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.

algebraic geometrynumber theory

Audience: researchers in the topic

MIT number theory seminar

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