McGill University, December 4 - 7, 2015
In this talk, I will explain that the local density function with respect to the size of a residue field, when we vary a base complete local ring to its finite unramified extension, is a rational function (i.e. polynomial divided by another polynomial).
This is joint work with Michel Waldschmidt. For each $n \geq 3$, we exhibit new families of Thue-Mahler equations having only trivial solutions. Furhermore, we produce an effective upper bound for the number of these solutions.
We make progress on this conjecture by giving a "subconvex" bound on the size of the 2-torsion of the class group of a number field in terms of its discriminant, for any value of n. The proof is surprisingly elementary, and we give several applications of this result stemming from the case of cubic fields, including improved bounds on the number of A4 fields, and on the number of integer points an elliptic curve can have.
Along the way, we prove a a surprising result on the shape of the lattice of the ring of integers of a number field. Namely, we show that such a lattice is very limited in how `skew' it can be.