Computational Aspects of Hyperelliptic Curves

伯克利加州大学伯克利分校Bjorn Poonen 98 01104在这个项目中，Poonen教授将组装一个超椭圆曲线表，其雅可比曲线的导体很小。该表应该包括其他数据，例如雅可比的Mordell-Weil群的结构和生成元，雅可比的自同态环，以及关于坏归约的素数的信息。不幸的是，没有已知的方法来计算表的各个部分，而产生这样的方法是该项目的主要部分。计算机算法的发展对超椭圆曲线的启示应该与这些算法用来生成的表格一样多。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。在此视图中，最简单的曲线就像没有控制柄的球体的曲面；椭圆曲线有一个控制柄，而更复杂的曲线有几个控制柄。简单的曲线自古以来就被广泛研究，并被很好地理解。椭圆曲线有一种非常丰富的结构，至今仍相当神秘。今天，有大量系统的椭圆曲线列表，列出了关于它们最重要的特征的信息。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共同的性质，但有更多的手柄。目前，数学家只知道如何计算超椭圆曲线的一些重要特征，但普宁教授对如何计算更难以捉摸的部分有扎实的想法。他的表格将是数学家的重要参考。

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.