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Let $X$ be a smooth scheme over an algebraically closed field. When $X$ isproper, it was proved in \cite{me1} that the moduli of $\ell$-adic continuousrepresentations of $\pi_1^\et(X)$, $\LocSys(X)$, is representable by a(derived) $\Ql$-analytic space. However, in the non-proper case one cannotexpect that the results of \cite{me1} hold mutatis mutandis. Instead, assuming$\ell$ is invertible in $X$, one has to bound the ramification at infinity ofthose considered continuous representations. The main goal of the current textis to give a proof of such representability statements in the open case. Wealso extend the representability results of \cite{me1}. More specifically,assuming $X$ is assumed to be proper, we show that $\LocSys(X)$ admits acanonical shifted symplectic form and we give some applications of suchexistence result.

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