Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Bounding the $ j$-invariant of integral points on $ X_{\mathrm{ns}}^{+}(p)$


Authors: Aurélien Bajolet and Min Sha
Journal: Proc. Amer. Math. Soc. 142 (2014), 3395-3410
MSC (2010): Primary 11G16, 11J86; Secondary 14G35, 11G50
DOI: https://doi.org/10.1090/S0002-9939-2014-12100-9
Published electronically: June 25, 2014
MathSciNet review: 3238416
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For any prime $ p\ge 7$, by using Baker's method we obtain two explicit bounds in terms of $ p$ for the $ j$-invariant of an integral point on $ X_{\mathrm {ns}}^{+}(p)$ which is the modular curve of level $ p$ corresponding to the normalizer of a non-split Cartan subgroup of $ \mathrm {GL}_2(\mathbb{Z}/p\mathbb{Z})$.


References [Enhancements On Off] (What's this?)

  • [1] Lars V. Ahlfors, Complex analysis. An introduction to the theory of analytic functions of one complex variable; International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1978. MR 510197 (80c:30001)
  • [2] A. Bajolet and Yu. Bilu, Computing integral points on $ X_{\mathrm {ns}}^{+}(p)$, preprint, 2012. arXiv:1212.0665
  • [3] Burcu Baran, A modular curve of level 9 and the class number one problem, J. Number Theory 129 (2009), no. 3, 715-728. MR 2488598 (2010f:11095), https://doi.org/10.1016/j.jnt.2008.09.013
  • [4] Burcu Baran, Normalizers of non-split Cartan subgroups, modular curves, and the class number one problem, J. Number Theory 130 (2010), no. 12, 2753-2772. MR 2684496 (2011i:11083), https://doi.org/10.1016/j.jnt.2010.06.005
  • [5] Yuri Bilu, Effective analysis of integral points on algebraic curves, Israel J. Math. 90 (1995), no. 1-3, 235-252. MR 1336325 (96e:11082), https://doi.org/10.1007/BF02783215
  • [6] Yuri Bilu and Guillaume Hanrot, Solving Thue equations of high degree, J. Number Theory 60 (1996), no. 2, 373-392. MR 1412969 (97k:11040), https://doi.org/10.1006/jnth.1996.0129
  • [7] Yuri Bilu and Marco Illengo, Effective Siegel's theorem for modular curves, Bull. Lond. Math. Soc. 43 (2011), no. 4, 673-688. MR 2820153 (2012h:11095), https://doi.org/10.1112/blms/bdq130
  • [8] Yuri Bilu and Pierre Parent, Runge's method and modular curves, Int. Math. Res. Not. IMRN 9 (2011), 1997-2027. MR 2806555 (2012m:11083), https://doi.org/10.1093/imrn/rnq141
  • [9] Yuri Bilu and Pierre Parent, Serre's uniformity problem in the split Cartan case, Ann. of Math. (2) 173 (2011), no. 1, 569-584. MR 2753610 (2012a:11077), https://doi.org/10.4007/annals.2011.173.1.13
  • [10] Yu. Bilu, P. Parent, and M. Rebolledo, Rational points on $ X_0^+ (p^r)$, Ann. Inst. Fourier 63 (2013), no. 3, 957-984. MR 3137477
  • [11] Antone Costa and Eduardo Friedman, Ratios of regulators in totally real extensions of number fields, J. Number Theory 37 (1991), no. 3, 288-297. MR 1096445 (92j:11138), https://doi.org/10.1016/S0022-314X(05)80044-7
  • [12] Kurt Heegner, Diophantische Analysis und Modulfunktionen, Math. Z. 56 (1952), 227-253 (German). MR 0053135 (14,725j)
  • [13] M. A. Kenku, A note on the integral points of a modular curve of level $ 7$, Mathematika 32 (1985), no. 1, 45-48. MR 817106 (87d:11040), https://doi.org/10.1112/S0025579300010846
  • [14] Daniel S. Kubert and Serge Lang, Modular units, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 244, Springer-Verlag, New York, 1981. MR 648603 (84h:12009)
  • [15] Stéphane Louboutin, Upper bounds on $ \vert L(1,\chi )\vert $ and applications, Canad. J. Math. 50 (1998), no. 4, 794-815. MR 1638619 (99f:11149), https://doi.org/10.4153/CJM-1998-042-2
  • [16] E. M. Matveev, An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers. II, Izv. Ross. Akad. Nauk Ser. Mat. 64 (2000), no. 6, 125-180 (Russian, with Russian summary); English transl., Izv. Math. 64 (2000), no. 6, 1217-1269. MR 1817252 (2002e:11091), https://doi.org/10.1070/IM2000v064n06ABEH000314
  • [17] Jean-Pierre Serre, Lectures on the Mordell-Weil theorem. Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt. With a foreword by Brown and Serre. Aspects of Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1997. MR 1757192 (2000m:11049)
  • [18] M. Sha, Bounding the $ j$-invariant of integral points on modular curves, Int. Math. Res. Notes, to appear. arXiv:1208.1337
  • [19] Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Kanô Memorial Lectures, No. 1, Publications of the Mathematical Society of Japan, Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. MR 0314766 (47 #3318)
  • [20] Carl Ludwig Siegel, Über einige Anwendungen diophantischer Approximationen, Abh. Pr. Akad. Wiss. (1929), no. 1 (=Ges. Abh. I, 209-266, Springer, 1966.) MR 0197270 (33 #5441)
  • [21] Carl Ludwig Siegel, Zum Beweise des Starkschen Satzes, Invent. Math. 5 (1968), 180-191 (German). MR 0228465 (37 #4045)
  • [22] Michel Waldschmidt, Diophantine approximation on linear algebraic groups, Transcendence properties of the exponential function in several variables. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 326, Springer-Verlag, Berlin, 2000. MR 1756786 (2001c:11075)
  • [23] Lawrence C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982. MR 718674 (85g:11001)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11G16, 11J86, 14G35, 11G50

Retrieve articles in all journals with MSC (2010): 11G16, 11J86, 14G35, 11G50


Additional Information

Aurélien Bajolet
Affiliation: Institut de Mathématiques de Bordeaux, Université Bordeaux 1, 33405 Talence cedex, France
Email: Aurelien.Bajolet@math.u-bordeaux1.fr

Min Sha
Affiliation: Institut de Mathématiques de Bordeaux, Université Bordeaux 1, 33405 Talence cedex, France
Email: shamin2010@gmail.com

DOI: https://doi.org/10.1090/S0002-9939-2014-12100-9
Keywords: Modular curves, non-split Cartan, $j$-invariant, Baker's method
Received by editor(s): March 14, 2012
Received by editor(s) in revised form: August 13, 2012, and October 30, 2012
Published electronically: June 25, 2014
Additional Notes: The second author was supported by the China Scholarship Council.
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society