Arithmetic gauge theory: A brief introduction

Kim, Minhyong

doi:10.1142/9789813278677_0004

Cited by 5 publications

(6 citation statements)

References 51 publications

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“…In some earlier papers, a preliminary attempt to define and compute arithmetic analogues of Chern–Simons functions was made [1, 3, 7, 8, 10]. Also, moduli spaces of ‘arithmetic gauge fields’ have been applied to Diophantine geometry [2, 11]. One of the obstructions to developing a full‐fledged arithmetic topological field theory based on Chern–Simons theory is that natural arithmetic dualities involve a sheaf

\mu _n

\hat{{\mathbb {Z}}}(1)

, which are not trivialisable in general.…”

Section: Towards Arithmetic Bf Theorymentioning

confidence: 99%

A note on abelian arithmetic BF‐theory

Carlson

Kim

2022

Bulletin of London Math Soc

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\mu _n

\hat{{\mathbb {Z}}}(1)

, which are not trivialisable in general.…”

Section: Towards Arithmetic Bf Theorymentioning

confidence: 99%

A note on abelian arithmetic BF‐theory

Carlson

Kim

2022

Bulletin of London Math Soc

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“…giving the generalisation of Selmer groups envisaged in [Kim,§10]. As phrased, this assumes that the primes in S do not divide ℓ, but primes dividing ℓ can be allowed if we incorporate crystalline data as in Example 5.18 below.…”

Section: Weighted Shifted Poisson Structuresmentioning

confidence: 99%

“…In [Kim,§10], Kim outlined an approach to interpreting Selmer groups as Lagrangian intersections of suitable moduli spaces, and proposed various generalisations. The main purpose of this paper is to provide the necessary foundations to make constructions of this sort precise in a derived geometric setting.…”

Section: Introductionmentioning

confidence: 99%

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

Pridham¹

2022

Preprint

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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...

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“…The inspiration for these constructions comes from a wonderful idea of M. Kim [Kim15] (see also [CKKPY16]) who, guided by the folkore analogy between 3-manifolds and rings of integers in number fields and between knots and primes, gave a construction of an arithmetic Chern-Simons invariant for finite gauge group. He also suggested ( [Kim18]) to look for more general Chern-Simons type theories in number theory that resemble the corresponding theories in topology and mathematical physics. An important ingredient of classical Chern-Simons theory is the symplectic structure on the character variety of a closed orientable surface: When the surface is the boundary of a 3-manifold, the Chern-Simons construction gives a section of a line bundle over the character variety.…”

Section: =mentioning

confidence: 99%

Volume and symplectic structure for l-adic local systems

Pappas

2020

Preprint

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We introduce a notion of volume for an ℓ-adic local system over an algebraic curve and, under some conditions, give a symplectic form on the rigid analytic deformation space of the corresponding geometric local system. These constructions can be viewed as arithmetic analogues of the volume and the Chern-Simons invariants of a representation of the fundamental group of a 3-manifold which fibers over the circle and of the symplectic form on the character varieties of a Riemann surface. We show that the absolute Galois group acts on the deformation space by conformal symplectomorphisms which extend to an ℓ-adic analytic flow. We also prove that the locus of the deformation space over which the local system suitably descends is the critical set of a collection of rigid analytic functions. The vanishing cycles of these functions give additional invariants. ContentsIntroduction 1 1. Preliminaries 6 2. K 2 invariants and 2-forms 10 3. Chern-Simons and volume 15 4. Representations of étale fundamental groups 25 5. Deformations and lifts 32 6. The symplectic nature of the Galois action 37 7. Appendix: Interpolation of iterates and flows 44 References 47 Notations: Throughout the paper N denotes the non-negative integers and ℓ is a prime. We denote by F ℓ the finite field of ℓ elements and by Z ℓ , resp. Q ℓ , the ℓ-adic integers, resp. ℓ-adic numbers. We fix an algebraic closure Qℓ of Q ℓ and we denote by | | ℓ (or simply | |), resp. v ℓ , the ℓ-adic absolute value, resp. ℓ-adic valuation on Qℓ , normalized so that |ℓ| ℓ = ℓ −1 , v ℓ (ℓ) = 1. We will denote by F a field of characteristic ℓ which is algebraic over the prime field F ℓ and by W (F), or simply W , the ring of Witt vectors with coefficients in F. If k is a field of characteristic = ℓ we fix an algebraic closure k. We denote by k sep the separable closure of k in k, by k(ζ ℓ ∞ ) = ∪ n≥1 k(ζ ℓ n ) the subfield of k sep generated over k by the (primitive) ℓ n -th roots of unity ζ ℓ n and by χ cycl : Gal(k sep /k) → Z × ℓ the cyclotomic character defined by σ(ζ ℓ n ) = ζ χ cycl (σ) ℓ nfor all n ≥ 1. We set G k = Gal(k sep /k). Finally, we will denote by ( ) * the Pontryagin dual, by ( ) ∨ the linear dual, and by ( ) × the units.

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Arithmetic gauge theory: A brief introduction

Cited by 5 publications

References 51 publications

A note on abelian arithmetic BF‐theory

A note on abelian arithmetic BF‐theory

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

Volume and symplectic structure for l-adic local systems

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