Constructing a Random Model for the Action of Frobenius on Fundamental Groups of Curves
Author(s)
Afton, Santana
Advisor(s)
Litt, Daniel
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Abstract
For a smooth proper curve $X$ defined over $\mathbb{F}_p$, work of de Jong and Gaitsgory have shown that any representation
\vspace{-2mm}
\[
\pi_1(X) \to \mathrm{GL}_n(\mathbb{F}_{\ell}(\!(t)\!))
\]
has finite image when restricted to $\pi_1(\overline{X})$. We create a random model of $\pi_1(X)$ by utilizing non-geometric automorphisms of $\pi_1(\overline{X})$, and prove that such behavior is generic when the image is solvable. Furthermore, this is proven using results we obtain about pro-$\ell$ free groups and pro-$\ell$ surface groups solely using group-theoretic techniques combined with the Weil conjectures. We end by discussing connections this work has to various topics in arithmetic geometry and geometric group theory.
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Date
2024-07-27
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Text
Resource Subtype
Dissertation