Equivalence of Real Elliptic Curves - Part I - Linear EquivalenceRose Math Seminar (2004)
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.
- real elliptic curve
Publication DateOctober 6, 2004
LocationRose-Hulman Institute of Technology, Terre Haute, IN
Citation InformationSean A Broughton. "Equivalence of Real Elliptic Curves - Part I - Linear Equivalence" Rose Math Seminar (2004)
Available at: http://works.bepress.com/allen_broughton/83/
Creative Commons License
This work is licensed under a Creative Commons CC_BY-NC-SA International License.