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Let $X \subset ℝ^{n}$ be a compact subanalytic surface. This paper shows that, in a suitable sense, there are very few rational points of $X$ that do not lie on some connected semialgebraic curve contained in $X$ .

Rational points on algebraic varieties over a generalized real closed field: a model theoretic approach.

Eberhard Becker, Bill Jacob (1985)

Journal für die reine und angewandte Mathematik

Rational points on certain quintic hypersurfaces

Maciej Ulas (2009)

Acta Arithmetica

Rational points on compact subgroups of algebraic groups. (Points rationnels sur les sous-groupes compacts des groupes algébriques.)

Bertrand, Daniel (1995)

Experimental Mathematics

Rational points on curves

Michael Stoll (2011)

Journal de Théorie des Nombres de Bordeaux

This is an extended version of an invited lecture I gave at the Journées Arithmétiques in St. Étienne in July 2009.We discuss the state of the art regarding the problem of finding the set of rational points on a (smooth projective) geometrically integral curve $C$ over $ℚ$ . The focus is on practical aspects of this problem in the case that the genus of $C$ is at least $2$ , and therefore the set of rational points is finite.

Rational points on elliptic curves over Q in elementary abelian 2-extensions of Q.

Martin Lorenz, Michael Laskia (1984)

Journal für die reine und angewandte Mathematik

Rational points on K3 surfaces: A new canonical height.

Joseph H. Silverman (1991)

Inventiones mathematicae

Rational points on K3 surfaces in ...x...x... .

A. Baragar (1996)

Mathematische Annalen

Rational points on modular curves and abelian varieties.

S. Kamienny (1985)

Journal für die reine und angewandte Mathematik

Rational points on some elliptic surfaces

Enrico Jabara (2012)

Acta Arithmetica

Rational points on some pencils of conics with 6 singular fibres

Peter Swinnerton-Dyer (1999)

Annales de la Faculté des sciences de Toulouse : Mathématiques

Rational points on the modular curves $X_{split} (p)$

Fumiyuki Momose (1984)

Compositio Mathematica

Rational points on twists of X₀(63)

Nils Bruin, Julio Fernández, Josep González, Joan-C. Lario (2007)

Acta Arithmetica

Rational points on $X_{0}^{+} (N)$ and quadratic $ℚ$ -curves

Steven D. Galbraith (2002)

Journal de théorie des nombres de Bordeaux

The rational points on $X_{0} (N) / W_{N}$ in the case where $N$ is a composite number are considered. A computational study of some of the cases not covered by the results of Momose is given. Exceptional rational points are found in the cases $N = 91$ and $N = 125$ and the $j$ -invariants of the corresponding quadratic $ℚ$ -curves are exhibited.

Rational points on $X_{0}^{+} (p)$ .

Galbraith, Steven D. (1999)

Experimental Mathematics

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