Special Values of p-adic L-Functions

现代代数数论的中心主题之一是，关于数学结构的局部信息经常可以以令人惊讶的方式拼凑在一起，从而产生关于该结构的全局信息。在数论中，从局部到全局的过程是深刻、神秘的，而且还没有被很好地理解。这一一般原理可以通过对L级数的值的精确猜想来具体化，这些值是根据数学结构的局部行为定义的某些函数。这些猜想表明，L级数在整数处的取值揭示了结构的整体算术不变量。虽然关于L级数的特值有很多非常普遍的猜想，但这些猜想只被证明是特例。本研究旨在推广这些结果，加深对数论这一重要领域的理解。数论中一些最重要和最著名的猜想与L级数的特殊值有关，包括Birch和Swinnerton-Dyer猜想，Stark，Beilinson猜想和Bloch和Kato猜想。例如，Birch和Swinnerton-Dyer猜想预测，L级数的S=1处的值包含关于有理数上椭圆曲线的点集大小的信息。L级数是由模p上的整数p上的椭圆曲线上的点数定义的。在许多情况下，所讨论的L级数有另一种表现形式，称为p-进L级数，它是通过应用与标准欧几里得几何中所遇到的不同的分析概念来定义的。经典和p元L级数的特值公式在现代数论研究中占有重要地位。在这个项目中，研究者将继续研究关于p-进L函数的特殊值的猜想。这些方法涉及研究Eisenstein级数的某些显式族，并建立在岩泽主要猜想的基础上。该项目旨在开发新的技术来证明p-进特值猜想，以及各种推广和求精。

One of the central themes in modern algebraic number theory is that local information about a mathematical structure can often be pieced together in surprising ways to yield global information about the structure. The passage from local to global in number theory is deep, mysterious, and not well-understood. This general philosophy can be made concrete through precise conjectures on values of L-series, which are certain functions defined in terms of the local behavior of mathematical structures. These conjectures assert that the values of L-series at integers reveal global arithmetic invariants of the structures. While there are plentiful and very general conjectures regarding special values of L-series, only special cases of the conjectures have been proven. This research project aims to generalize these results and deepen understanding in this important area of number theory.Some of the most important and well-known conjectures in number theory concern the special values of L-series, including the conjectures of Birch and Swinnerton-Dyer, Stark, Beilinson, and Bloch and Kato. For example, the Birch and Swinnerton-Dyer Conjecture predicts that the value at s=1 of the L-series defined in terms of the number of points on an elliptic curve over the integers modulo p for all primes p carries information on the size of the set of points of the elliptic curve over the rational numbers. In many cases, the L-series in question have alternate manifestations known as p-adic L-series, which are defined by applying a different notion of analysis from that encountered in standard Euclidean geometry. Special-value formulae for classical and p-adic L-series hold an important place at the center of modern number theory research. In this project, the investigator will continue to study conjectures on the special values of p-adic L-functions. The methods involve studying certain explicit families of Eisenstein series and build on work on the Iwasawa Main Conjecture. The project aims to develop new techniques to prove p-adic special-value conjectures, as well as various generalizations and refinements.