Speaker: David Roberts (University of Minnesota, Morris)

Title: Wild Ramification in Hypergeometric Motives

Abstract:

The bulk of my talk will be an overview of the current state of knowledge of wild ramification in general hypergeometric motives at a fixed prime $p$.  The presentation will be as elementary and visual as possible, using p-adic ordinals of  field discriminants of trinomials $x^n - n t x + (n-1) t$ and their underlying Galois theory as a continuing example.  It will be revealed that the general situation is very complicated, but exhibits enough patterns that one can still reasonably hope for a universal formula identifying all numerical invariants of wild p-adic ramification in all hypergeometric motives.

If one restricts to the case where $\operatorname{ord}_p(t)$ is coprime to $p$ then the situation simplifies considerably.  The ramp conjecture of Section 13 of my survey on Hypergeometric Motives with  Fernando Rodriguez Villegas predicts conductor exponents.  I will conclude with a new refinement of the ramp conjecture that predicts, via Feynman-like diagrams, how the conductor exponents decompose as a sum of slopes. The refinement reveals much more structure than the original ramp conjecture, and I hope will point the way to a proof.