Computational Aspects of Hyperelliptic Curves

在这个项目中，Poonen教授将装配一个雅可比矩阵具有小导体的超椭圆曲线表。表中还应包括雅可比矩阵的modell - weil群的结构和生成子、雅可比矩阵的自同态环、bad约简的素数等信息。不幸的是，没有已知的方法来计算表的某些部分，而生成这样的方法是该项目的主要部分。计算机算法的发展应该对超椭圆曲线和使用这些算法生成的表格有同样多的了解。在一种观点中，曲线的形状就像带有手柄的球体表面。在这种观点中，最简单的曲线就像一个没有手柄的球体表面；椭圆曲线只有一个柄，更复杂的曲线有几个柄。自古以来，人们就对简单曲线进行了广泛的研究，并对其有了很好的理解。椭圆曲线具有非常丰富的结构，这仍然是相当神秘的。今天，有大量系统的椭圆曲线列表，并将其最重要的特征信息制成表格。这个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相同的性质，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教授对如何计算更难以捉摸的部分有着坚实的想法。他的表格将是数学家的重要参考资料。

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.