Shifted symplectic structures on derived analytic moduli of $\ell$-adic local systems and 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-étale 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 Gm-invariants.In particular, this establishes the existence of shifted symplectic and Lagrangian structures on derived moduli stacks of ℓ-adic constructible complexes on smooth varieties via Poincaré duality, and on derived moduli stacks of ℓ-adic Galois representations via Tate and Poitou-Tate duality; the latter proves a conjecture of Minhyong Kim.1 for instance BGLr is 2-shifted symplectic J.P.PRIDHAM suitable F -valued sheaves on X with an (n − 2m)-shifted symplectic structure when X is proper (Examples 3.13), or an (n + 1 − 2m)-shifted Lagrangian structure in general (Examples 3.21).The algebraic closure hypothesis is necessary for the examples in §3 because of the Tate twists featuring in duality theorems, so the purpose of Section 4 is to develop a generalisation of the theory of shifted symplectic structures to address cases where the dualising bundle is non-trivial. The considerations introduced here work equally well in algebraic and analytic settings, applying to moduli of maps from spaces which are not Calabi-Yau but have a dualising line bundle, and this weighted theory can be thought of as a special case of the theory of P-shifted symplectic structures from the seminal manuscript [BG]. The idea is to characterise the moduli space as the G m -invariant locus of a natural G m -equivariant formal thickening, with that thickening carrying a shifted symplectic structure of non-zero weight with respect to the G m -action. The resulting constructions have a similar flavour to Iwasawa theory, but with G m -actions rather than Ẑ * -actions. Remark 5.13 describes local forms for weighted shifted symplectic spaces in terms of twisted shifted cotangent bundles, and §4.4 establishes weighted representability results.Section 5 contains the main applications of the paper, constructing weighted shifted symplectic (Examples 5.12) and Lagrangian (Examples 5.17) structures on a wide range of derived analytic moduli stacks of pro-étale sheaves. Poincaré duality for smooth proper schemes and local Tate duality for local fields tends to give rise to symplectic structures for moduli of sheaves, while Poincaré duality for smooth schemes and Poitou-Tate duality for number fields give rise to Lagrangian structures. There are various ways to combine these, including Lagrangian intersections yielding the Selmer-type constructions envisaged by Kim (Example 5.17.( 7)).In Section 6, we set up the theory of shifted Poisson structures in the weighted setting. The main result is Theorem 6.8 and its generalisation in §6.6.2, ext...