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# 14H52: Elliptic Curves

## Introduction

This is a fascinating area of algebraic geometry dealing with nonsingular curves of genus 1 -- in English, solutions to equations y^2 = x^3 + A x + B. It turns out to have important connections to number theory and in particular to factorization of ordinary integers (and thus to cryptography). Also, what appear to be simple Diophantine equations often lead to elliptic curves. Through Riemann surfaces it has connections to topology; through modular forms and zeta functions to analysis. Elliptic curves also played a role in the recent resolution of the conjecture known as Fermat's Last Theorem.

## Subfields

Parent field: 14H - Algebraic Curves

## Textbooks, reference works, and tutorials

Online texts on elliptic curves by James Milne and Ian Connell

## Software and tables

If you have Maple, get Connell's APECS package, which does nearly everything anyone knows an algorithm for on an elliptic curve. Here's a link to APECS.

If you need to compute ranks of elliptic curves over the rationals, you definitely need Cremona's MWRANK. See his book or his web site for more information.

Others:

• SIMATH is particularly useful for finding all integral points on curves.
• "Eliptic", a public domain implementation of elliptic curves public key cryptography. [not tested]
• For comic relief you might want this UBASIC program written to look for rational points on a certain elliptic curve with a square x coordinate. (Turns out to be none)

## Selected topics at this site

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Last modified 2000/01/26 by Dave Rusin. Mail: