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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 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 > 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 .

Rational torsion points on Jacobians of modular curves

Hwajong Yoo (2016)

Acta Arithmetica

Let p be a prime greater than 3. Consider the modular curve X₀(3p) over ℚ and its Jacobian variety J₀(3p) over ℚ. Let (3p) and (3p) be the group of rational torsion points on J₀(3p) and the cuspidal group of J₀(3p), respectively. We prove that the 3-primary subgroups of (3p) and (3p) coincide unless p ≡ 1 (mod 9) and 3 ( p - 1 ) / 3 ≡ 1 ( m o d p ) .

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