let $x$ be a smooth affine variety over a finite field of characteristic $p$. the dwork-monsky trace formula is a fundamental tool in understanding the $l$-functions of $p$-adic representations of $\pi_1(x)$. we extend this result to the study of representations valued in a higher-dimensional local ring $r$. the special case $r=\mathbb{z}_p[[t]]$ arises naturally in the study of \'etale $\mathbb{z}_p$-towers over $x$. time permitting, we discuss some spectral-halo type results and conjectures describing the $p$-adic variation of slopes in certain $\mathbb{z}_p$-towers.