The curve and the Langlands program Laurent Fargues (CNRS/IMJ) t'etlp.at µ Îà t.at KozoI 2in spoliais Ü c Gallotta
The curve Defined and studied in our joint work with Fontaine Two aspects : compact p-adic Riemann surface / algebraic curve Starting datum : F F p perfectoid field p=variable, Y = {0 < p < 1} open punctured disk, adic space W (O F )[ 1 p, 1 [ϖ] ] = { n [x n ]p n x n F, sup n } x n < + O(Y ) completion of the preceding. Fontaine s ring. No explicit formula for elements of O(Y )
The curve Y ϕ Frobenius X = Y /ϕ Z compact adic space /Q p O(1) line bundle /X automorphic factor j(ϕ, p) = p Schematical curve X = Proj ( d 0 O(Y ) ϕ=pd ) GAGA type morphism X X
The curve Theorem (F.-Fontaine) The curve is a curve : X is a Dedekind scheme, X noetherian regular dimension 1 adic space Perfectoid residue fields : x X, k(x) Q p perfectoid, [k(x) : F ] < + Ô X,x = B + dr (k(x)) X is complete : deg(x) := [k(x) : F ] f Q p (X), deg(divf ) = 0
The curve X \ {x} = Spec(B x ), B x Dedekind, P.I.D. if F alg. closed. (B x, ord x ) not euclidean i.e. H 1 (O( 1)) 0 F alg. closed. X = untilts of F up to Frob.
Vector bundles on the curve GAGA verified : Bun X BunX X complete good Harder-Narasimhan reduction theory µ = deg rk λ Q O(λ) stable slope λ vector bundle, λ = d h, pushforward of O(d) via cyclic cover Y /ϕ hz Y /ϕ Z Theorem (F.-Fontaine) F alg. closed {λ 1 λ n λ i Q} Bun X / (λ 1,..., λ n ) [ i O(λ i )].
Vector bundles on the curve : applications Quick simple proofs of the two fundamentals theorems of p-adic Hodge theory : weakly admissible admissible (Colmez-Fontaine) de Rham pot. semi-stable (Berger) F = C p, X Gal(Q p Q p) X fixed. Study Galois equivariant vector bundles and their modifications at Look at modifications of vector bundles ϕ-modules over A inf Scholze-Weinstein
G-bundles G reductive group /Q p, Q p := Q un p, σ = Frob, B(G) = G( Q p )/σ conj, b gbg σ b G( Q p ) E b = Y ϕ G G-bundle /X, ϕ acts on G via conjugation by bσ Theorem (F) F alg. closed B(G) Hét 1 (X, G) [b] [E b ] Dictionary : reduction theory (Atiyah-Bott) for G-bundles / Kottwitz description of B(G). Example : E b semi-stable b is basic (isoclinic)
G-bundles : applications (Chen, F, Shen) : Proof of F.-Rapoport conjecture on period spaces (p-adic analogs of Griffith s period spaces). Example : computation of p-adic period spaces for SO(2, n 2) Construction of local Shimura varieties (Scholze) Sht(G, b, µ) as moduli of modifications E 1 E b Newton stratification of Hodge-Tate flag manifold (Caraiani-Scholze), modification E 1,x, x flag manifold, E 1,x E b for some b stratification by the set of such [b]
The stack Bun G of G-bundles/curve Introduced to formulate a geometrization conjecture of the local Langlands correspondence ϕ }{{} F ϕ }{{} Langlands parameter perverse sheaf on Bun G S = F p -perfectoid space X S adic space/q p = family of curves (X k(s) ) s S Bun G = v-stack on Perf Fp of G-bundles/curve B(G) Bun G Bun G = Bun c1=α G α π 1 (G) Γ
The stack Bun G Bun c 1=0,ss G = [ /G(Q p ) ] open Bun G Aut(trivial G-bundle) = G(Q p ) topological group classical case G algebraic group More generally α π 1 (G) Γ Bun c 1=α,ss G = [ /J b (Q p ) ] b basic, J b inner form of G In general each H.N. stratum is a classifying stack [ /Jb ] J b = Jb 0 J b(q p ), Jb 0 =unipotent diamond
The geometrization conjecture ϕ = discrete Langlands parameter Conjecture ϕ : W Qp L G ϕ F ϕ S ϕ -equivariant perverse Hecke eigensheaf on Bun G s.t. the stalks of F ϕ at semi-stable points gives local Langlands + internal structure of L-packets for all extended pure inner forms of G
The geometrization conjecture For this : Need to give a meaning to perverse sheaf on Bun G Need to give a meaning to the Hecke eigensheaf property : establish geometric Satake in this context joint work with Scholze : give a precise statement of the conjecture + construction of the local Langlands correspondence π ϕ π à la V. Lafforgue using Bun G
Constructible and perverse étale sheaves on Bun G Joint with Scholze. Λ {F l, Q l } étale coefficients, l p Bun G is a (cohomologically) smooth stack of dimension 0 Good notion of constructible sheaf on Bun G = reflexive sheaves w.r.t. Verdier duality Fiberwise criterion of constructibility in terms of representation theory : b G( Q p ), the stalk at the point given by b is an admissible representation of J b (Q p )
Geometric Satake Div 1 = Spa(Q p ) /ϕ Z sheaf of deg. 1 effective div. on the curve Gr B dr G Spa(Q p ) Scholze s B dr affine grassmanian, Gr B dr G /ϕz Div 1 Beilinson-Drinfeld type affine Grassmanian Theorem (F.-Scholze) Geometric Satake holds for Gr B dr G, Satake category Rep( L G, Λ).
Construction of Langlands parameters Factorization enhancement : I finite set Gr B dr G,I (Spa(Q p ) ) I + factorization property when I varies. Hecke I Bun G Bun G (Spa(Q p ) ) I W Rep( L G I, Λ) IC W kernel on Hecke I via geo Satake
Construction of Langlands parameters Coupled with V. Lafforgue strategy (global function field) we construct local Langlands }{{} π irred. rep. of G(Qp) ϕ π }{{} semi-simple Langlands parameter Very little known about this correspondence : surjectivity, finiteness of fibers? Geometrization conjecture goes in the other direction Langlands parameter representation and would give this + internal structure of L-packets
Back to the geometrization conjecture The GL 1 -case. Classically : Abel-Jacobi morphism locally trivial fibration in simply connected alg. var. (projective spaces) in high degree Reduced to the following theorem Theorem (F) For d 3, the Abel-Jacobi morphism AJ d : Div d Pic d is a pro-étale locally trivial fibration in simply connected diamonds. Here Div d = Hilbert diamond = (Div 1 ) d /S d with Div 1 = Spa(Q p ) /ϕ Z.
That s only the beginning of the story!