Mathematical Sciences: Arguments of Gauss Sums

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This award supports the research of Professor Grant to work in number theory, specifically, the theory of Gauss sums. Gauss sums are ubiquitous in number theory, playing a role in class field theory. The argument of the quadratic Gauss sum was found by Gauss himself in 1805 after a four-year search. No progress was made on the arguments of cubic Gauss sums for 170 years, when Cassels and Mathews related the argument to elliptic functions. Professor Grant proposes to study the argument of quintic Gauss sums using the arithmetic of curves of genus 2. Specifically, he plans to apply his previous work on the arithmetic of curves of genus 2 and their Jacobians. This is research in the field of number theory. Number theory starts with the whole numbers and questions such as the divisibility of one whole number by another. It is among the oldest fields of mathematics and it was originally pursued for purely aesthetic reasons. However, within the last half century, it hlas become an essential tool in developing new algorithms for computer science and new error correcting codes for electronics.