Zeta functions everywhere!

23.05.2018, 14:00  –  Haus 9, Raum 2.22
Institutskolloquium

Pierrette und Philippe Cassou-Noguès (University of Bordeaux)

14:00: Philippe Cassou-Nogues (Bordeaux): Zeta functions and Galois module structures.
15:00: Tea and Coffee break
15:30: Pierrette Cassou-Nogues (Bordeaux): Zeta functions and roots of Bernstein polynomials.

Abstracts:

Philippe Cassou-Nogues (Bordeaux): Zeta functions and Galois module structures.
In this talk I shall address a question raised by A. Frohlich in the seventies, putting it in a historical context. The underlying problem being of arithmetic nature, I shall first recall some basic concepts such as that of a Galois extension. Here is the question: starting from a Galois extension N/Q, does the ring of integers O_N in N have a Z-basis of the form {a^γ}, γ running over Γ=Gal(N/Q)?
This question was answered positively by Hilbert when Γ is commutative. As I shall explain in this talk, the obstruction to the existence of such a basis, when Γ is non commutative, comes - as conjectured by Frohlich - from the existence of zeroes at s = 1/2 of some analytic functions attached to the extension. If time permits, I shall mention some of the techniques developped to solve this problem, since they have since then led to new research in Number Theory and Arithmetic Geometry.

Pierrette Cassou-Nogues (Bordeaux): Zeta functions and roots of Bernstein polynomials.
This talk which takes place in the p-adic setup - and for which I will provide the basic concepts needed here - addresses rationality issues.
In 1966, Borewicz and Safarevic conjectured that the generating series P(t) for the number of solutions of a certain polynomial equation over a p-adic field K, is a rational fonction of t. Igusa proved this rationality property in the 70's using the Igusa Zeta function Z_K(s) and the equality P(t) = (1-tZ_K(s))/(1-t) where t = q^-s. When K = R or C, Igusa conjectured that the poles of Z_K(s) are roots of the Bernstein polynomial of f.
This conjecture was proven for n = 2 by F. Loeser in 1988. We shall discuss it in the frame of the motivic Zeta function (defined by Denef and Loeser) which is easier to handle than the Igusa Zeta function. For this purpose, weshall use a very old tool, the Newton polygon, in order to give an idea of the ideas underlying the proof.