Elliptic Curves

The topic of elliptic curves is an exciting one for a variety of reasons. For one thing, elliptic curves are a meeting ground where ideas from geometry, number theory, topology, group theory, and complex analysis all come together and interact in a rich and satisfying way. For another, the subject abounds with unsolved problems, many of which are accessible to the novice mathematician. Two recent developments have brought elliptic curves to the attention of the mathematical public. One is the recent proof by Gerd Faltings of the Mordell conjecture, which involves three types of curves: rational curves (genus 0), elliptic curves (genus I), and curves of genus at least 2. The other is the use of elliptic curves in the problem of factoring large numbers and determining whether or not they are prime. This, in turn, has important applications in the construction of public key codes [Deneen 19871.