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Functions and differentials on the non-split Cartan modular curve of level 11


Authors: Julio Fernández and Josep González
Journal: Math. Comp. 86 (2017), 437-454
MSC (2010): Primary 11F46, 14G35, 14Q05; Secondary 14H45, 11F03
DOI: https://doi.org/10.1090/mcom/3109
Published electronically: April 13, 2016
MathSciNet review: 3557806
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Abstract | References | Similar Articles | Additional Information

Abstract: The genus $ 4$ modular curve $ X_{ns}(11)$ attached to a non-split Cartan group of level $ 11$ admits a model defined over $ \mathbb{Q}$. We compute generators for its function field in terms of Siegel modular functions. We also show that its Jacobian is isomorphic over $ \mathbb{Q}$ to the new part of the Jacobian of the classical modular curve $ X_0(121)$.


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  • [Bar10] 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
  • [BGGP05] Matthew H. Baker, Enrique González-Jiménez, Josep González, and Bjorn Poonen, Finiteness results for modular curves of genus at least 2, Amer. J. Math. 127 (2005), no. 6, 1325-1387. MR 2183527 (2006i:11065)
  • [Che98] Imin Chen, The Jacobians of non-split Cartan modular curves, Proc. London Math. Soc. (3) 77 (1998), no. 1, 1-38. MR 1625491 (99m:11068), https://doi.org/10.1112/S0024611598000392
  • [Cre97] J. E. Cremona, Algorithms for modular elliptic curves, 2nd ed., Cambridge University Press, Cambridge, 1997. MR 1628193 (99e:11068)
  • [DFGS14] Valerio Dose, Julio Fernández, Josep González, and René Schoof, The automorphism group of the non-split Cartan modular curve of level 11, J. Algebra 417 (2014), 95-102. MR 3244639, https://doi.org/10.1016/j.jalgebra.2014.05.036
  • [Gal96] S. Galbraith,
    Equations for modular curves,
    PhD thesis, University of Oxford, 1996.
  • [Gon91] Josep Gonzàlez Rovira, Equations of hyperelliptic modular curves, Ann. Inst. Fourier (Grenoble) 41 (1991), no. 4, 779-795 (English, with French summary). MR 1150566 (93g:11064)
  • [Gon12] Josep González, Equations of bielliptic modular curves, JP J. Algebra Number Theory Appl. 27 (2012), no. 1, 45-60. MR 3086199
  • [Hal98] Emmanuel Halberstadt, Sur la courbe modulaire $ X_{\text {nd\'ep}}(11)$, Experiment. Math. 7 (1998), no. 2, 163-174 (French, with English and French summaries). MR 1677158 (99m:11062)
  • [KL81] Daniel S. Kubert and Serge Lang, Modular units, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 244, Springer-Verlag, New York-Berlin, 1981. MR 648603 (84h:12009)
  • [Lig77] Gérard Ligozat, Courbes modulaires de niveau $ 11$, Modular functions of one variable, V (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976) Lecture Notes in Math., vol. 601, Springer, Berlin, 1977, pp. 149-237 (French). MR 0463118 (57 #3079)
  • [Maz77] B. Mazur, Rational points on modular curves, Modular functions of one variable, V (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976) Lecture Notes in Math., vol. 601, Springer, Berlin, 1977, pp. 107-148. MR 0450283 (56 #8579)
  • [Maz91] B. Mazur, Number theory as gadfly, Amer. Math. Monthly 98 (1991), no. 7, 593-610. MR 1121312 (92f:11077), https://doi.org/10.2307/2324924
  • [Ogg74] Andrew P. Ogg, Hyperelliptic modular curves, Bull. Soc. Math. France 102 (1974), 449-462. MR 0364259 (51 #514)
  • [Ser72] Jean-Pierre Serre, Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259-331 (French). MR 0387283 (52 #8126)
  • [Ser89] Jean-Pierre Serre, Lectures on the Mordell-Weil theorem, Aspects of Mathematics, E15, Friedr. Vieweg & Sohn, Braunschweig, 1989. Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt. MR 1002324 (90e:11086)
  • [Shi95] Mahoro Shimura, Defining equations of modular curves $ X_0(N)$, Tokyo J. Math. 18 (1995), no. 2, 443-456. MR 1363479 (96j:11085), https://doi.org/10.3836/tjm/1270043475

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Additional Information

Julio Fernández
Affiliation: Departament de Matemàtiques, Universitat Politècnica de Catalunya, EPSEVG, Avinguda Víctor Balaguer 1, 08800 Vilanova i la Geltrú, Spain
Email: julio.fernandez.g@upc.edu

Josep González
Affiliation: Departament de Matemàtiques, Universitat Politècnica de Catalunya, EPSEVG, Avinguda Víctor Balaguer 1, 08800 Vilanova i la Geltrú, Spain
Email: josep.gonzalez@upc.edu

DOI: https://doi.org/10.1090/mcom/3109
Received by editor(s): December 3, 2014
Received by editor(s) in revised form: June 10, 2015
Published electronically: April 13, 2016
Additional Notes: The authors were partially supported by DGICYT Grant MTM2015-66180-R
Article copyright: © Copyright 2016 American Mathematical Society

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