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Math-2469 · mathematics · Algebraic geometry

The Restricted Geometric Langlands Conjecture

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Let $k$ be algebraically closed, let $X/k$ be a smooth projective geometrically connected curve, and let $G$ be reductive, with $\operatorname{char}(k)$ greater than the Coxeter number of $G$. In the corresponding derived $\ell$-adic setting, prove the canonical equivalence $\operatorname{Shv}_{\operatorname{Nilp}}(\operatorname{Bun}_G)\simeq\operatorname{IndCoh}_{\operatorname{Nilp}}(\operatorname{LocSys}^{\operatorname{restr}}_{\widehat G})$.

Formulation references: [S1], [S2].

Short English statement

Let $k$ be algebraically closed, let $X/k$ be a smooth projective geometrically connected curve, and let $G$ be reductive, with $\operatorname{char}(k)$ greater than the Coxeter number of $G$. In the corresponding derived $\ell$-adic setting, prove the canonical equivalence $\operatorname{Shv}_{\operatorname{Nilp}}(\operatorname{Bun}_G)\simeq\operatorname{IndCoh}_{\operatorname{Nilp}}(\operatorname{LocSys}^{\operatorname{restr}}_{\widehat G})$.

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2469-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2469

The equivalence retains four characteristic conditions in the finite-field regime and additional very-good and Weyl-order assumptions over arbitrary algebraically closed bases. The shorter Coxeter-number condition in this problem is not by itself identified with all those hypotheses for every reductive group.

The Restricted Geometric Langlands Conjecture

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 014; 2026-09-24), the asserted result is that restricted geometric Langlands has full spectral support in positive characteristic in the two stated characteristic regimes, with additional very-good and Weyl-order conditions over arbitrary algebraically closed fields.

Mathematical statement of the claimed result

Manuscript 1.
Let $k$ be algebraically closed of characteristic $p>0$, $\ell\ne p$, $E=\overline{\mathbb Q}_{\ell}$, $X/k$ smooth projective connected and $G/k$ connected reductive. Require the four common conditions: an invariant nondegenerate Lie-algebra form nondegenerate on every Levi center, Chevalley restriction for every Levi, semisimple Lie-algebra centralizers being Levi, and nilpotent elements over every extension lying in parabolic unipotent-radical Lie algebras. Regime A additionally requires $k=\overline{\mathbb F}_q$ and descent of $X$ and split $G$ to $\mathbb F_q$. Regime B instead permits arbitrary algebraically closed $k$, but requires $p$ very good for $G$ and $p\nmid|W_G|$. Very good excludes primes dividing $n+1$ for $A_n$, $2$ for $B,C,D$, $2,3$ for $G_2,F_4,E_6,E_7$, and $2,3,5$ for $E_8$.

Let $Y=\operatorname{LS}_{\widehat G}^{\rm restr}(X)$ be the restricted spectral prestack, parametrizing right-t-exact symmetric monoidal functors from dual-group representations to $\operatorname{QCoh}(T)\otimes_E\operatorname{QLisse}_E(X)$ on derived affine $T$. The claim says the previously occurring open-and-closed union $Y'$ equals all of $Y$, giving a $\operatorname{QCoh}(Y)$-linear equivalence \[ \mathbb L_G^{\rm restr}:\operatorname{Shv}_{\rm Nilp}(\operatorname{Bun}_G(X)) \xrightarrow{\sim}\operatorname{IndCoh}_{\rm Nilp}(Y). \] The automorphic and spectral categories each retain nilpotent singular support. This positive-characteristic result is restricted to the two stated regimes.

Outline of the argument

For manuscript 1 (family 014; 2026-09-24), a hypothetical missing component is transported along a trait to a geometric specialization. Levi reduction and transverse Hecke modifications provide cotangent tests whose nonvanishing prevents that spectral summand from being annihilated.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 014; 2026-09-24).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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