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）的方程的解，以及素数（它们是整数的组成部分）的分布。第一个问题的一个例子是Birch-Swinnerton-Dyer猜想，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zeta函数零点分布的黎曼假设来高精度地回答。请注意，这是克莱研究所的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。我们的工作涉及构成这两个问题的一些成分。事实上，我们得到的例子是广义的Birch-Swinnerton-Dyer代数方程以外的椭圆曲线。对这些一般性的知识知之甚少，甚至获得例子也是一项艰巨的任务。这些例子还提供了关于黎曼zeta函数的特殊值及其推广，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），例如它们的零点的分布。我们结合联合收割机技术从许多领域的数学，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.******