Past Number Theory Seminar

It is a wide open question whether the set of rational points on a smooth quartic surface in projective three-space can be nonempty, yet finite. In this talk I will treat the case of diagonal quartics V, which are given by: a x^4 + b y^4 + c z^4 + d w^4 = 0 for some nonzero rational a,b,c,d. I will assume that the product abcd is a square and that V contains at least one rational point P. I will prove that if none of the coordinates of P is zero, and P is not contained in one of the 48 lines on V, then the set of rational points on V is dense. This is based on joint work with Adam Logan and David McKinnon.

Let N(A) be the number of integer solutions of x^2 + y^2

I will explain the statement of a generalization of Serre's conjecture on mod p Galois representations to the context of Hilbert modular forms. The emphasis will be on the recipe for the set of possible weights (formulated by Buzzard, Jarvis and myself, and partly proved by Gee) and its behavior in some special cases.