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On a class of elliptic functions associated with imaginary quadratic fields

Author(s): Li-Chien Shen
Journal: Proc. Amer. Math. Soc. 132 (2004), 463-471.
MSC (2000): Primary 33E05
Posted: August 28, 2003
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Abstract: Let $-D$ be the field discriminant of an imaginary quadratic field. We construct a class of elliptic functions associated naturally with the quadratic field $Q(\sqrt {-D})$ which, combined with the general theory of elliptic functions, allows us to provide a unified theory for two fundamental results (one classical and one due to Ramanujan) about the elliptic functions.

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H. Cohn, Advanced Number Theory, Dover, New York, 1980. MR 82b:12001

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B. C. Berndt, S. Bhargava and F. Garvan, Ramanujan's theories of elliptic functions to alternative bases, Trans. Amer. Math. Soc. 347(11) (1995), 4163-4244. MR 97h:33034

4.
Li-Chien Shen, On an identity of Ramanujan based on the hypergeometric series $_{2}F_{1}(\tfrac{1}{3}, \tfrac{2}{3};\tfrac{1}{2}; x)$, J. of Number Theory 69(2) (1998), 125-134. MR 99d:11042

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Li-Chien Shen, On a class of $q$-series related to quadratic forms, Bulletin of the Institute of Mathematics, Academia Sinica, 26(2) (1998), 111-126. MR 99h:11091

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E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th edition, Cambridge University Press, London, 1958. MR 31:2375

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Additional Information:

Li-Chien Shen
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611-2082
Email: shen@math.ufl.edu

DOI: 10.1090/S0002-9939-03-07259-9
PII: S 0002-9939(03)07259-9
Keywords: Elliptic function, character, class number, quadratic field, discriminant
Received by editor(s): October 3, 2002
Posted: August 28, 2003
Communicated by: David E. Rohrlich
Copyright of article: Copyright 2003, American Mathematical Society

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