## Experimental Mathematics

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

### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 19 November 2008

**Permanent link to this document**

https://projecteuclid.org/euclid.em/1227118970

**Mathematical Reviews number (MathSciNet)**

MR2433884

**Zentralblatt MATH identifier**

1218.11065

**Subjects**

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

**Keywords**

Rational points curves solvability local-to-global obstruction descent

#### Citation

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