Mike Roth (Queen's University)

Number Theory Seminar

Monday, March 25th, 2024

Time: 2:30 p.m.  Place: Jeffery Hall, Room 202

Speaker: Mike Roth (Queen's University)

Title: Galois groups as monodromy groups in étale cohomology

Abstract: This talk is a companion to the talk of David Nguyen earlier in the term. That talk concerned estimating the size of certain trigonometric sums, and the method was to interpret those sums as coming from étale sheaves on an open subset of P^1, and then use the weight machinery of étale cohomology.

In the talk I will try and give a simple introduction to the idea of a sheaf of locally constant sections over a curve, and related ideas in the purely topological case, and then say how those notions can be expressed in terms of representations of Galois groups in the characteristic p case. Hopefully there will be time to explain the idea of the ‘weights’ of a sheaf, and the weights of the action on cohomology. Finally, I hope to briefly discuss Grothendieck’s viewpoint of ’sheaves as functions’, and so return to the problem of estimating trigonometric sums.

None of these interpretations or constructions are new. They are all part of the beautiful synthesis of number theory and geometry that is étale cohomology, as envisioned by Grothendieck, and as developed by Grothendieck, Artin, Deligne, and collaborators in the 1960’s and 70’s.