Chen, Imin; Glebov, Gleb On Chudnovsky-Ramanujan type formulae. (English) Zbl 1422.11243 Ramanujan J. 46, No. 3, 677-712 (2018). Summary: In a well known 1914 paper, S. Ramanujan in 1914 [Q. J. Math., Oxf. (2) 45, 350–372 (1914; JFM 45.1249.01), see also Collected papers of Srinivasa Ramanujan, Cambridge Univ. Press, Cambridge (1927; JFM 53.0030.02), pp. 23–39] gave a number of rapidly converging series for \(1/\pi \) which are derived using modular functions of higher level. D. V. Chudnovsky and G. V. Chudnovsky in their 1988 paper [Ramanujan revisited, Proc. Conf., Urbana-Champaign/Illinois 1987, 375–472 (1988; Zbl 0647.10002)] derived an analogous series representing \(1/\pi \) using the modular function \(J\) of level 1, which results in highly convergent series for \(1/\pi \), often used in practice. In this paper, we explain the Chudnovsky method in the context of elliptic curves, modular curves, and the Picard-Fuchs differential equation. In doing so, we also generalize their method to produce formulae which are valid around any singular point of the Picard-Fuchs differential equation. Applying the method to the family of elliptic curves parameterized by the absolute Klein invariant \(J\) of level 1, we determine all Chudnovsky-Ramanujan type formulae which are valid around one of the three singular points: \(0, 1, \infty \). Cited in 2 Documents MSC: 11Y60 Evaluation of number-theoretic constants 14H52 Elliptic curves 14K20 Analytic theory of abelian varieties; abelian integrals and differentials 33C05 Classical hypergeometric functions, \({}_2F_1\) Keywords:elliptic curves; elliptic functions; Dedekind eta function; Eisenstein series; hypergeometric function; Picard-Fuchs differential equation Citations:Zbl 0647.10002; JFM 45.1249.01; JFM 53.0030.02 PDFBibTeX XMLCite \textit{I. Chen} and \textit{G. Glebov}, Ramanujan J. 46, No. 3, 677--712 (2018; Zbl 1422.11243) Full Text: DOI arXiv References: [1] Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (1999) · Zbl 0920.33001 [2] Archinard, N, Exceptional sets of hypergeometric series, J. Number Theory, 101, 244-269, (2003) · Zbl 1125.11043 [3] Borwein, J., Borwein, P.: Pi and the AGM. Wiley, New York (1987) · Zbl 0611.10001 [4] Chan, HH; Verrill, H, The apéry numbers, the almkvist-zudilin numbers and new series for \(1/π \), Math. Res. Lett., 16, 405-420, (2009) · Zbl 1193.11038 [5] Chowla, S; Selberg, A, On epstein’s zeta-function, J. Reine Angew. Math., 227, 86-110, (1967) · Zbl 0166.05204 [6] Chudnovsky, DV; Chudnovsky, GV; Andrews, GE (ed.); Askey, RA (ed.); Berndt, BC (ed.); Ramanathan, KG (ed.); Rankin, RA (ed.), Approximation and complex multiplication according to Ramanujan, 375-472, (1988), Boston [7] Chudnovsky, D.V., Chudnovsky, G.V.: Use of computer algebra for Diophantine and differential equations, In: Chudnovsky, D.V., Jenks, R.D. (eds.) Computer Algebra, Lecture Notes in Pure and Appl. Math. 113, pp. 1-81. Dekker, New York (1989) · Zbl 0685.10024 [8] Cox, D.: Primes of the Form \(x^2 + ny^2\): Fermat, Class Field Theory, and Complex Multiplication, 2nd edn. Wiley, New York (2013) · Zbl 1275.11002 [9] Fricke, R., Klein, F.: Vorlesungen über die Theorie der elliptischen Modulfunctionen. Teubner, Leipzig (1890) [10] Greenhill, A.G.: The Applications of Elliptic Functions. Macmillan and Co., New York (1892) · JFM 24.0410.02 [11] Kummer, E.E.: Über die hypergeometrische Reihe. J. Reine Angew. Math. 15, 39-83, 127-172 (1836) [12] Lang, S.: Elliptic Functions, 2nd edn. Springer, New York (1987) · Zbl 0615.14018 [13] Ramanujan, S, Modular equations and approximations to \(π \), Q. J. Math. (Oxford), 45, 350-372, (1914) · JFM 45.1249.01 [14] Ramanujan, S, On certain arithmetical functions, Trans. Camb. Phil. Soc., 22, 159-184, (1916) [15] Ramanujan, S.: Collected Papers. Cambridge University Press, Cambridge (1927) · JFM 53.0030.02 [16] Serre, J.-P.: Congruences et formes modulaire (d’après H. P. F. Swinnerton-Dyer), Séminaire Bourbaki, 24e année (1971/1972), Exp. No. 416, Lecture Notes in Math. 317, pp. 319-338. Springer, Berlin (1973) · Zbl 1193.11038 [17] Silverman, J.H.: The Arithmetic of Elliptic Curves, 2nd edn. Springer, Dordrecht (2009) · Zbl 1194.11005 [18] Weber, H.: Lehrbuch der Algebra, vol. III, 2nd edn. Vieweg, Braunschwieg (1908) · Zbl 1125.11043 [19] Whittaker, E.T., Watson, G.N.: A Course in Modern Analysis, 2nd edn. Cambridge University Press, Cambridge (1915) · JFM 45.0433.02 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.