Deciding Existence of Rational Points on Curves: An Experiment

Abstract

In this paper we gather experimental evidence related to the question of deciding whether a curve has a rational point. We consider all genus-$2$ curves over $\Bbb Q$ given by an equation $y^2 = f(x)$ with $f$ a square-free polynomial of degree 5 or 6, with integral coefficients of absolute value at most 3. For each of these roughly 200,000 isomorphism classes of curves, we decide whether there is a rational point on the curve by a combination of techniques that are applicable to hyperelliptic curves in general.

In order to carry out our project, we have improved and optimized some of these techniques. For 2 of the curves, our result is conditional on the Birch and Swinnerton-Dyer conjecture or on the generalized Riemann hypothesis.