Elliptic and Modular Functions
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In most of our previous work in [AEC], the major theorems have been of the form “Let E/K be an elliptic curve. Then E/K has such-and-such a property. ” In this chapter we will change our perspective and consider the set of elliptic curves as a whole. We will take the collection of all (isomorphism classes of) elliptic curves and make it into an algebraic curve, a so-called modular curve. Then by studying functions and differential forms on this modular curve, we will be able to make deductions about elliptic curves. Further, the Fourier coefficients of these modular functions and modular forms turn out to be extremely interesting in their own right, especially from a number-theoretic viewpoint. We will be able to prove some of their properties in the last part of the chapter.
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