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Complex Multiplication

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Part of the Lecture Notes in Mathematics book series (LNM,volume 320)

Abstract

The purpose of this article is to explain the main ideas of the topic with neither proofs, nor historical remarks. The exposition is divided into three parts. No modular functions appear in Part I except that the values of the j-function are needed as invariants of elliptic curves. Part II relies on the theory of Hecke operators (on cusp forms of weight 2), while Part III concerns the modular functions of arbitrary level. Almost nothing is new in the sense that most of the results, with proofs, can be found in [3], [4], [5], but I have attempted to present the material as accessibly (and perhaps dogmatically) as possible. The formulation in Part II is essentially equivalent to, but somewhat different from, what was done in [3, §7.5].

Keywords

  • Elliptic Curve
  • Complex Multiplication
  • Elliptic Curf
  • Abelian Variety
  • Number Field

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes by Willem Kuyk

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References

  1. M. DEURING, Die Zetafunktion einer algebraischen Kurve vom Gesiechte Eins, I,II,III,IV, Nachr. Akad. Wiss. Göttingen, (1953) 85–94, (1955) 13–42, (1956) 37–76, (1957) 55–80.

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  2. E. HECKE, Zur Theorie der elliptischen Modulfunktionen, Math. Ann., 97 (1926), 210–242 (= Math. Werke, 428–460).

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  3. G. SHIMURA, Introduction to the arithmetic theory of automorphic functions, Publ. Math. Soc. Japan, No. 11, Iwanami Shoten and Princeton Univ. Press, 1971.

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  4. G. SHIMURA, On elliptic curves with complex multiplication as factors of the jacobians of modular function fields, Nagoya Math. J. 43 (1971), 199–208.

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  5. G. SHIMURA, Class fields over real quadratic fields and Hecke operators, Ann. of Math. 95 (1972), 130–190.

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  6. G. SHIMURA, On the factors of the jacobian variety of a modular function field, to appear.

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  7. H. WEBER, Lerhbuch der Algebra III.

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© 1973 Springer-Verlag Berlin Heidelberg

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Shimura, G. (1973). Complex Multiplication. In: Kuijk, W. (eds) Modular Functions of One Variable I. Lecture Notes in Mathematics, vol 320. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-38509-7_2

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  • DOI: https://doi.org/10.1007/978-3-540-38509-7_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-06219-6

  • Online ISBN: 978-3-540-38509-7

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