Number Theory Seminar: Dan Collins

"Which numbers (and in particular, which primes) are sums of two rational cubes" is a classical and still-not-entirely-solved Diophantine problem. I'll talk about how it turns into a problem about rational points on elliptic curves, and how it can then be attacked using the modern machinery of the arithmetic of elliptic curves. In particular, proving that certain primes can be written as a sum of two cubes can be accomplished by constructing a Heegner-type point and proving it's nonzero. This is a subtle question and has been carried out in different ways by Elkies and by Dasgupta-Voight. My work (in process) gives a new method to carry this out, based on a new construction I've given of a certain type of anticyclotomic p-adic L-function. Following the new format for the number theory seminar, this talk will consist of two 45-minute parts. The first 45 minutes will be expository and is intended to be accessible for graduate students in number theory. There will be a short break (when people are welcome to leave), and the second 45 minutes will be at a higher level.