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arXiv · 1909.01447

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

James Upton
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Abstract

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 $X=\mathbb{A}^1$.

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James Upton. 2019-09-03. Weak Analytic Geometry and a Trace Formula for Families of $p$-adic Representations. https://arxiv.org/abs/1909.01447

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