Fargues–Scholze Correspondence Explained

• The Fargues–Scholze correspondence offers a geometric realization of semisimple local Langlands parameters through sheaves on $G$-bundles over the Fargues–Fontaine curve over a nonarchimedean local field E.
• Key components include the $G(E)$-representations, Hecke eigendata, and L-parameters, using geometric Saitake, Hecke correspondences, and excursion operators.
• The Fargues–Fontaine classification involves the replacement of the quotient space $\operatorname{Spec} \breve E/\phi^{\mathbb Z}$ with a relative curve over perfectoid spaces, incorporating vector bundles and divisors with Weil group actions.

The Fargues–Scholze correspondence is a geometric realization of semisimple local Langlands parameters through sheaves on the stack of $G$-bundles over the Fargues–Fontaine curve. Its organizing principle is

$\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$

Here $E$ is a nonarchimedean local field with residue field $\mathbb F_q$ of characteristic $p>0$, $G/E$ is a connected reductive group, $\ell\neq p$, and $\widehat G$ is the Langlands dual group. The construction uses $\ell$-adic sheaves, geometric Satake, Hecke correspondences, excursion operators, and the spectral action of perfect complexes on the stack of $L$-parameters. The geometric framework and the assignment of a continuous semisimple parameter to every irreducible smooth representation are established; the full categorical local Langlands equivalence remains conjectural in general (Fargues et al., 2021).

1. Geometric foundations

The Fargues–Fontaine curve replaces the quotient $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$0, whose étale fundamental group is essentially the Weil group $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$1, by a relative curve over perfectoid spaces. For a perfectoid space $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$2 over $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$3, one constructs

$\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$4

with Frobenius $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$5, and defines

$\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$6

The divisor space

$\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$7

parametrizes degree-one Cartier divisors on $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$8, equivalently untilts over $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$9, and satisfies

$E$0

Thus the curve supplies both vector bundles, which encode representation-theoretic data, and degree-one divisors, whose local systems carry Weil-group actions (Fargues et al., 2021).

The Fargues–Fontaine classification identifies isocrystals with vector bundles. If $E$1 is an isocrystal, its associated bundle is obtained by descent: $E$2 For an algebraically closed complete field $E$3, every vector bundle decomposes as

$E$4

For a reductive group $E$5, a $E$6-bundle is equivalently an exact tensor functor

$E$7

and the moduli stack is

$E$8

The geometric points of $E$9 are classified by Kottwitz’s set $\mathbb F_q$0: $\mathbb F_q$1 Consequently,

$\mathbb F_q$2

The stratum indexed by $\mathbb F_q$3 is governed by the $\mathbb F_q$4-centralizer $\mathbb F_q$5, and for basic $\mathbb F_q$6 one has

$\mathbb F_q$7

The derived category of sheaves on this stratum recovers the derived category of smooth $\mathbb F_q$8-representations. In this way, $\mathbb F_q$9 simultaneously contains the representation theories of $p>0$0 and of its relevant inner forms and Levi forms.

2. Sheaves and representation theory

For a $p>0$1-algebra $p>0$2, Fargues and Scholze construct the derived $p>0$3-category

$p>0$4

of lisse $p>0$5-adic sheaves. For torsion coefficients this uses the étale theory of diamonds; for $p>0$6-coefficients it requires solid sheaves in condensed mathematics because $p>0$7 is a $p>0$8-stack with non-quasicompact stabilizers.

The category is assembled from the $p>0$9-strata. For each $G/E$0,

$G/E$1

where the right-hand side is the derived category of smooth representations. In particular, the geometrically trivial stratum yields a fully faithful embedding

$G/E$2

A smooth representation $G/E$3 of $G/E$4 is therefore regarded as a sheaf supported on

$G/E$5

followed by extension to $G/E$6.

The compact objects of $G/E$7 are characterized stratumwise: their restrictions to the categories $G/E$8 are compact and vanish for all but finitely many $G/E$9. Compact objects on each stratum are generated by compact inductions

$\ell\neq p$0

for open pro-$\ell\neq p$1 subgroups $\ell\neq p$2. Universally locally acyclic objects correspond stratumwise to admissible representations. Bernstein–Zelevinsky duality extends to compact objects, giving an involutive duality

$\ell\neq p$3

This is the representation-theoretic content of the geometrization: the entire collection of extended inner forms is encoded by one sheaf category, while the Harder–Narasimhan stratification records the decomposition into representation categories.

3. Hecke operators and geometric Satake

The global Hecke stack classifies tuples

$\ell\neq p$4

where $\ell\neq p$5 and $\ell\neq p$6 are $\ell\neq p$7-bundles, $\ell\neq p$8 is a degree-one divisor, and

$\ell\neq p$9

is a modification meromorphic along $\widehat G$0. It defines

$\widehat G$1

Bounded Hecke correspondences are indexed by dominant cocharacters $\widehat G$2 and have proper maps on bounded pieces. Their associated functors are of the form

$\widehat G$3

For a finite set $\widehat G$4, the Beilinson–Drinfeld Grassmannian and local Hecke stack are

$\widehat G$5

The Satake category consists of complexes that are relative perverse, flat over $\widehat G$6, universally locally acyclic over $\widehat G$7, and have quasicompact support. Convolution is defined by

$\widehat G$8

while fusion, obtained by allowing legs to collide, supplies a symmetric monoidal structure.

The fiber functor is geometric cohomology: $\widehat G$9 Tannakian reconstruction identifies the resulting dual group with $\ell$0, equipped with its Weil action, up to an explicit cyclotomic normalization. The Satake equivalence is

$\ell$1

For $\ell$2, the corresponding Satake sheaf $\ell$3 defines a Hecke functor

$\ell$4

The Weil action is geometric: it arises because the modification point varies over $\ell$5, whose fundamental group is $$W_E (<a href="/papers/2102.13459" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Fargues et al., 2021</a>).</p> <p>Geometric Eisenstein series provide a parallel mechanism for principal-series representations. For a quasi-split group with simply connected derived group, a normalized Eisenstein functor</p> <p>$$\ell$6

is constructed from $\ell$7 to $\ell$8. The factor $\ell$9 incorporates the half-modulus character and makes the construction compatible with Verdier duality and normalized parabolic induction. For sufficiently regular torus parameters, $L$0 is a Hecke eigensheaf with eigenvalue induced from

$L$1

Its restrictions to unramified strata are expressed through normalized parabolic inductions and Weyl conjugates of the corresponding character (Hamann, 2022).

4. Excursion operators and semisimple parameters

Hecke functors alone provide dual-group representations and Weil actions; their tensor structure produces excursion operators. Let $L$2 be a representation of $L$3, and let

$L$4

be $L$5-invariant maps. For $L$6, the associated operator on a Schur-irreducible object $L$7 is

$L$8

By Schur’s lemma, this operator is scalar. The collection of scalars defines a character of the excursion algebra, and Tannakian reconstruction produces a unique continuous semisimple parameter

$L$9

It is characterized by the identity

$\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$00

for every excursion datum.

For an irreducible smooth representation $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$01 of $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$02, viewed as a sheaf $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$03 on $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$04, one defines

$\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$05

This yields semisimple $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$06-parameters for all irreducible smooth representations without assuming the full local Langlands correspondence. The construction is compatible with local class field theory for tori, twisting, central characters, contragredients, products, Weil restriction, maps of reductive groups inducing adjoint isomorphisms, normalized parabolic induction, and the known correspondence for $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$07 and its inner forms (Fargues et al., 2021).

The associated spectral Bernstein center is

$\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$08

For suitable coefficients, Fargues and Scholze construct canonical maps

$\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$09

The composite sends a function on the parameter stack to a central operator on smooth representations, and its action on an irreducible $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$10 is evaluation at $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$11.

In equal characteristic, local-global compatibility with V. Lafforgue’s global correspondence identifies the Fargues–Scholze semisimple parameter of a local component with the semisimplified restriction of the global parameter. The same work identifies the Fargues–Scholze and Genestier–Lafforgue semisimple correspondences and supplies a canonical non-semisimple lift in positive characteristic (Li-Huerta, 2023).

5. The spectral action and the categorical conjecture

The excursion construction is part of a stronger action. Let

$\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$12

be the stack of $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$13-parameters. Under the condition

$\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$14

Fargues and Scholze construct an action

$\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$15

For a perfect complex $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$16 on $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$17, the action produces an exact endofunctor $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$18 on compact sheaves. The universal $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$19-torsor identifies representations of the $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$20-group with vector bundles on $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$21, and the action of these vector bundles recovers the Hecke operators: $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$22

The categorical local Langlands conjecture predicts, for quasisplit $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$23 with a chosen Whittaker datum, an equivalence

$\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$24

The spectral side consists of bounded coherent complexes with quasicompact support and nilpotent singular support. The Whittaker sheaf $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$25, corresponding to

$\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$26

is expected to correspond to the structure sheaf of $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$27. Equivalently, the spectral-action functor

$\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$28

should be given by

$\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$29

with a fully faithful right adjoint on compact objects.

The conjecture predicts that simple spectral objects consist of a Frobenius-semisimple parameter together with an irreducible representation of its centralizer or component group. This is intended to recover the internal structure of $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$30-packets. The full equivalence, the matching of $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$31-structures, and the complete description of singular support remain conjectural in the original framework (Fargues et al., 2021).

Subsequent work proves the categorical statement on the supercuspidal locus for odd unramified unitary and unitary similitude groups, constructs the corresponding Hecke eigensheaves, and establishes the strongest Kottwitz conjecture, including non-minuscule cocharacters (Meli et al., 2022). A generic categorical correspondence has also been constructed for generic Bernstein blocks of a large class of quasisplit groups, including quasisplit classical groups and $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$32, through fully faithful functors into ind-coherent sheaves on arithmetic parameter stacks (Helm et al., 8 Jun 2026).

6. Comparisons, applications, and extensions

The semisimple nature of the original construction makes comparison with classical local Langlands a central problem. For $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$33 and its nonsplit inner form $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$34, the Fargues–Scholze parameters agree after semisimplification with the Gan–Takeda and Gan–Tantono parameters for all irreducible smooth representations when $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$35 is unramified and $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$36 (Hamann, 2021). For odd unramified unitary and unitary similitude groups, they agree with the correspondences of Mok, KMSW, and Bertoloni Meli–Nguyen (Meli et al., 2022). Analogous compatibility has been proved for pure inner forms of unramified special orthogonal and unitary groups, with the Fargues–Scholze parameter also resolving the $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$37-conjugacy ambiguity in even orthogonal local Langlands (Peng, 6 Mar 2025).

For parahoric-spherical representations, the Fargues–Scholze parameter is characterized by formal properties and agrees at Frobenius with Haines’s parahoric Satake parameter: $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$38 The uniqueness argument reduces the correspondence to torus parameters, parabolic induction, central isogenies, products, Weil restriction, and trivial representations of anisotropic inner forms of $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$39 (Li, 2022).

The framework also controls local Shimura varieties and local shtukas. Their compactly supported cohomology carries commuting actions of inner-form groups and Weil groups, and is expressed through Hecke functors on $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$40. This yields finiteness theorems, Kottwitz-type formulas, and cohomological realizations of $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$41-parameters. The spectral action has applications to torsion vanishing for Shimura varieties: after localization at a generic Hecke eigensystem, perverse $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$42-exactness of Hecke operators and the geometry of Igusa varieties produce bounds such as

$\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$43

under the stated hypotheses (Hamann et al., 2023). For Hodge-type Shimura varieties, functorial Igusa stacks yield a sheaf on $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$44 whose spectral action gives compatibility with semisimple local Langlands, Eichler–Shimura relations, and further torsion-vanishing results (Daniels et al., 2024).

The correspondence is compatible with close fields in the semisimple setting. For sufficiently ramified $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$45-adic fields and the corresponding equal-characteristic local field, the mod-$\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$46 Fargues–Scholze correspondence is compatible with Deligne–Kazhdan equivalences on representations and parameters at bounded congruence level. With $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$47-coefficients, the comparison is established after restriction to suitable wild-inertia subgroups (Li-Huerta, 2024). In characteristic $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$48, local-global compatibility and global shtuka uniformization recover a canonical full parameter from the semisimplified one and identify the Fargues–Scholze and Genestier–Lafforgue constructions (Li-Huerta, 2023).

The ordinary stack $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$49 sees extended pure inner forms. To incorporate all rigid inner forms, including nontrivial inner forms of $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$50, an enlarged stack $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$51 is defined using the Kaletha gerbe over the Fargues–Fontaine curve. Its strata are indexed by an extended Kottwitz set $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$52 and contain the representation categories of the corresponding rigid inner forms. The enlarged dual group $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$53 and parameter stack $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$54 support an extended spectral action

$\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$55

The torus case is proved, and for connected center with $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$56 the extended categorical conjecture reduces to the original one (Dillery et al., 3 Aug 2026).

For elliptic semisimple parameters, the spectral action yields canonical projectors indexed by $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$57-conjugacy classes in the component group quotient $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$58. The resulting summands of the Fargues–Scholze packet character space are stable for the associated endoscopic data, extending the stable case to all semisimple classes (Kazhdan et al., 1 Mar 2025). More generally, conjectural constructions seek to recover Arthur packets from singularities of the parameter stack and to relate coherent or ind-coherent spectral objects to packet enhancements (Tam, 25 Aug 2026).

The established content of the Fargues–Scholze correspondence is therefore the geometric construction of $\boxed{\text{%%%%113%%%%-representations are sheaves on %%%%114%%%%, and Hecke eigendata are %%%%115%%%%-parameters.}}$59, geometric Satake, Hecke and excursion operators, the spectral action, semisimple parameters, spectral Bernstein-center maps, and numerous comparison and cohomological theorems. The principal unresolved statement remains the full categorical equivalence between sheaves on the automorphic stack and coherent or ind-coherent sheaves with the prescribed nilpotent singular support on the parameter stack.