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2008 Deciding Existence of Rational Points on Curves: An Experiment
Nils Bruin, Michael Stoll
Experiment. Math. 17(2): 181-189 (2008).


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.


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Nils Bruin. Michael Stoll. "Deciding Existence of Rational Points on Curves: An Experiment." Experiment. Math. 17 (2) 181 - 189, 2008.


Published: 2008
First available in Project Euclid: 19 November 2008

zbMATH: 1218.11065
MathSciNet: MR2433884

Primary: 11D41 , 11G30 , 11Y50
Secondary: 14G05 , 14G25 , 14H25 , 14H45 , 14Q05

Keywords: curves , descent , local-to-global obstruction , rational points , solvability

Rights: Copyright © 2008 A K Peters, Ltd.


Vol.17 • No. 2 • 2008
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