Curves of genus greater than one

Abstract

In this chapter we shall look at an area which has attracted a growing amount of attention in recent years, namely work on curves of genus greater than one. This has come about as the theory of elliptic curves (or curves of genus one) has reached a remarkably advanced state. Work with curves of genus two has been proceeding for over a century but only in the last few years have there been a lot of arithmetical calculations with such curves. There are three basic diophantine questions one can ask about a curve of genus greater than one: Can one compute C (ℚ), the set of rational points? By a theorem of Faltings this set is known to be finite. Can one compute C (ℤ), the set of integral points? Can one compute the explicit structure of the Jacobian of C ? The Jacobian is an algebraic group which is associated with the curve. We shall return to this below. Clearly if we can give an efficient algorithmic answer to the first question then we already have an efficient algorithm for the second. For the current state of knowledge on curves of genus two the reader should consult [26] and [156]. Curves of higher genus have attracted attention from theoreticians but little is known of how to solve the basic diophantine problems on such curves efficiently.