Elliptic Curve Theory and Applications: Special Topics in Elliptic Curves

​Introductory course on elliptic curves, free and open to the public. This is a technical course, a light background in math will be useful but not necessary.

​For our last session on Sunday we will explore a sample range of special topics in elliptic curves, including some open problems in mathematics. This session is more of a survey than a continuation of the previous material. Newcomers welcome!

​We’ll apply the foundations we’ve established to look at properties such rank, torsion, and the Mordell–Weil theorem. Then we will study elliptic curve isogenies and isogeny based cryptography as it relates to post-quantum security. After, we’ll look at complex multiplication and the special elliptic curves that admit unusually rich symmetries. Finally we’ll talk about the Birch–Swinnerton-Dyer conjecture, and some of the major open problems that continue to make elliptic curves central to modern mathematics.

​This final seminar brings the series full circle: from an accessible geometric object to questions that reach into some of the deepest unresolved mathematics. We’ll enjoy this series finale with pizza 🍕🍷

​Seminar 4: Special Topics in Elliptic Curves

1. ​Elliptic Curves over ℚ :

   • ​Rank

   • ​Torsion

   • ​Mordell–Weil Theorem

2. ​Isogenies and Endomorphisms

3. ​Complex Multiplication and Special Elliptic Curves

4. ​Cryptography

5. ​Birch–Swinnerton-Dyer Conjecture

6. ​Open Problems

​Readings:

1. ​Joseph H. Silverman & John T. Tate, "Rational Points on Elliptic Curves" textbook chapters 1-3

2. ​Whitfield Diffie & Martin Hellman, “New Directions in Cryptography” (1976)

3. ​Neal Koblitz, “Elliptic Curve Cryptosystems,” Mathematics of Computation 48 (1987), 203–209.

4. ​Victor S. Miller, “Use of Elliptic Curves in Cryptography,” CRYPTO '85, pp. 417–426.