Computational Aspects of Hyperelliptic Curves

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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.