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I posted this question in Math StackExchange a couple years ago; due to the recent surge in interest, and following the feedback of several users, I've decided to cross-post it here. I apologize for ...

Let $k$ be a finite field and $k_E:=k((X))$ denote the field of Laurent series over $k$. We define a Frobenius endomorphism on $k_E$ via $f(X)\mapsto f(X^p)$. We choose a lift $\varphi:k_E^{sep}\...

Let $\Omega$ be a completion of an algebraic closure of $\mathbb F_q\left(\left(\frac1T\right)\right)$ for the valuation $-\deg$. Consider two matrices $M_1,M_2$ in $\mathcal M_2(\Omega)$ that are $\...