Mahler measure and curves over finite fields

数论是研究与整数有关的问题：1,2,3,4，…****数论中的两个中心问题是涉及整数（或有理数）的方程的求解，以及素数的分布（素数是整数的组成部分）。第一个问题的一个例子是Birch- Swinnerton-Dyer猜想，它与椭圆曲线的算术有关。第二个问题可以通过证明关于黎曼ζ函数零点分布的黎曼假设得到高精度的解答。****请注意，这是克莱研究所七个千禧年问题中的两个，每个问题的奖金为一百万美元。这两个问题在其他领域都有影响，比如密码学。****我们的工作涉及到构成这两个问题的一些因素。事实上，我们得到了一些猜想的例子，这些猜想是Birch- Swinnerton-Dyer猜想对椭圆曲线以外的方程的推广。对于这些普遍的猜想，我们知道的不多，甚至连获得例子都是一项艰巨的任务。这些例子也提供了关于黎曼ζ函数的特殊值以及它的推广，l函数的信息。我们还研究了某些zeta函数的性质（与某些方程的解的数量直接相关），例如它们的零的分布。****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.******