Computational Aspects of Hyperelliptic Curves

摘要 Bjorn Poonen加州大学伯克利分校98 01104 在这个项目中，Poonen教授将组装一个表的超椭圆曲线的雅可比有小导体。该表应包括其他数据，如雅可比矩阵的Mordell-Weil群的结构和生成元，雅可比矩阵的自同态环，以及关于坏约化的素数的信息。不幸的是，没有已知的方法来计算表的部分，并产生这样的方法是该项目的主要部分。 计算机算法的发展应该像这些算法用来产生的表格一样揭示超椭圆曲线。 在一个视图中，曲线的形状类似于附加了控制柄的球体的曲面。 在这种观点中，最简单的曲线就像没有手柄的球面;椭圆曲线有一个手柄，而更复杂的曲线有几个手柄。 简单曲线自古以来就被广泛研究，并且非常容易理解。 椭圆曲线具有非常丰富的结构，仍然相当神秘。今天有广泛的系统列表的椭圆曲线表与信息，他们最重要的特点。 该项目的主要目标是开始为一类更复杂的曲线构建表格。超椭圆曲线与椭圆曲线有许多共同的性质，但有更多的句柄。 目前，数学家们只知道如何计算超椭圆曲线的一些重要特征，但Poonen教授对如何获得更难以捉摸的部分有着坚实的想法。 他的表将是一个重要的参考数学家。

ABSTRACT Bjorn Poonen University of California, Berkeley 98 01104 In this project, Professor Poonen will assemble a table of hyperelliptic curves whose Jacobians have small conductor. The table should include other data such as the structure and generators of the Mordell-Weil group of the Jacobian, the endomorphism ring of the Jacobian, and information about the primes of bad reduction. Unfortunately there are no known methods for calculating parts of the table, and producing such methods is the major part of the project. The development of computer algorithms should shed as much light on hyperelliptic curves as the tables these algorithms are used to produce. In one view, curves are shaped like the surface of a sphere with handles attached. In this view, the simplest curves are like the surface of a sphere without a handle; elliptic curves have one handle, and more complicated curves have several. The simple curves have been extensively studied since ancient times and are pretty well understood. Elliptic curves have a very rich structure that is still rather mysterious. Today there are extensive systematic lists of elliptic curves tabulated with information about their most important features. The main goal of this project is to start building tables for a class of more complicated curves. Hyperelliptic curves share many properties with elliptic curves, but have more handles. Right now, mathematicians only know how to calculate some of the important features of hyperelliptic curves, but Professor Poonen has solid ideas about how to get at the more elusive parts. His tables will be an important reference for mathematicians.