Equivalence of Real Elliptic Curves

S. Allen Broughton

Abstract

This web page contains the notes and supporting scripts for a talk given in the Rose-Hulman Mathematics faculty-student seminar.

Rose-Hulman Mathematics Faculty Student Seminar - Winter 2004-05

First talk

Title: Equivalence of Real Elliptic Curves - Part I - Linear Equivalence
Abstract:
This is the first of several talks on elliptic curves given by Allen Broughton and Ken McMurdy. In the two talks by Allen Broughton a complete answer will be given to a question posed by Ken McMurdy during his job talk last spring. What is the moduli space of real elliptic curves like? Since then a complete answer has been worked out and it is surprisingly simple.

In the first talk a basic introduction to real elliptic curves will be given -- starting from definitions, smoothness, projective completion, the geometry of the group law, the geometry of tangents and inflection points and ending up with the notions of embedded linear equivalence, normal Weierstrass form, and linear classification. The main result is that there are two families of curves each depending on a single real parameter. Each curve in one family has one component and each curve in the other family has two components*. The talk does not use calculations more complex than high school algebra and the geometric concepts that we cover in our multi-variable calculus course (except a smidgen of topology at one point). There will be lots of pictures.

*Well that statement is almost true . The explanation of almost true will be given in the second talk, which will cover the complexifications of real elliptic curves, real forms of complex elliptic curves, the moduli space complex elliptic curves, and the automorphism groups of curves.

Second Talk

Title: Equivalence of Real Elliptic Curves - Part II - Birational Equivalence
Abstract:  This second talk on real elliptic curves will complete the picture of birational equivalence of real elliptic curves by looking at the complex elliptic curve defined by the original curve. The complex curve is called a complexification of the real curve and the real curve is called a real form of the complex curve. The complex curve is a torus and it interesting to visualize the real forms as curves on the torus. We will spend most of the talk exploring the very interesting relationship among the real forms, mirror reflections on the torus, and the automorphisms of the complex curve.

Non-isomorphic real curves can have can have isomorphic complexifications. The main result we will show is that each complex elliptic curve defined by real equations has exactly two real forms which are birationally inequivalent. The most interesting part is that there is exactly one complex elliptic curve that has a real form with one component and another real form with two components.

We will not use any calculations more complex than high school algebra and nor any geometric concepts beyond what we cover in our multi-variable calculus course. The calculations are made quite easy by using the Weierstrass form discussed in the first talk. The first part of the talk will be a recap of the first talk in the context of complex elliptic curves. There will be lots of pictures.

Materials


Email:  allen.broughton@rose-hulman.edu
Webpage:  http://www.rose-hulman.edu/~brought/
This page last updated on 29 Dec 09.