Shifted symplectic structures on derived analytic moduli of $\ell$-adic local systems and Galois representations

TL;DR

This paper develops a framework for non-Archimedean derived analytic geometry to construct shifted symplectic structures on moduli stacks of $ ext{ell}$-adic local systems and Galois representations, confirming conjectures and linking to duality theories.

Contribution

It introduces weighted shifted symplectic structures in derived non-Archimedean geometry and applies them to establish symplectic structures on moduli of $ ext{ell}$-adic sheaves and Galois representations, proving a conjecture of Minhyong Kim.

Findings

Existence of shifted symplectic structures on derived moduli stacks of $ ext{ell}$-adic complexes.

Construction of Lagrangian structures via Poincaré duality.

Confirmation of Kim's conjecture on Galois representations.

Abstract

We develop a characterisation of non-Archimedean derived analytic geometry based on dg enhancements of dagger algebras. This allows us to formulate derived analytic moduli functors for many types of pro-\'etale sheaves, and to construct shifted symplectic structures on them by transgression using arithmetic duality theorems. In order to handle duality functors involving Tate twists, we introduce weighted shifted symplectic structures on formal weighted moduli stacks, with the usual moduli stacks given by taking $\mathbb{G}_m$-invariants. In particular, this establishes the existence of shifted symplectic and Lagrangian structures on derived moduli stacks of $\ell$-adic constructible complexes on smooth varieties via Poincar\'e duality, and on derived moduli stacks of $\ell$-adic Galois representations via Tate and Poitou--Tate duality; the latter proves a conjecture of Minhyong Kim. Unweighted shifted symplectic and Lagrangian structures are also established for $\ell$-adic Galois representations of cyclotomic fields $K(\mu_{\ell^{\infty}})$, subject to additional constraints related to Iwasawa theory; these derived moduli stacks yield a non-abelian analogue of Selmer complexes, with $0$-shifted symplectic structures related to generalised Cassels--Tate pairings.