Rational points on X 0 + ( p r )

Yuri Bilu[1]; Pierre Parent[2]; Marusia Rebolledo[3]

  • [1] IMB, Université Bordeaux 1 351 cours de la Libération 33405 Talence CEDEX, FRANCE
  • [2] IMB, Université Bordeaux 1 351 cours de la Libération 33405 Talence CEDEX FRANCE
  • [3] Université Blaise Pascal Clermont-Ferrand 2 Laboratoire de Mathématiques Campus universitaire des Cézeaux 63177 Aubière FRANCE

Annales de l’institut Fourier (2013)

  • Volume: 63, Issue: 3, page 957-984
  • ISSN: 0373-0956

Abstract

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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 .

How to cite

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Bilu, Yuri, Parent, Pierre, and Rebolledo, Marusia. "Rational points on $X_0^+ (p^r )$." Annales de l’institut Fourier 63.3 (2013): 957-984. <http://eudml.org/doc/275469>.

@article{Bilu2013,
abstract = {Using the recent isogeny bounds due to Gaudron and Rémond we obtain the triviality of $X_0^+ (p^r )(\mathbb\{Q\})$, for $\{r&gt;1\}$ and $p$ a prime number exceeding $2\cdot 10^\{11\}$. This includes the case of the curves $X_\{\mathrm\{split\}\} (p)$. We then prove, with the help of computer calculations, that the same holds true for $p$ in the range $11\le p\le 10^\{14\}$, $p\ne 13$. 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 $p^r=13^2$.},
affiliation = {IMB, Université Bordeaux 1 351 cours de la Libération 33405 Talence CEDEX, FRANCE; IMB, Université Bordeaux 1 351 cours de la Libération 33405 Talence CEDEX FRANCE; Université Blaise Pascal Clermont-Ferrand 2 Laboratoire de Mathématiques Campus universitaire des Cézeaux 63177 Aubière FRANCE},
author = {Bilu, Yuri, Parent, Pierre, Rebolledo, Marusia},
journal = {Annales de l’institut Fourier},
keywords = {Elliptic curves; modular curves; rational points; Runge’s method; isogeny bounds; Gross-Heegner points; elliptic curves; Runge's method},
language = {eng},
number = {3},
pages = {957-984},
publisher = {Association des Annales de l’institut Fourier},
title = {Rational points on $X_0^+ (p^r )$},
url = {http://eudml.org/doc/275469},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Bilu, Yuri
AU - Parent, Pierre
AU - Rebolledo, Marusia
TI - Rational points on $X_0^+ (p^r )$
JO - Annales de l’institut Fourier
PY - 2013
PB - Association des Annales de l’institut Fourier
VL - 63
IS - 3
SP - 957
EP - 984
AB - Using the recent isogeny bounds due to Gaudron and Rémond we obtain the triviality of $X_0^+ (p^r )(\mathbb{Q})$, for ${r&gt;1}$ and $p$ a prime number exceeding $2\cdot 10^{11}$. This includes the case of the curves $X_{\mathrm{split}} (p)$. We then prove, with the help of computer calculations, that the same holds true for $p$ in the range $11\le p\le 10^{14}$, $p\ne 13$. 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 $p^r=13^2$.
LA - eng
KW - Elliptic curves; modular curves; rational points; Runge’s method; isogeny bounds; Gross-Heegner points; elliptic curves; Runge's method
UR - http://eudml.org/doc/275469
ER -

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