Mahler measure and curves over finite fields

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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.