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The eigencurve is a powerful tool introduced by Coleman and Mazur to study$p$-adic families of overconvergent modular forms. In this article, weintroduce an analogous set of tools for understanding families of"overconvergent" $p$-adic representations of $\pi_1(X)$, where $X$ is a smoothaffine variety over a finite field of characteristic $p$. Our main theorem is atrace formula relating the $L$-function of such a family to the geometry of asequence of associated eigenvarieties. In the case of a single $p$-adicrepresentation, 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 appliedby Davis, Wan, and Xiao to prove a spectral halo decomposition of theeigencurve attached to $\mathbb{Z}_p$-towers over $X=\mathbb{A}^1$.

Weak Analytic Geometry and a Trace Formula for Families of $p$-adic Representations