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The Galois Invariant Locus in the Berkovich Projective Line Rumely, Robert


This talk concerns joint work with Xander Faber. Let $K$ be a nonarchimedean local field of characteristic 0 and residue characteristic $p > 0.$ Let $q = p^f$ be the order of its residue field, and let $\mathbf{C}_K$ be the completion of an algebraic closure of $K$. The group of continuous automorphisms $Gal_c(\mathbf{C}_K/K)$ acts on the Berkovich Projective Line $\bf{P}^1_{\mathbf{C}_K}$. We show that the Galois invariant locus in $\bf{P}^1_{\mathbf{C}_K}$ is a densely branched tree which properly contains the path-closure of $\mathbf{P}^1(K)$, and is contained in a tube of path-distance radius $1/(p-1)*[1 + 1/(p-1)]$ around the path-closure. The radius can probably be improved to $ 1/(p-1)$. The Galois invariant locus has $q+1$ branches at each type II point in the locus corresponding to a disc $D(a,p^b) $, with $b$ rational, and no other branches. We construct a conjecturally dense subset of the Galois invariant locus. We also establish a conjecture of Benedetto, that each Galois invariant point is defined over a totally ramified extension of $K$.

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