Explicit methods for rational points on curves

One of the “big problems” of number theory is to understand the set of rational points on a variety, or equivalently, the rational solutions to a system of polynomial equations. Despite thousands of years of research, we are very far from having a general method for solving all such problems. There is even some evidence that deciding the existence of a rational solution is an undecidable problem (the corresponding problem for integers, “Hilbert’s Tenth Problem”, was proved to be undecidable by Martin Davis, Hilary Putnam, Julia Robinson [6]. and Yuri Matijasevič [11]. Therefore many researchers have tried to solve special cases of the general problem. One way to subdivide the task is to classify varieties by their dimension, which can be defined as the dimension of the complex analytic space whose underlying set is the set of complex number solutions to the system. This space will be a complex manifold if the equations satisfy the differential criterion for smoothness. The study of this complex analytic space is useful for more than just classification: it was discovered in the 20th century that the geometry of this space has a profound influence on the set of rational points. The rational points on 0-dimensional varieties are easy to understand. By suitable projection, one reduces to the problem of understanding the rational roots of a polynomial in one variable with rational coefficients, and there are elementary methods for understanding these. Rational points on curves (1-dimensional varieties X) are already much harder: there is still no general algorithm for determining the set of rational points that has been proved to determine the rational points in every case. One can reduce to the case where the curve is smooth, projective, and geometrically integral, or equivalently, where the corresponding complex analytic space is a compact Riemann surface; from now on we will assume this. Then one can subdivide the problem further, according to the topological genus g of the compact Riemann surface. This nonnegative integer g can also be defined algebraically as the dimension of the space of regular differentials, or the dimension of the sheaf cohomology space H(X,OX). Major number-theoretic breakthroughs of the 20th century have given us a qualitative understanding of the set X(Q) of rational points on a (smooth, projective, geometrically integral) curve as above. Many of these results generalize to the case where the field Q of rational numbers is replaced by a finite extension, or even some other types of fields, but for simplicity we will discuss the case of Q. In the case g = 0, the problem of deciding whether X(Q) is nonempty is equivalent to the problem of deciding whether a quadratic form in three variables represents 0, and a criterion for this in terms of congruences goes back to work of Legendre. Moreover, if a rational point exists, then X is isomorphic to the projective line P over Q, and hence the rational solutions may be parametrized. For instance, the special case of (the projective closure of) the curve x + y = 1 yields the familiar parametrization of Pythagorean triples.