Masters Thesis

Data of Modular Curves

A moduli problem seeks to find a bijection between a class of objects and a topological space that describes the parameters of the class of objects. We will present the moduli problem for a type of curve used in cryptography, elliptic curves. The topological space describing elliptic curves is the quotient of the complex plane by the action of matrices in SL_2(Z), which we call a modular curve. Taking a quotient of the upper half of the complex plane by subgroups of SL_2(Z) also give moduli spaces of elliptic curves but include some extra structure. There are special points on modular curves, which we will discuss and give methods for finding.

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