Class Numbers of Global Function Fields

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Rosen 9626914 Professors Rosen and J. Hoffstein developed, some years ago, average value theorems for Dirichlet L-series formed with quadratic characters over rational function fields. Taking the special value at s=1 allows one to derive average value theorems for class numbers of hyperelliptic curves. If the constant field is a finite field with q elements, the (q-1)'st roots of unity are in the base field. If l is a prime dividing q-1, then every cyclic extension of the base field of degree l is a Kummer extension. Therefore, it seems likely that one can average Dirichlet L-series (and certain products of such series) formed with characters of order l. Also, by taking the special values at s=1, one can hope to derive class number averages as done in the quadratic case. Professor Rosen has begun work on this project and published some promising results. He will work on a number of specific problems raised by these general considerations. This research falls into the general mathematical field of Number Theory. Number theory has its historical roots in the study of the whole numbers, addressing such problems as identifying prime numbers and decomposing whole numbers as products of primes. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.