，使用各种数学领域的方法，包括表示理论和代数几何；这项研究是由英国皇家学会资助的。我的团队已经发现了几个欧拉系统的新例子，包括一个与西格尔模形式有关的例子，它可能会对2属代数曲线的算法产生非常有趣的结果。然而，我们对这些对象的理解有一个明显的差距，那就是在很多情况下，我们无法证明新对象不为零。在欧拉系统的早期构造中，这种输入是由称为“显式互易律”的定理提供的，该定理将欧拉系统与l函数的值联系起来。提出的研究目标是证明一些新发现的欧拉系统的显互易律。直到最近，这个问题似乎完全无法解决；但最近文森特·皮罗尼（Vincent Pilloni）在p进自同构形式理论上的突破，提出了一种解决这个问题的策略。

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.