Rational points on Atkin-Lehner quotients of Shimura curves of discriminant p q

Florence Gillibert[1]

  • [1] IMB Bordeaux I 351, cours de la Libération 33405 Talence (France)

Annales de l’institut Fourier (2013)

  • Volume: 63, Issue: 4, page 1613-1649
  • ISSN: 0373-0956

Abstract

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Let p and q be two distinct prime numbers, and X p q / w q be the quotient of the Shimura curve of discriminant p q by the Atkin-Lehner involution w q . We describe a way to verify in wide generality a criterion of Parent and Yafaev to prove that if p and q satisfy some explicite congruence conditions, known as the conditions of the non ramified case of Ogg, and if p is large enough compared to q , then the quotient X p q / w q has no rational point, except possibly special points.

How to cite

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Gillibert, Florence. "Points rationnels sur les quotients d’Atkin-Lehner de courbes de Shimura de discriminant $pq$." Annales de l’institut Fourier 63.4 (2013): 1613-1649. <http://eudml.org/doc/275488>.

@article{Gillibert2013,
abstract = {Soient $p$ et $q$ deux nombres premiers distincts et $X^\{pq\}/w_q$ le quotient de la courbe de Shimura de discriminant $pq$ par l’involution d’Atkin-Lehner $w_q$. Nous décrivons un moyen permettant de vérifier un critère de Parent et Yafaev en grande généralité pour prouver que si $p$ et $q$ satisfont des conditions de congruence explicites, connues comme les conditions du cas non ramifié de Ogg, et si $p$ est assez grand par rapport à $q$, alors le quotient $X^\{pq\}/w_q$ n’a pas de point rationnel non spécial.},
affiliation = {IMB Bordeaux I 351, cours de la Libération 33405 Talence (France)},
author = {Gillibert, Florence},
journal = {Annales de l’institut Fourier},
keywords = {Shimura curves; rational points; Gross vectors; Atkin-Lehner involutions},
language = {fre},
number = {4},
pages = {1613-1649},
publisher = {Association des Annales de l’institut Fourier},
title = {Points rationnels sur les quotients d’Atkin-Lehner de courbes de Shimura de discriminant $pq$},
url = {http://eudml.org/doc/275488},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Gillibert, Florence
TI - Points rationnels sur les quotients d’Atkin-Lehner de courbes de Shimura de discriminant $pq$
JO - Annales de l’institut Fourier
PY - 2013
PB - Association des Annales de l’institut Fourier
VL - 63
IS - 4
SP - 1613
EP - 1649
AB - Soient $p$ et $q$ deux nombres premiers distincts et $X^{pq}/w_q$ le quotient de la courbe de Shimura de discriminant $pq$ par l’involution d’Atkin-Lehner $w_q$. Nous décrivons un moyen permettant de vérifier un critère de Parent et Yafaev en grande généralité pour prouver que si $p$ et $q$ satisfont des conditions de congruence explicites, connues comme les conditions du cas non ramifié de Ogg, et si $p$ est assez grand par rapport à $q$, alors le quotient $X^{pq}/w_q$ n’a pas de point rationnel non spécial.
LA - fre
KW - Shimura curves; rational points; Gross vectors; Atkin-Lehner involutions
UR - http://eudml.org/doc/275488
ER -

References

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  1. Siegfried Bosch, Werner Lütkebohmert, Michel Raynaud, Néron models, 21 (1990), Springer-Verlag, Berlin Zbl0705.14001MR1045822
  2. Nils Bruin, E. Victor Flynn, Josep González, Victor Rotger, On finiteness conjectures for endomorphism algebras of abelian surfaces, Math. Proc. Cambridge Philos. Soc. 141 (2006), 383-408 Zbl1116.14042MR2281405
  3. P. Clark, Local and global points on moduli spaces of potentially quaternionic abelian surfaces, (2003) 
  4. P. Deligne, M. Rapoport, Les schémas de modules de courbes elliptiques, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) (1973), 143-316. Lecture Notes in Math., Vol. 349, Springer, Berlin Zbl0281.14010MR330050
  5. Bas Edixhoven, On Néron models, divisors and modular curves, J. Ramanujan Math. Soc. 13 (1998), 157-194 Zbl0931.11021MR1666374
  6. Benedict H. Gross, Heights and the special values of L -series, Number theory (Montreal, Que., 1985) 7 (1987), 115-187, Amer. Math. Soc., Providence, RI Zbl0623.10019MR894322
  7. A. Grothendieck, M. Raynaud, D. Rim, Groupes de monodromie en géométrie algébrique (SGA7-I), 288 (1972), Lecture Notes in Math., Springer Zbl0237.00013MR354656
  8. Bruce W. Jordan, Ron A. Livné, On the Néron model of Jacobians of Shimura curves, Compositio Math. 60 (1986), 227-236 Zbl0609.14018MR868139
  9. Aristides Kontogeorgis, Victor Rotger, On the non-existence of exceptional automorphisms on Shimura curves, Bull. Lond. Math. Soc. 40 (2008), 363-374 Zbl1151.11026MR2418792
  10. Wenzhi Luo, Dinakar Ramakrishnan, Determination of modular forms by twists of critical L -values, Invent. Math. 130 (1997), 371-398 Zbl0905.11024MR1474162
  11. S. Molina, Specialisation of Heegner points and applications, (2010) 
  12. A. P. Ogg, Mauvaise réduction des courbes de Shimura, Séminaire de théorie des nombres, Paris 1983–84 59 (1985), 199-217, Birkhäuser Boston, Boston, MA Zbl0581.14024MR902833
  13. Pierre J. R. Parent, Towards the triviality of X 0 + ( p r ) ( ℚ ) for r &gt; 1 , Compos. Math. 141 (2005), 561-572 Zbl1167.11310MR2135276
  14. Pierre J. R. Parent, Andrei Yafaev, Proving the triviality of rational points on Atkin-Lehner quotients of Shimura curves, Math. Ann. 339 (2007), 915-935 Zbl1129.14036MR2341907
  15. K. A. Ribet, On modular representations of Gal ( Q ¯ / Q ) arising from modular forms, Invent. Math. 100 (1990), 431-476 Zbl0773.11039MR1047143
  16. Victor Rotger, Which quaternion algebras act on a modular abelian variety ?, Math. Res. Lett. 15 (2008), 251-263 Zbl1226.11067MR2385638
  17. Victor Rotger, Alexei Skorobogatov, Andrei Yafaev, Failure of the Hasse principle for Atkin-Lehner quotients of Shimura curves over ℚ , Mosc. Math. J. 5 (2005), 463-476, 495 Zbl1087.11042MR2200761
  18. Goro Shimura, Introduction to the arithmetic theory of automorphic functions, (1971), Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo Zbl0221.10029MR314766
  19. Joseph H. Silverman, The arithmetic of elliptic curves, 106 (1992), Springer-Verlag, New York Zbl0585.14026MR1329092
  20. Joseph H. Silverman, Advanced topics in the arithmetic of elliptic curves, 151 (1994), Springer-Verlag, New York Zbl0911.14015MR1312368
  21. C. de Vera, Victor Rotger, Galois representations over fields of moduli and rational points on Shimura curves Zbl1300.11070
  22. Marie-France Vignéras, Arithmétique des algèbres de quaternions, 800 (1980), Springer, Berlin Zbl0422.12008MR580949

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