Further properties of a function of Ogg and Ligozat. (English) Zbl 1238.11068
Summary: Certain identities of Ramanujan may be succinctly expressed in terms of the rational function \(\breve{g}_{\chi} = \breve{f}_{\chi}-\frac{1}{\breve {f}_{\chi}}\) on the modular curve \(X_{0}(N)\), where \(\breve{f}_{\chi} = w_{N}f_{\chi}\) and \(f _{\chi }\) is a certain modular unit on the Nebentypus cover \(X_{\chi}(N)\) introduced by Ogg and Ligozat [in: B. Mazur, Publ. Math., Inst. Hautes Étud. Sci. 47, 33–186 (1977), p. 107–108].for prime \(N \equiv 1 \pmod 4\) and \(w_{N}\) is the Fricke involution. These correspond to levels \(N = 5,13\), where the genus \(g_{N}\) of \(X_{0}(N)\) is zero. In this paper we study slightly more general kind of relations for each \(\breve{g}_{\chi}\) such that \(X_{0}(N)\) has genus \(g_{N}=1,2,\) and also for each \(h_{\chi} = g_{\chi}+\breve{g}_{\chi}\) such that the Atkin-Lehner quotient \(X_{0}^{+}(N)\) has genus \(g_{N}^{+} = 1,2\). Finally we study the normal closure of the field of definition of the zeros of the latter.
MSC:
| 11G16 | Elliptic and modular units |
| 20H05 | Unimodular groups, congruence subgroups (group-theoretic aspects) |
| 11G18 | Arithmetic aspects of modular and Shimura varieties |
References:
| [1] | Berndt, B.: Ramanujan’s Notebooks Part III. Springer, New York (1991) · Zbl 0733.11001 |
| [2] | Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system I: the user language. J. Symb. Comp. 24(3-4), 235–265 (1997). http://www.maths.usyd.edu.au:8000/u/magma/ · Zbl 0898.68039 · doi:10.1006/jsco.1996.0125 |
| [3] | Cremona, J.E.: Elliptic curves of conductor 0,000. http://www.maths.nott.ac.uk/personal/jec/ftp/data/ · JFM 01.0130.02 |
| [4] | Csirik, J.: The kernel of the Eisenstein ideal, U.C. Berkeley Ph.D. thesis (1999), v+49 · Zbl 1004.11035 |
| [5] | Darmon, H.: Stark-Heegner points over real quadratic fields. In: Number Theory (Tiruchirapalli, 1996). Contemporary Mathematics, vol. 210, pp. 41–69. Am. Math. Soc., Providence (1998) · Zbl 0923.11099 |
| [6] | Grayson, D.R., Stillman, M.E.: Macaulay 2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay |
| [7] | Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 4th edn. Oxford University Press, London (1960) · Zbl 0086.25803 |
| [8] | Jenkins, M.: Proof of an arithmetical theorem leading, by means of Gauss’ fourth demonstration of Legendre’s law of reciprocity, to the extension of that law. Proc. Lond. Math. Soc. 2, 29–32 (1867) · JFM 02.0092.03 |
| [9] | Kubert, D.S., Lang, S.: Units in the modular function field. II. A full set of units. Math. Ann. 218(2), 175–189 (1975) · Zbl 0311.14005 · doi:10.1007/BF01370818 |
| [10] | Kubert, D.S., Lang, S.: Modular Units. Grundlehren der Mathematischen Wissenschaften, vol. 244. Springer, New York (1981) · Zbl 0492.12002 |
| [11] | Mazur, B.: Modular curves and the Eisenstein ideal. Publ. Math. IHES 47, 33–186 (1977) · Zbl 0394.14008 · doi:10.1007/BF02684339 |
| [12] | Ogg, A.P.: Rational points on certain elliptic modular curves. In: Analytic Number Theory, Proc. Sympos. Pure Math., vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972, pp. 221–231. Am. Math. Soc., Providence (1973) |
| [13] | The PARI Group, Bordeaux, PARI/GP, Version 2.1.5. http://www.parigp-home.de/ |
| [14] | Serre, J.P.: A Course in Arithmetic. Springer, New York/Heidelberg (1973) · Zbl 0256.12001 |
| [15] | Shimura, G.: Introduction to the Arithmetic Theory of Automorphic Functions. Princeton University Press, Princeton (1994) · Zbl 0872.11023 |
| [16] | Singmaster, D.: Problem 1654. Math. Mag. 75(4), 317 (2002) · doi:10.2307/3219173 |
| [17] | Sun, Z.: Products of binomial coefficients modulo p 2. Acta Arith. 97(1), 87–98 (2001) · Zbl 0986.11009 · doi:10.4064/aa97-1-5 |
| [18] | Tate, J.T.: Les conjectures de Stark sur les fonctions L d’Artin en s=0. In: Progress in Mathematics, vol. 47. Birkhäuser, Boston (1984). Lecture notes edited by Dominique Bernardi and Norbert Schappacher · Zbl 0545.12009 |
| [19] | Zagier, D.: Modular points, modular curves, modular surfaces and modular forms. In: Workshop Bonn 1984. Lecture Notes in Mathematics, vol. 1111, pp. 225–248. Springer, Berlin (1985) · Zbl 0594.14022 |
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