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A note on automorphic forms of weight one and weight three

Author: Peter F. Stiller
Journal: Trans. Amer. Math. Soc. 291 (1985), 503-518
MSC: Primary 11F12; Secondary 14D05
MathSciNet review: 800250
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Abstract: In this paper the author develops an interesting relationship between classical automorphic forms of weights one and three, and the solutions of certain second order differential equations related to elliptic (modular) surfaces. In particular for a cusp form of weight three, it is shown that the special values of the associated Dirichlet series can be determined from the periods of an inhomogeneous differential equation, or what is the same thing, the monodromy of an associated third order differential equation. Explicit examples are provided for principal congruence subgroups $ \Gamma (N)$ with $ N \equiv 0\,\operatorname{mod} \,4$.

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  • [1] Bateman Manuscript Project, Higher transcendental functions, Vol. 3, McGraw-Hill, New York, 1953, pp. 20-23.
  • [2] Pierre Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, Vol. 163, Springer-Verlag, Berlin-New York, 1970 (French). MR 0417174
  • [3] M. Eichler, Quadratische Formen und Modulfunktionen, Acta Arith. 4 (1958), 217–239 (German). MR 0096635
  • [4] L. R. Ford, Automorphic functions, McGraw-Hill, New York, 1929, p. 99.
  • [5] R. Fricke and F. Klein, Vorlesung über die Theorie der elliptischen Modulfunktionen, Teubner, Leipzig, 1890.
  • [6] P. Griffiths, Differential equations on algebraic varieties, Princeton lectures, unpublished.
  • [7] E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944. MR 0010757
  • [8] C. Jordan, Cours d'analyse, Gauthier-Villars, Paris, 1909.
  • [9] Nicholas M. Katz and Tadao Oda, On the differentiation of de Rham cohomology classes with respect to parameters, J. Math. Kyoto Univ. 8 (1968), 199–213. MR 0237510
  • [10] Nicholas M. Katz, 𝑝-adic properties of modular schemes and modular forms, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Springer, Berlin, 1973, pp. 69–190. Lecture Notes in Mathematics, Vol. 350. MR 0447119
  • [11] K. Kodaira, On compact analytic surfaces. II, Ann. of Math. (2) 77 (1963), 563-626.
  • [12] Serge Lang, Introduction to modular forms, Springer-Verlag, Berlin-New York, 1976. Grundlehren der mathematischen Wissenschaften, No. 222. MR 0429740
  • [13] H. Petersson, Über die Kongruenzgrupen der Strufe 4, J. Reine Angew. Math. 212 (1963), 64-72.
  • [14] Alain Robert, Elliptic curves, Lecture Notes in Mathematics, Vol. 326, Springer-Verlag, Berlin-New York, 1973. Notes from postgraduate lectures given in Lausanne 1971/72. MR 0352107
  • [15] Jean-Pierre Serre, Congruences et formes modulaires [d’après H. P. F. Swinnerton-Dyer], Séminaire Bourbaki, 24e année (1971/1972), Exp. No. 416, Springer, Berlin, 1973, pp. 319–338. Lecture Notes in Math., Vol. 317 (French). MR 0466020
    Ž. P. Serr, Congruences and modular forms, Uspehi Mat. Nauk 28 (1973), no. 2(170), 183–196 (Russian). Translated from the French (Séminaire Bourbaki, 24e année (1971/1972), Exp. No. 416, pp. 319–338, Lecture Notes in Math., Vol. 317, Springer, Berlin, 1973) by Ju. I. Manin. MR 0466021
  • [16] Goro Shimura, Sur les intégrales attachées aux formes automorphes, J. Math. Soc. Japan 11 (1959), 291–311 (French). MR 0120372,
  • [17] -, Introduction to the arithmetic theory of automorphic forms, Princeton Univ. Press, Princeton, N. J., 1971.
  • [18] Tetsuji Shioda, On elliptic modular surfaces, J. Math. Soc. Japan 24 (1972), 20–59. MR 0429918,
  • [19] Peter F. Stiller, Differential equations associated with elliptic surfaces, J. Math. Soc. Japan 33 (1981), no. 2, 203–233. MR 607940,
  • [20] Peter F. Stiller, Elliptic curves over function fields and the Picard number, Amer. J. Math. 102 (1980), no. 4, 565–593. MR 584462,
  • [21] -, Automorphic forms and the Picard number of an elliptic surface, Aspects of Math. E, Vol. E5, Vieweg, Braunschweig, 1984.
  • [22] Peter Stiller, Special values of Dirichlet series, monodromy, and the periods of automorphic forms, Mem. Amer. Math. Soc. 49 (1984), no. 299, iv+116. MR 743544,
  • [23] André Weil, Elliptic functions according to Eisenstein and Kronecker, Springer-Verlag, Berlin-New York, 1976. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 88. MR 0562289
    A. Veĭl′, \cyrÈllipticheskie funktsii po Èĭzenshteĭnu i Kronekeru, Izdat. “Mir”, Moscow, 1978 (Russian). Translated from the English by Ju. I. Manin. MR 0562290

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Keywords: Elliptic surface, automorphic form, $ q$-expansion, $ K$-equations, Gauss-Manin connection, monodromy
Article copyright: © Copyright 1985 American Mathematical Society