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Rational points on a subanalytic surface

Jonathan Pila (2005)

Annales de l’institut Fourier

Let X ⊂ 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 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 the unit sphere

Eric Schmutz (2008)

Open Mathematics

It is known that the unit sphere, centered at the origin in ℝn, has a dense set of points with rational coordinates. We give an elementary proof of this fact that includes explicit bounds on the complexity of the coordinates: for every point ν on the unit sphere in ℝn, and every ν > 0; there is a point r = (r 1; r 2;…;r n) such that: ⊎ ‖r-v‖∞ < ε.⊎ r is also a point on the unit sphere; Σ r i 2 = 1.⊎ r has rational coordinates; r i = a i b i for some integers a i, b i.⊎ for all i , 0 ⩽ a i ⩽ b i ⩽ ( 32 1 / 2 l o g 2 n ε ) 2 l o g 2 n . One consequence of this...

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 r )

Yuri Bilu, Pierre Parent, Marusia Rebolledo (2013)

Annales de l’institut Fourier

Using the recent isogeny bounds due to Gaudron and Rémond we obtain the triviality of X 0 + ( p r ) ( ℚ ) , for r &gt; 1 and  p a prime number exceeding 2 · 10 11 . This includes the case of the curves X split ( p ) . We then prove, with the help of computer calculations, that the same holds true for  p in the range 11 ≤ p ≤ 10 14 , p ≠ 13 . The combination of those results completes the qualitative study of rational points on X 0 + ( p r ) undertook in our previous work, with the only exception of  p r = 13 2 .

Currently displaying 101 – 120 of 547