Tripod-Degrees

Abstract

Let $p$, $l$ be distinct prime numbers. A tripod-degree over $p$ at $l$ is defined to be an $l$-adic unit obtained by forming the image, by the $l$-adic cyclotomic character, of some continuous automorphism of the geometrically pro-$l$ fundamental group of a split tripod over a finite field of characteristic $p$. The notion of a tripod-degree plays an important role in the study of the geometrically pro-$l$ anabelian geometry of hyperbolic curves over finite fields, e.g., in the theory of cuspidalizations of the geometrically pro-$l$ fundamental groups of hyperbolic curves over finite fields. In the present paper, we study the tripod-degrees. In particular, we prove that, under a certain condition, the group of tripod-degrees over $p$ at $l$ coincides with the closed subgroup of the group of $l$-adic units topologically generated by $p$. As an application of this result, we also conclude that, under a certain condition, the natural homomorphism from the group of automorphisms of the split tripod to the group of outer continuous automorphisms of the geometrically pro-$l$ fundamental group of the split tripod that lie over the identity automorphism of the absolute Galois group of the basefield is surjective.