Experimental Mathematics

Deciding Existence of Rational Points on Curves: An Experiment

Nils Bruin and Michael Stoll

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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.

Article information

Experiment. Math., Volume 17, Issue 2 (2008), 181-189.

First available in Project Euclid: 19 November 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11D41: Higher degree equations; Fermat's equation 11G30: Curves of arbitrary genus or genus = 1 over global fields [See also 14H25] 11Y50: Computer solution of Diophantine equations
Secondary: 14G05: Rational points 14G25: Global ground fields 14H25: Arithmetic ground fields [See also 11Dxx, 11G05, 14Gxx] 14H45: Special curves and curves of low genus 14Q05: Curves

Rational points curves solvability local-to-global obstruction descent


Bruin, Nils; Stoll, Michael. Deciding Existence of Rational Points on Curves: An Experiment. Experiment. Math. 17 (2008), no. 2, 181--189. https://projecteuclid.org/euclid.em/1227118970

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