Weak admissibility of exponentially twisted cohomology associated with some nondegenerate functions
Peijiang Liu
TL;DR
This work ties exponentially twisted cohomology to p-adic Hodge theory by constructing generalized filtered $\Phi$-modules over a Frobenius extension $K_{*}/K_{*}$ and analyzing their weak admissibility. It develops the Newton polyhedron framework for nondegenerate Laurent polynomials $f$, introducing the NP-module $V_{\mathrm{NP}}$ with dual Frobenius structures and a Newton filtration, and proves that the specialization map between exponentially twisted de Rham and rigid cohomology is an isomorphism in many cases. Under the condition $p\neq 2$ and $\ell_{\mathrm{HT}}(V_{\mathrm{dR}},F^{\ast}_{\mathrm{irr}})\le p-2$, the associated filtered $\Phi$-module is weakly admissible, with combinatorial methods to compute the HT-length; the NP-agreeable framework plays a central role in establishing these results. The paper thereby provides a Newton-above-Hodge interpretation for exponential sums, clarifies the relationship between irregular Hodge theory and $p$-adic Hodge-type filtrations, and offers concrete examples and questions for extending weak admissibility to more general geometric and arithmetic settings.
Abstract
In this article, we study the filtered $Φ$-modules canonically attached to the exponentially twisted cohomology associated with some nondegenerate functions. Inspired by $p$-adic Hodge theory, we conjecture that those filtered $Φ$-modules are weakly admissible. We show that this expectation is correct under some assumptions using the theory of Adolphson and Sperber.