On Heterogeneous Spaces

Let C be a curve of genus g ^ 2 defined over a number field k. Faltings's proof of the Mordell conjecture [5] guarantees that C(k) is finite, yet no practical effective procedure is known for bounding either the heights of the ^-rational points or their number. If n: C -* D is a /c-morphism to another curve, then rj carries /c-points to /r-points. The problem of computing C{k) is thus reduced to determining D(k). However, such a cover is unlikely to exist if the genus of D is greater than 0, since the Jacobian of D would have to be isogenous to a factor in the Jacobian of C. On the other hand, there are arbitrarily many curves which cover C. Unfortunately, there is no a priori way to determine over which field the inverse image of A>rational points will be defined. When n: D -* C is an unramified cover, a classical theorem of Weil and Chevalley [2] determines a finite extension k' of/: such that n~{C(k)) is contained in D{k'). Since D might well cover another curve E, knowledge of the arithmetic of E can be used to study the arithmetic of C. Indeed, this method was used by Chabouty [1] to bound the number of integer points on elliptic curves, and by Kubert and Lang [7] to study rational points on hyperelliptic and superelliptic curves. But in practice, for producing curves on which one can be certain that all rational points are known, the Weil-Chevalley theorem is difficult to use. Extending the groundfield makes it harder to compute rational points. The purpose of this paper is to introduce a method by which all the rational points on certain curves can be found, not only in theory, but also in practice. We introduce certain auxiliary curves called heterogeneous spaces. These are unramified geometrically abelian covers n.D -*• C. The main result is Theorem 1.4, which produces finite sets of heterogeneous spaces r\i:Di-*C such that every A>point of C is the image of a /:-point on one of the Dv Often, each D{ will cover another curve E{, whose arithmetic is known. In this way, we can sometimes determine C{k) completely without extending the ground field. The basic results are presented in the first section, and are not very difficult to prove. The interest in the method comes from its practical applications. In Section 2 we find heterogeneous spaces which are double covers of hyperelliptic curves, and use them in Sections 3 and 4 to find all the rational points on certain curves of genus 2 and 3. Section 4 also contains the equations needed to carry out a three-descent on a general elliptic curve with a rational three-torsion point. In the final section, we use