Fricke’s two-valued modular equations
HTML articles powered by AMS MathViewer
- by Harvey Cohn PDF
- Math. Comp. 51 (1988), 787-807 Request permission
Abstract:
The modular equation of order b is a polynomial relation between $j(z)$ and $j(z/b)$, which has astronomically large coefficients even for small values of b. Fricke showed that a two-valued relation exists for 37 small values of b. This relation would have much smaller coefficients and would also be convenient for finding singular moduli. Although Fricke produced no two-valued relations explicitly (no doubt because of the tedious amount of algebraic manipulation), they are now found by use of MACSYMA. For 31 cases ranging from $b = 2$ to 49, Fricke provided the equations necessary to generate the relations (with two corrections required). The remaining six cases (of order 39, 41, 47, 50, 59, 71) require an extension of Fricke’s methods, using the discriminant function, theta functions, and power series approximations.References
- A. O. L. Atkin and J. Lehner, Hecke operators on $\Gamma _{0}(m)$, Math. Ann. 185 (1970), 134–160. MR 268123, DOI 10.1007/BF01359701 B. C. Berndt, A. J. Biagioli & J. M. Purtilo, "Ramanujan’s modular equation of large prime degree." (Manuscript.) W. E. H. Berwick, "Modular invariants expressible in terms of quadratic and cubic irrationalities," Proc. London Math. Soc., v. 28, 1927, pp. 53-69.
- B. J. Birch, Some calculations of modular relations, Modular functions of one variable, I (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Mathematics, Vol. 320, Springer, Berlin, 1973, pp. 175–186. MR 0332658
- Harvey Cohn, Introduction to the construction of class fields, Cambridge Studies in Advanced Mathematics, vol. 6, Cambridge University Press, Cambridge, 1985. MR 812270 H. Cohn, "Singular moduli and modular equations for Fricke’s cases," Proc. Bowdoin AMS Summer Institute 1987. (Manuscript.) R. Fricke, Die elliptischen Funktionen und ihre Anwendungen II, Leipzig, 1922. R. Fricke, Lehrbuch der Algebra III (Algebraische Zahlen), Braunschweig, 1928.
- Benedict H. Gross and Don B. Zagier, On singular moduli, J. Reine Angew. Math. 355 (1985), 191–220. MR 772491
- Erich Kaltofen and Noriko Yui, Explicit construction of the Hilbert class fields of imaginary quadratic fields with class numbers $7$ and $11$, EUROSAM 84 (Cambridge, 1984) Lecture Notes in Comput. Sci., vol. 174, Springer, Berlin, 1984, pp. 310–320. MR 779136, DOI 10.1007/BFb0032853
- M. A. Kenku, The modular curve $X_{0}(39)$ and rational isogeny, Math. Proc. Cambridge Philos. Soc. 85 (1979), no. 1, 21–23. MR 510395, DOI 10.1017/S0305004100055444 F. Klien & R. Fricke, Vorlesungen über die Theorie der elliptischen Modulfunctionen, Vol. II, Leipzig, 1892, pp. 445-474.
- Gérard Ligozat, Courbes modulaires de genre $1$, Supplément au Bull. Soc. Math. France, Tome 103, no. 3, Société Mathématique de France, Paris, 1975 (French). Bull. Soc. Math. France, Mém. 43. MR 0417060
- B. Mazur and P. Swinnerton-Dyer, Arithmetic of Weil curves, Invent. Math. 25 (1974), 1–61. MR 354674, DOI 10.1007/BF01389997
- Morris Newman, Construction and application of a class of modular functions, Proc. London Math. Soc. (3) 7 (1957), 334–350. MR 91352, DOI 10.1112/plms/s3-7.1.334
- Morris Newman, Conjugacy, genus, and class numbers, Math. Ann. 196 (1972), 198–217. MR 311573, DOI 10.1007/BF01428049
- Andrew P. Ogg, Hyperelliptic modular curves, Bull. Soc. Math. France 102 (1974), 449–462. MR 364259 B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, New York, 1984. J. Vélu, "Les points rationnels de ${X_0}(37)$," Bull. Soc. Math. France, Mém. 37, Paris, 1974, pp. 169-179.
- Noriko Yui, Explicit form of the modular equation, J. Reine Angew. Math. 299(300) (1978), 185–200. MR 476642, DOI 10.1515/crll.1978.299-300.185
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Math. Comp. 51 (1988), 787-807
- MSC: Primary 11F03; Secondary 11F27
- DOI: https://doi.org/10.1090/S0025-5718-1988-0935079-4
- MathSciNet review: 935079