Class Numbers of Global Function Fields

Rosen教授和J. Hoffstein教授几年前发展了在有理函数域上由二次元构成的Dirichlet l级数的平均值定理。取s=1处的特殊值，可以导出超椭圆曲线类数的平均值定理。如果常域是一个有q个元素的有限域，则（q-1） st个单位根在基域中。如果l是一个素数除q-1，则l次基域的每一个循环扩展都是Kummer扩展。因此，似乎可以对具有l阶特征的Dirichlet l系列（以及此类系列的某些乘积）求平均值。此外，通过取s=1处的特殊值，可以期望得到二次情况下的类数平均值。Rosen教授已经开始了这个项目的工作，并发表了一些有希望的结果。他将研究这些一般性考虑所提出的若干具体问题。本研究属于数论的一般数学领域。数论的历史根源在于对整数的研究，解决诸如识别素数和将整数分解为素数的乘积等问题。它是数学中最古老的分支之一，人们为了纯粹的美学原因而追求了许多世纪。然而，在过去的半个世纪里，它已经成为数据传输和处理以及通信系统等各种应用领域不可或缺的工具。

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.