Points rationnels sur les quotients d’Atkin-Lehner de courbes de Shimura de discriminant pq  [ Rational points on Atkin-Lehner quotients of Shimura curves of discriminant pq ]
Annales de l'Institut Fourier, Volume 63 (2013) no. 4, p. 1613-1649
Let p and q be two distinct prime numbers, and X pq /w q be the quotient of the Shimura curve of discriminant pq 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 pq /w q has no rational point, except possibly special points.
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.
DOI : https://doi.org/10.5802/aif.2810
Classification:  10X99,  14A12,  11L05
Keywords: Shimura curves, rational points, Gross vectors, Atkin-Lehner involutions 
@article{AIF_2013__63_4_1613_0,
     author = {Gillibert, Florence},
     title = {Points rationnels sur les quotients d'Atkin-Lehner de courbes de Shimura de discriminant $pq$},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {63},
     number = {4},
     year = {2013},
     pages = {1613-1649},
     doi = {10.5802/aif.2810},
     zbl = {06359596},
     mrnumber = {3137362},
     language = {fr},
     url = {https://aif.centre-mersenne.org/item/AIF_2013__63_4_1613_0}
}

Points rationnels sur les quotients d’Atkin-Lehner de courbes de Shimura de discriminant $pq$. Annales de l'Institut Fourier, Volume 63 (2013) no. 4, pp. 1613-1649. doi : 10.5802/aif.2810. https://aif.centre-mersenne.org/item/AIF_2013__63_4_1613_0/

[1] Bosch, Siegfried; Lütkebohmert, Werner; Raynaud, Michel Néron models, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Tome 21 (1990) | MR 1045822 | Zbl 0705.14001

[2] Bruin, Nils; Flynn, E. Victor; González, Josep; Rotger, Victor On finiteness conjectures for endomorphism algebras of abelian surfaces, Math. Proc. Cambridge Philos. Soc., Tome 141 (2006) no. 3, pp. 383-408 | Article | MR 2281405 | Zbl 1116.14042

[3] Clark, P. Local and global points on moduli spaces of potentially quaternionic abelian surfaces, Harvard University (2003) (Ph. D. Thesis)

[4] Deligne, P.; Rapoport, M. Les schémas de modules de courbes elliptiques, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Springer, Berlin (1973), p. 143-316. Lecture Notes in Math., Vol. 349 | MR 330050 | Zbl 0281.14010

[5] Edixhoven, Bas On Néron models, divisors and modular curves, J. Ramanujan Math. Soc., Tome 13 (1998) no. 2, pp. 157-194 | MR 1666374 | Zbl 0931.11021

[6] Gross, Benedict H. Heights and the special values of L-series, Number theory (Montreal, Que., 1985), Amer. Math. Soc., Providence, RI (CMS Conf. Proc.) Tome 7 (1987), pp. 115-187 | MR 894322 | Zbl 0623.10019

[7] Grothendieck, A.; Raynaud, M.; Rim, D. Groupes de monodromie en géométrie algébrique (SGA7-I), Lecture Notes in Math., Springer Tome 288 (1972) | MR 354656

[8] Jordan, Bruce W.; Livné, Ron A. On the Néron model of Jacobians of Shimura curves, Compositio Math., Tome 60 (1986) no. 2, pp. 227-236 | Numdam | MR 868139 | Zbl 0609.14018

[9] Kontogeorgis, Aristides; Rotger, Victor On the non-existence of exceptional automorphisms on Shimura curves, Bull. Lond. Math. Soc., Tome 40 (2008) no. 3, pp. 363-374 | Article | MR 2418792 | Zbl 1151.11026

[10] Luo, Wenzhi; Ramakrishnan, Dinakar Determination of modular forms by twists of critical L-values, Invent. Math., Tome 130 (1997) no. 2, pp. 371-398 | Article | MR 1474162 | Zbl 0905.11024

[11] Molina, S. Specialisation of Heegner points and applications, Universitat Politècnica de Catalunya (2010) (Ph. D. Thesis)

[12] Ogg, A. P. Mauvaise réduction des courbes de Shimura, Séminaire de théorie des nombres, Paris 1983–84, Birkhäuser Boston, Boston, MA (Progr. Math.) Tome 59 (1985), pp. 199-217 | MR 902833 | Zbl 0581.14024

[13] Parent, Pierre J. R. Towards the triviality of X 0 + (p r )(ℚ) for r>1, Compos. Math., Tome 141 (2005) no. 3, pp. 561-572 | Article | MR 2135276 | Zbl 1167.11310

[14] Parent, Pierre J. R.; Yafaev, Andrei Proving the triviality of rational points on Atkin-Lehner quotients of Shimura curves, Math. Ann., Tome 339 (2007) no. 4, pp. 915-935 | Article | MR 2341907 | Zbl 1129.14036

[15] Ribet, K. A. On modular representations of Gal (Q ¯/Q) arising from modular forms, Invent. Math., Tome 100 (1990) no. 2, pp. 431-476 | Article | MR 1047143 | Zbl 0773.11039

[16] Rotger, Victor Which quaternion algebras act on a modular abelian variety ?, Math. Res. Lett., Tome 15 (2008) no. 2, pp. 251-263 | Article | MR 2385638 | Zbl 1226.11067

[17] Rotger, Victor; Skorobogatov, Alexei; Yafaev, Andrei Failure of the Hasse principle for Atkin-Lehner quotients of Shimura curves over ℚ, Mosc. Math. J., Tome 5 (2005) no. 2, p. 463-476, 495 | MR 2200761 | Zbl 1087.11042

[18] Shimura, Goro Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo (1971) (Kanô Memorial Lectures, No. 1) | MR 314766 | Zbl 0221.10029

[19] Silverman, Joseph H. The arithmetic of elliptic curves, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 106 (1992) (Corrected reprint of the 1986 original) | MR 1329092 | Zbl 0585.14026

[20] Silverman, Joseph H. Advanced topics in the arithmetic of elliptic curves, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 151 (1994) | Article | MR 1312368 | Zbl 0911.14015

[21] De Vera, C.; Rotger, Victor Galois representations over fields of moduli and rational points on Shimura curves (preprint available at : http ://www-ma2.upc.edu/vrotger/docs/students/dV-R.pdf)

[22] Vignéras, Marie-France Arithmétique des algèbres de quaternions, Springer, Berlin, Lecture Notes in Mathematics, Tome 800 (1980) | MR 580949 | Zbl 0422.12008