Moduli of $\ell$-adic pro-\'etale local systems for smooth non-proper schemes

TL;DR

This paper extends the moduli space theory of $ ext{ell}$-adic local systems to smooth non-proper schemes, establishing their representability and symplectic structure, with implications for understanding ramification at infinity.

Contribution

It proves the representability of moduli of $ ext{ell}$-adic local systems on non-proper schemes and extends symplectic structure results to this broader context.

Findings

Representability of $ ext{ell}$-adic local systems on non-proper schemes.

Existence of a canonical shifted symplectic form on the moduli space.

Application to bounding ramification at infinity.

Abstract

Let $X$ be a smooth scheme over an algebraically closed field. When $X$ is proper, it was proved in \cite{me1} that the moduli of $\ell$-adic continuous representations of $\pi_1^\et(X)$, $\LocSys(X)$, is representable by a (derived) $\Ql$-analytic space. However, in the non-proper case one cannot expect that the results of \cite{me1} hold mutatis mutandis. Instead, assuming $\ell$ is invertible in $X$, one has to bound the ramification at infinity of those considered continuous representations. The main goal of the current text is to give a proof of such representability statements in the open case. We also extend the representability results of \cite{me1}. More specifically, assuming $X$ is assumed to be proper, we show that $\LocSys(X)$ admits a canonical shifted symplectic form and we give some applications of such existence result.