Mahler measure and curves over finite fields

有限域上的马勒测量和曲线

基本信息

批准号：
355412-2013
负责人：
Lalin, Matilde
金额：
$ 1.68万
依托单位：
Université de Montréal
依托单位国家：
加拿大
项目类别：
Discovery Grants Program - Individual
财政年份：
2017
资助国家：
加拿大
起止时间：
2017-01-01 至 2018-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635507
关键词：
Mahler measure curves over finite

项目摘要

Number Theory is the study of questions related to the integral numbers: 1, 2, 3, 4, ...Two central questions in Number Theory are the resolution of equations involving integral (or rational) numbers, and the distribution of prime numbers (which are the building blocks of integral numbers). An example of the first question is the Birch--Swinnerton-Dyer conjecture, related to the arithmetic of elliptic curves. The second question can be answered with high precision by proving the Riemann hypothesis about the distribution of the zeros of the Riemannn zeta function.Note that these are two of the seven Millennial Problems of the Clay Institute, with a prize of one million US dollars each. Both problems have implications in other fields, like cryptography, for example.Our work relates some of the ingredients that compose both of these problems. In fact we obtain examples of conjectures that are generalizations of the Birch--Swinnerton-Dyer conjectures for equations other than elliptic curves. Not much is known about these general conjectures, and even obtaining examples is a hard task. The examples also yield information about special values of the Riemann zeta function as well as its generalizations, L-functions. We also study properties of certain zeta functions (directly related to the number of solutions of certain equations), such as the distribution of their zeroes.We combine techniques from many areas of mathematics, such as combinatorics, algebraic geometry, complex analysis, topology, graph theory, differential equations, etc.

数论是研究与整数相关的问题：1、2、3、4...数论中的两个核心问题是涉及整数（或有理数）的方程的求解，以及素数的分布（它们是整数的组成部分）。第一个问题的一个例子是 Birch--Swinnerton-Dyer 猜想，与椭圆曲线的运算有关。第二个问题可以通过证明关于黎曼 zeta 函数零点分布的黎曼假设来高精度回答。请注意，这是克莱研究所七个千禧年问题中的两个，每个奖金为 100 万美元。这两个问题对其他领域都有影响，例如密码学。我们的工作涉及构成这两个问题的一些因素。事实上，我们获得了猜想的例子，这些猜想是椭圆曲线以外的方程的 Birch--Swinnerton-Dyer 猜想的推广。人们对这些普遍猜想知之甚少，甚至获得例子也是一项艰巨的任务。这些示例还提供了有关黎曼 zeta 函数及其泛化 L 函数的特殊值的信息。我们还研究某些 zeta 函数的性质（与某些方程的解的数量直接相关），例如它们的零点分布。我们结合了许多数学领域的技术，例如组合学、代数几何、复分析、拓扑、图论、微分方程等