|Date: Friday, December 08, 2017
Location: 1866 East Hall (3:00 PM to 4:00 PM)
Title: The conductor of a Galois group
Abstract: Let D be the Galois group of an extension of local fields. Here by a local field we mean a finite extension of a p-adic field. In this talk we will define (Artin) conductors, which are number-theoretic invariants associated to the linear representations of D. Having defined things locally at each prime, the theory readily globalizes to define the conductor of a representation of a Galois group G of number fields. The conductor of a sum of representations is the product of the corresponding conductors, so it suffices to consider the irreducible representations. A way to consider all the irreducibles at once is to take the conductor of the regular representation of G, which turns out to be the discriminant of the extension -- the so-called Fuhrerdiskriminantenproduktformel.
Along with conductors we will construct the Artin representation of D. All the computations will be done in terms of characters so that at the end of the day we will have only the character of the Artin representation; curiously, no direct construction of this representation is known. From the formula mentioned above, it turns out that the exponents of the prime divisors of the discriminant are given by the dimensions of the Artin representations.
It will be helpful to have some background in algebraic number theory, especially the theory surrounding the ramification of primes, but mostly just for the sake of orientation and motivation. I will do my best to define everything.
Speaker: Andrew Odesky
Institution: University of Michigan