CMIIMSC 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 ladic Laurent series Leo Gratien ENS. 171023 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 socalled "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.
