The Arithmetic of Elliptic Curves

Elliptic curves have played a central role in number theory and algebraic geometry for the last 100 years. Elliptic curves were crucial in Wiles’ proof of Fermat’s Last Theorem, they provide the fastest known algorithms for factoring integers, and there are cryptosystems based on their arithmetic. Elliptic curves arise in the congruent number problem, in the search for Mersenne primes, and a myriad of other applications. They sit at the confluence of geometry, algebra, and number theory. They are also just plain fun. There are many unanswered questions about elliptic curves that involve some of the deepest mathematics of our times, and will provide us a springboard for active mathematical investigation and adventure. (MATH 0121)