Seminario / Dan Collins
on Thursday, December 6, at 11.00 sharp, in Aula C, Dan Collins will talk about:
TITLE: Heegner points for x3+y3=p and p-adic L-functions
ABSTRACT:: "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 algebraic number theory. 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. I'll describe an approach to this using p-adic L-functions, and talk about my strategy (not yet complete) for carrying this out using my work on that topic - and explain where the difficulties in finishing the analysis are.