Past Number Theory Seminar

We recall that an elliptic curve is a curve of genus one with a rational point on it. Certain algorithms for determining the structure of the group of rational points on an elliptic curve produce a whole set of curves of genus one and then require that we determine which of these curves has a rational point.

Unfortunately no algorithm which has been proved to terminate is known for doing this. Such an algorithm or proof would probably have profound implications for the study of elliptic curves and may shed light on the Birch and Swinnerton-Dyer conjecture.

This talk will be about joint work with Samir Siksek (Warwick) on the development of a new algorithmic criterion for determining that a given curve of genus one has no rational points. Both the theory behind the criterion and recent attempts to make the criterion computationally practical, will be detailed.

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.