On l-independence for the etale cohomology of rigid spaces over local fields

TL;DR

This paper proves that the trace of the Weil group action on the l-adic etale cohomology of rigid spaces over local fields is an integer and independent of l under certain conditions, extending previous results to more general cases.

Contribution

It establishes l-independence of traces for etale cohomology of rigid spaces, including the case of smooth spaces and those over fields of characteristic zero, using modified and new techniques.

Findings

Trace of Weil group action is an integer.

Trace is independent of l for smooth rigid spaces.

Results extend to certain non-smooth cases using finiteness theorems.

Abstract

We investigate the action of the Weil group on the compactly supported l-adic etale cohomology groups of rigid spaces over a local field. We prove that the alternating sum of the traces of the action is an integer and is independent of l when either the rigid space is smooth or the characteristic of the base field is equal to 0. We modify the argument of T. Saito to prove a result on l-independence for nearby cycle cohomology, which leads to our l-independence result for smooth rigid spaces. In the general case, we use the finiteness theorem of R. Huber, which requires the restriction on the characteristic of the base field.