Mahler measure and curves over finite fields

数论是研究与整数有关的问题：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(素数是整数的基石)。第一个问题的一个例子是与椭圆曲线的算术有关的Birch-Swinnerton-Dyer猜想。通过证明关于黎曼Zeta函数零点分布的黎曼假设，第二个问题可以高精度地回答。 请注意，这是粘土研究所七个千禧年问题中的两个，每个问题的奖金为100万美元。这两个问题在其他领域都有影响，比如密码学。 我们的工作将构成这两个问题的一些因素联系起来。事实上，我们得到了一些猜想的例子，它们是除椭圆曲线以外的方程的Birch-Swinnerton-Dyer猜想的推广。人们对这些普遍的猜想知之甚少，即使是获得例子也是一项艰巨的任务。这些例子还给出了关于Riemann Zeta函数及其推广L函数的特殊值的信息。我们还研究了某些Zeta函数的性质(与某些方程的解的个数直接相关)，例如它们的零点分布。 我们结合了许多数学领域的技术，如组合学、代数几何、复分析、拓扑学、图论、微分方程式等。

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.