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 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 42 of the curves, our result is conditional on the Birch and Swinnerton-Dyer conjecture or on the generalized Riemann hypothesis.

Keywords:

• Primary 11D41, 11G30, 11Y50
• Secondary 14G05, 14G25, 14H25, 14H45, 14Q05
• Rational points
• curves
• solvability
• local-to-global obstruction
• descent