  The eigencurve is a powerful tool introduced by Coleman and Mazur to study
$p$-adic families of overconvergent modular forms. In this article, we
introduce an analogous set of tools for understanding families of
"overconvergent" $p$-adic representations of $\pi_1(X)$, where $X$ is a smooth
affine variety over a finite field of characteristic $p$. Our main theorem is a
trace formula relating the $L$-function of such a family to the geometry of a
sequence of associated eigenvarieties. In the case of a single $p$-adic
representation, our result reduces to the well known trace formula of Monsky.
We apply our theory to the study of $T$-adic exponential sums attached to
$\mathbb{Z}_p$-towers over $X$. Special cases of this theory have been applied
by Davis, Wan, and Xiao to prove a spectral halo decomposition of the
eigencurve attached to $\mathbb{Z}_p$-towers over $X=\mathbb{A}^1$.
