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arXiv · 1610.04969

Local arboreal representations

Jacqueline Anderson·Spencer Hamblen·Bjorn Poonen·Laura Walton
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Abstract

Let $K$ be a field complete with respect to a discrete valuation $v$ of residue characteristic $p$. Let $f(z) \in K[z]$ be a separable polynomial of the form $z^\ell-c.$ Given $a \in K$, we examine the Galois groups and ramification groups of the extensions of $K$ generated by the solutions to $f^n(z)=a$. The behavior depends upon $v(c)$, and we find that it shifts dramatically as $v(c)$ crosses a certain value: $0$ in the case $p \nmid \ell$, and $-p/(p-1)$ in the case $p=\ell$.

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Jacqueline Anderson, Spencer Hamblen, Bjorn Poonen, Laura Walton. 2020-07-02. Local arboreal representations. https://doi.org/10.1093/imrn%2Frnx054

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