Jump sets in local fields
Carlo Pagano (Leiden University)
Wednesday 9th May, 2018 16:00-17:00 Maths 110
In this talk I will explain the role of jump sets, simple combinatorial objects, in answering basic questions regarding the arithmetic of local fields. Namely we will firstly see how jump sets allow us to parametrize the possible structure of filtered groups occurring as multiplicative groups of local fields with their natural filtration coming from the valuation of the field. This theory will be explained by means of analogy with the more elementary classification of automorphism types in finite abelian p-groups, for some prime p. The analogy will lead us to a classification of such structure by means of orbits of certain p-adic groups of filtered automorphism acting on ”free-filtered modules”. We will see how one can put a natural measure on the set of local fields (with given degree and ramification degree), by a classical formula of Serre, and how we have a natural measure on the possible filtered modules, simply by using the Haar measure of the corresponding orbit via the above classification. It turns out that there is equidistribution of the isomorphisms classes of filtered modules among the set of principal units of local fields (weighted with Serre’s Mass formula). I will explain some of the ideas of the proof of this theorem, and in particular the role played in it by a simple combinatorial Markov process called “the shooting game”. Following the shooting game closely one is lead to explicit invariants of a large class of Eisenstein polynomials: one can see the structure of the filtered unit group encoded in the valuation of the coefficients of the polynomials. I will explain how this links to some basic invariant of ramification theory. Finally it will be shown the role played by the unit group of a local field K, viewed as a filtered group, in the description of the abelian extensions of K, and how, using jump sets and the theory of filtered modules described above, we can answer basic questions regarding the ramification of such extensions. No technical working knowledge in the arithmetic of local fields will be necessary to follow this talk, since the material will be introduced from scratch.