Points rationnels sur les quotients d’Atkin-Lehner de courbes de Shimura de discriminant $pq$

Gillibert, Florence

Annales de l'Institut Fourier, Tome 63 (2013) no. 4, pp. 1613-1649.

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.

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.

Classification : 10X99, 14A12, 11L05
Mot clés : courbes de Shimura, points rationnels, vecteurs de Gross, involutions d’Atkin-Lehner
Keywords: Shimura curves, rational points, Gross vectors, Atkin-Lehner involutions

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

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

[3] Clark, P. Local and global points on moduli spaces of potentially quaternionic abelian surfaces, Harvard University (2003) (Ph. D. Thesis Available at http://math.uga.edu/∼pete/thesis.pdf)

[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 | Zbl

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

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

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

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

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

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

[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 (Progr. Math.), Volume 59, Birkhäuser Boston, Boston, MA, 1985, pp. 199-217 | MR | Zbl

[13] Parent, Pierre J. R. Towards the triviality of $X_0^{+}\left(p^r\right)\left(\mathbb{Q}\right)$ for $r>1$, Compos. Math., Volume 141 (2005) no. 3, pp. 561-572 | DOI | MR | Zbl

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

[15] Ribet, K. A. On modular representations of $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ arising from modular forms, Invent. Math., Volume 100 (1990) no. 2, pp. 431-476 | DOI | MR | Zbl

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

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

[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 | Zbl

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

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

[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, Lecture Notes in Mathematics, 800, Springer, Berlin, 1980 | MR | Zbl