Tripod-degrees

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.