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
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Notes by Willem Kuyk
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References
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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
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