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arxiv: 2304.06622 · v2 · submitted 2023-04-13 · 🧮 math.RT · math.NT

Character Sheaves on Tori over Local Fields

Tanmay Deshpande , Saniya Wagh This is my paper

Pith reviewed 2026-05-24 09:42 UTC · model grok-4.3

classification 🧮 math.RT math.NT
keywords character sheavesmultiplicative local systemslocal Langlands correspondencetori over local fieldsGreenberg functorNéron modelinertial parameterspro-algebraic groups
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The pith

A canonical isomorphism equates the group of multiplicative local systems on L^+T to inertial local Langlands parameters for any torus T.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper first applies the Greenberg functor to the connected Néron model of a torus T over a complete discrete valuation field to obtain the pro-algebraic group L^+T. It computes the fundamental group of L^+T and then defines multiplicative local systems on this group. The central result is a canonical isomorphism of abelian groups identifying these local systems with the inertial local Langlands parameters for T. The construction follows Serre's earlier treatment for the multiplicative group and recovers the classical Langlands correspondence for tori through the sheaf-function dictionary.

Core claim

We construct a canonical isomorphism of abelian groups between the group of multiplicative local systems on L^+T and inertial local Langlands parameters for T, where L^+T is the connected commutative pro-algebraic group over the residue field obtained by applying the Greenberg functor to the connected Néron model of T.

What carries the argument

The pro-algebraic group L^+T obtained from the Greenberg functor applied to the connected Néron model of T, on which multiplicative local systems are defined and compared to inertial parameters.

If this is right

  • The isomorphism extends Serre's computation for the multiplicative group to arbitrary tori.
  • It recovers the classical local Langlands correspondence for tori via the sheaf-function correspondence.
  • Multiplicative local systems on L^+T supply a geometric model for the inertial part of the correspondence.
  • The construction gives a uniform description of both sides as abelian groups.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same Greenberg-functor construction could be tested on non-toral groups to see whether analogous isomorphisms appear.
  • Functoriality of the isomorphism under morphisms of tori would be a natural next check.
  • Explicit calculations for split and anisotropic tori of small dimension could verify the groups match in rank and torsion.

Load-bearing premise

The Greenberg functor applied to the connected Néron model produces a connected commutative pro-algebraic group L^+T whose fundamental group and multiplicative local systems behave in the way required for the isomorphism.

What would settle it

An explicit torus T for which the abelian group of multiplicative local systems on the associated L^+T fails to be isomorphic to the group of inertial local Langlands parameters for T.

read the original abstract

Let $\breve{K}$ be a complete discrete valuation field with an algebraically closed residue field ${k}$ and ring of integers $\breve{{O}}$. Let $T$ be a torus defined over $\breve{K}$. Let $L^+T$ denote the connected commutative pro-algebraic group over ${k}$ obtained by applying the Greenberg functor to the connected N\'eron model of $T$ over $\breve{{O}}$. Following the work of Serre for the multiplicative group, we first compute the fundamental group $\pi_1(L^+T)$. We then study multiplicative local systems (or character sheaves) on $L^+T$ and establish a local Langlands correspondence for them. Namely, we construct a canonical isomorphism of abelian groups between the group of multiplicative local systems on $L^+T$ and inertial local Langlands parameters for $T$. Finally, we relate our results to the classical local Langlands correspondence for tori over local fields due to Langlands, via the sheaf-function correspondence.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper defines L⁺T as the connected commutative pro-algebraic group over the algebraically closed residue field k obtained by applying the Greenberg functor to the connected Néron model of a torus T over a complete discrete valuation field Ǩ with ring of integers Ǒ. It computes the fundamental group π₁(L⁺T), extending Serre's computation in the Gₘ case, and constructs a canonical isomorphism of abelian groups between the multiplicative local systems (character sheaves) on L⁺T and the inertial local Langlands parameters for T. The results are related to Langlands' classical local Langlands correspondence for tori via the sheaf-function correspondence.

Significance. If the stated isomorphism holds, the work supplies a geometric construction realizing the inertial part of the local Langlands correspondence for tori in terms of character sheaves on a pro-algebraic group obtained from Néron models. It extends standard tools (Greenberg functor, Néron models, sheaf-function correspondence) and could serve as a template for analogous geometric correspondences in broader settings within the Langlands program.

minor comments (3)
  1. The abstract invokes the Greenberg functor and the connected Néron model to define L⁺T but does not specify the precise functorial properties used to guarantee that L⁺T is commutative and pro-algebraic; a brief reminder of these properties in §2 would improve readability.
  2. The computation of π₁(L⁺T) is stated to extend Serre's result for Gₘ, but the precise identification of the fundamental group with a quotient of the cocharacter lattice (or equivalent) is not previewed; adding this identification as an equation in the introduction would clarify the subsequent isomorphism.
  3. The sheaf-function correspondence relating the constructed isomorphism to Langlands' classical result is mentioned only at the end of the abstract; a short statement of the precise compatibility (e.g., which functions correspond to which parameters) would strengthen the final section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of the manuscript, including the accurate summary of the main results and the recommendation for minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; construction is self-contained

full rationale

The paper's central result is an explicit construction: apply the Greenberg functor to the connected Néron model of T to obtain L⁺T, extend Serre's computation of π₁ to this pro-algebraic group, then exhibit a canonical isomorphism between multiplicative local systems on L⁺T and inertial Langlands parameters for T, finally relating it to Langlands' classical correspondence via the sheaf-function correspondence. All steps invoke standard external tools (Greenberg functor, Néron models, Serre's Gₘ case, Langlands' result) without any self-citation load-bearing the isomorphism, without fitted parameters renamed as predictions, and without any definitional reduction of the claimed isomorphism to its own inputs. The derivation therefore remains independent of the paper's own equations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard constructions from algebraic geometry and number theory over local fields together with prior computations; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The Greenberg functor applied to the connected Néron model of T yields a connected commutative pro-algebraic group L^+T over k
    Invoked to define the object on which local systems are studied
  • standard math Serre's computation of the fundamental group for the multiplicative group extends to general tori
    Used as the starting point for computing π1(L^+T)

pith-pipeline@v0.9.0 · 5705 in / 1467 out tokens · 32085 ms · 2026-05-24T09:42:07.410950+00:00 · methodology

discussion (0)

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Reference graph

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