The Arithmetic of Elliptic Curves pp 146-170 | Cite as

# Elliptic Curves over ℂ

## Abstract

Evaluation of the integral giving arc-length on a circle, namely
\( \int {1\sqrt {1 - {x^2}} } dx \) , leads to an (inverse) trigonometric function. The analogous problem for the arc-length of an ellipse yields an integral which is not computable in terms of so-called “elementary” functions. Due to the indeterminacy in the sign of the square root, the study of such integrals over ℂ leads one to look at the Riemann surface on which they are most naturally defined. For the ellipse, this Riemann surface turns out to be the set of complex points on an elliptic curve *E*. We thus begin our study of elliptic curves over ℂ by studying certain *elliptic integrals*, which are line integrals on *E*(ℂ). (In fact, the reason that elliptic curves are so named is because they are the Riemann surfaces associated to the integrals for the arc-length of ellipses. In terms of their geometry, ellipses and elliptic curves actually have little in common, the former having genus 0 and the latter genus 1.)

## Keywords

Riemann Surface Meromorphic Function Elliptic Curve Elliptic Curf Elliptic Function## Preview

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