Tripod-degrees - Y.Hoshi

Jun 5, 2023  ·   In Kyoto Japan  ·   Event ``AHGT Seminar''

Abstract

Let \(p,\ l\) be distinct prime numbers. A tripod-degree over p at l is defined to be an \(\ell\)-adic unit obtained by forming the image, by the \(\ell\)-adic cyclotomic character, of some continuous automorphism of the geometrically pro-\(\ell\) fundamental group of a split tripod (i.e., a hyperbolic curve obtained by forming the complement in the projective line of three distinct rational points) over a finite field of characteristic \(p\). The notion of a tripod-degree plays an important role in the study of the geometrically pro-\(\ell\) anabelian geometry of hyperbolic curves over finite fields. In this talk, we completely determine the set of tripod-degrees under a certain condition with respect to the pair \((p, l)\). Moreover, we also discuss an application of this result to the study of the geometrically pro-\(\ell\) anabelian geometry of split tripods over finite fields.

1. Introduction to Combinatorial Anabelian Geometry (slides), Go Yamashita, from the 2022 Summer school
2. Overview of Combinatorial Anabelian Geometry (slides), Shinichi Mochizuki, from the 2021 workshop
3. Tripod degrees, Yuichiro Hoshi, RIMS preprint, May 2023