Galen Ballew

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# Elliptic Curves

25 May 2016

### Elliptic curves with complex multiplication give rise to tori over the complex numbers

Completed in spring 2016 at the Mathematical Computing Laboratory at UIC.

This repository contains the paper, poster, images, and 3d modeling files from my research and learning project on elliptic curves. I completed this project with James Duncan under the guidance of Cara Mullen (PhD) and Professor Alina Cojocaru (PhD).

##### Summary

Our research focuses on elliptic curves $$E$$ over $$\mathbb{Q}$$ with complex multiplication (by the maximal order of an imaginary quadratic field). Viewed over $$\mathbb{C}$$, each $$E$$ gives rise to two tori, defined by the generators $$\omega_1$$ and $$\omega_2$$ of the period lattice. These tori can be constructed virtually into a 3D mesh. Further, this mesh can be translated into gcode and printed using a 3D printer.