Number Theory Seminar: “everywhere unramified” objects in number theory and the cohomology of GLn(Z)

SEMINARS, NUMBER THEORY

Speaker:

Frank Calegari - University of Chicago

One theme in number theory is to study objects via their ramification: the discriminant of a number field, the conductor of an elliptic curve, the level of a modular form, and so on. There is, however, some particular interest in understanding objects which are “everywhere unramified” — and also understanding when such objects don’t exist. Such non-existence results are often the starting point for inductive arguments. For example, Minkowski’s theorem that there are no unramified extensions of Q can be used to prove the Kronecker-Weber theorem, and the vanishing of a certain space of modular forms is the starting point for Wiles’ proof of Fermat’s Last Theorem. In this talk, I will begin by describing many such vanishing results both in arithmetic and in the theory of automorphic forms, and how they are related by the Langlands program (sometimes only conjecturally). Then I will describe the construction of a new example of an automorphic form of level one and “weight zero”. This construction also gives the first non-zero classes in the cohomology of GLn(Z) (for some n) that come from “cuspidal” modular forms (for n > 0).

For more info, see https://ashvin-swaminathan.github.io/home/NTSeminar.html