CMI-IMSC Number Theory seminar
Date: Tuesday, 17 October 2023
Time: 3:30 - 4:30 PM
Venue: Lecture Hall 802
On the finiteness of geometric monodromy in l-adic Laurent series
Leo Gratien
ENS.
17-10-23

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Abstract

Epitomised by the work of Wiles, and later Deligne, we know that studying Galois representations of Q_l vector spaces is a fruitful approach in number theory. Namely, let X_0 denote a proper and smooth scheme over a field of characteristic p (say F_p, where p != l) and X its geometric realisation over the algebraic closure of F_p. A crucial question posed by Deligne is about the image of pi_1(X) inside a continuous representation of pi_1(X_0) in \bar{Q_l} : under an irreducibility hypothesis, this image is finite in GL_n\bar{Q_l}).

This talk will be about such a question, where we change the local field to F_l((t)), the Laurent series in t. This is essentially the only way to alter the original conjecture. Now completely settled, I will discuss the origin and framework of the so-called "de Jong's conjecture" ; the interplay with the finiteness of universal deformation rings; and the proof in the cases n=1,2. If time permits, I will say a word about the case n>2.