Deciding Existence of Rational Points on Curves: An Experiment

@article{Bruin2008DecidingEO,
  title={Deciding Existence of Rational Points on Curves: An Experiment},
  author={Nils Bruin and Michael Stoll},
  journal={Experimental Mathematics},
  year={2008},
  volume={17},
  pages={181 - 189}
}
  • N. Bruin, M. Stoll
  • Published 2008
  • Mathematics, Computer Science
  • Experimental Mathematics
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… Expand

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References

SHOWING 1-10 OF 34 REFERENCES
Finite coverings and rational points
Problem 3 is easy for a genus 0 curve. For a genus 1 curve, which we can turn into an elliptic curve by declaring P to be the origin, it comes down to the determination of the Mordell-Weil rank (andExpand
Heuristics for the Brauer–Manin Obstruction for Curves
  • B. Poonen
  • Mathematics, Computer Science
  • Exp. Math.
  • 2006
TLDR
If the conjecture holds, and if Tate– Shafarevich groups are finite, then there exists an algorithm to decide whether a curve over k has a k-point, and the Brauer– Manin obstruction to the Hasse principle for curves over the number fields is the only one. Expand
Towers of 2-covers of hyperelliptic curves
In this article, we give a way of constructing an unramified Galois-cover of a hyperelliptic curve. The geometric Galois-group is an elementary abelian 2-group. The construction does not make use ofExpand
Finite descent obstructions and rational points on curves
Let k be a number field and X a smooth projective k-variety. In this paper, we study the information obtainable from descent via torsors under finite k-group schemes on the location of the k-rationalExpand
Two-cover descent on hyperelliptic curves
TLDR
An algorithm is described that determines a set of unramified covers of a given hyperelliptic curve, with the property that any rational point will lift to one of the covers, if the algorithm returns an empty set. Expand
The Mordell-Weil sieve : proving non-existence of rational points on curves
We discuss the Mordell–Weil sieve as a general technique for proving results concerning rational points on a given curve. In the special case of curves of genus 2, we describe quite explicitly howExpand
THE BRAUER-MANIN OBSTRUCTION FOR CURVES
Let X be a smooth projective variety defined over a number field K. A fundamental problem in arithmetic geometry is to determine whether or not X has any K-rational points. In general this is a veryExpand
Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves
TLDR
This paper provides empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves and relates six quantities associated to a Jacobian over the rational numbers to the size of the Shafarevich-Tate group. Expand
Exhibiting SHA[2] on hyperelliptic Jacobians
Abstract We discuss approaches to computing in the Shafarevich–Tate group of Jacobians of higher genus curves, with an emphasis on the theory and practice of visualisation. Especially forExpand
The Hasse principle and the Brauer-Manin obstruction for curves
Abstract.We discuss a range of ways, extending existing methods, to demonstrate violations of the Hasse principle on curves. Of particular interest are curves which contain a rational divisor classExpand
...
1
2
3
4
...