UCLA. Number theory seminar. Local zeta functions for some Shimura. varieties with tame bad reduction. Thomas Haines University of Maryland

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1 UCLA Number theory seminar Local zeta functions for some Shimura varieties with tame bad reduction Thomas Haines University of Maryland April 22,

2 Local zeta functions Consider E/Q p, and a smooth d-dimensional variety X/E, with model over O E. Let p = prime ideal of O E. Definition We define Z p (s, X) to be 2d i=0 where det(1 Np s Φ p ; H i c (X E Q p, Q l ) Γ0 p) ( 1)i+1, l p, Φ p = geom. Frobenius, inertia subgroup = Γ 0 p Γ p := Gal( Q p /E) Good reduction implies: can forget ( ) Γ0 and pass to counting rational points over fields F q... 2

3 For varieties defined over number fields, take product of above over all finite places (and take something at infinite places...). We want to understand analytic properties, e.g. analytic continuation, functional equation, and special values. For latter two, must consider finite number of places with bad reduction. Definition above is likely correct : e.g., heuristic argument indicating ( ) Γ0 ensures functional equation...(later). Basic problem in Langlands program: for Shimura varieties, express Z p (s, X) in terms of local factors of automorphic L-functions. 3

4 Local L-functions π p = irred. admissible rep. of G(Q p ), having local Langlands parameter φ πp : W Qp SL 2 (C) L G := W Qp Ĝ. Let r = (r, V ) be an algebraic rep. of L G. Definition We define L p (s, π p, r) to be [ ] det(1 p s p 1/2 0 rφ πp (Φ 0 p 1/2 ); (kern) Γ0 ) 1, Φ W Qp a geom. Frobenius, N := rφ πp (1 V, [ ] 0 1 ), nilpotent operator on 0 0 Γ 0 -action on ker(n) V is via rφ πp restricted to Γ 0 id W Qp SL 2 (C). 4

5 Note π p spherical implies: (kern) Γ0 = V, and can express in terms of Satake parameter of π p : det(1 p s r(sat.(π p )); V ) 1, ( automorphic analogue of good reduction ). Tame bad reduction Take X over number field E. We say X has tame bad reduction at p provided that Γ 0 p acts unipotently on cohomology of X Ep Q p (action by functoriality). This will happen for Shimura varieties with Iwahori (more generally parahoric) level structure at p (later...). The automorphic analogue is: π f = v π v satisfies φ πp (Γ 0 p ) = id. (Thus, for any (r, V ) we have (kern) Γ0 = kern.) Conjecturally, this happens at least when π p has Iwahori fixed vectors. 5

6 Shimura varieties S K (C) = G(Q)\[G(A f ) X ]/K, a variety defined over reflex field E. S K = Sh(G, h, K), where h : C G R and X = the G(R)-conj. class of h. h µ, a minuscule cocharacter of GĒ. Iwahori level structure at p: K = K p K p, K p G(Q p ) an Iwahori subgroup. Consider PEL Shimura varieties S K (moduli spaces of abelian varieties with add. structure...). For p E dividing p, and E := E p, get a model over O E by posing suitable moduli problem over O E. We can do this in case of Iwahori level structure at p (but not deeper level structure). 6

7 ( ) Prototype: Y 0 (p), attached to Iwahori p GL 2 (Z p ). Moduli space: (E 1 E 2 ), degree p isogenies of elliptic curves. Special fiber is union of two smooth curves, intersecting transversally at the supersingular points. In general, the moduli space parametrizes chains (A 1, λ 1, ι 1, η) (A 2, λ 2, ι 2, η) of p-isogenies between polarized abelian varieties with additional structure (still makes sense over O E ; singularities in special fiber now much more complicated...) Goal: understand something about Z(s, S K ) in case of Iwahori level structure. First, we summarize Langlands strategy in case Y (N), where p N (good reduction). (Adding cusps gives X(N)...) 7

8 Langlands strategy, for Y (N), p N Here G = GL 2, and K p = GL 2 (Z p ). 1) Write Tr(Φ j ; H c (S K E Q p, Q l )) in form γ 0 γ,δ (vol)o γ (f p )TO δσ (φ j ) 2) Fundamental lemma: TO δσ (φ j ) = O Nδ (bφ j ) 3) use Arthur-Selberg trace formula γ 0 (vol)o γ0 (f) + = π m(π)tr π(f) +. Explanations: 1) (γ 0, γ, δ) G(Q) G(A p f ) G(Q p j), such that γ (resp. Nδ) is locally stably conjugate to γ 0. σ = can. gen. of Gal(Q p j/q p ), and TO δσ (φ) = G δσ \G(Q p j) φ(x 1 δσ(x)) dx dx δσ. 8

9 Use Lefschetz trace formula, and count points over F q using Honda-Tate theory: s.s. conjugacy classes γ 0 isogeny classes of elliptic curves. Then count points (E, η) where E ranges over a given isogeny class (η is a full level-n structure on E). φ j = char(gl 2 (Z p j) ( ) p 0 GL (Z p j)) H(GL 2 (Q p j)), 2) b : H(GL 2 (Q p j)) H(GL 2 (Q p )) is base change homomorphism for spherical Hecke algebras. Even with good reduction, get complications in higher dimensions from non-compactness and endoscopy. However, these problems don t arise for Kottwitz simple Shimura varieties (defined later...). Still, even for these varieties, new complications emerge when there is tame bad reduction. 9

10 Problems in bad reduction case (A) non-trival inertia action on H i (S K E Q p, Q l ). (B) Assume S K /O E proper. Then H i (S K E Q p, Q l ) = H i (S K OE F p, RΨ( Q l )), where RΨ( Q l ) D(S K F p ) is the sheaf of nearby cycles, a complex of sheaves whose complexity measures the singularities in the special fiber. To (temporarily) circumvent (A), we work with semi-simple trace: If V is an l-adic rep. of Γ := Gal( Q p /Q p ), finite Γ-stable filtration V k 1 V k V such that inertia Γ 0 acts through finite quotient on k gr k V. Then set Tr ss (Φ, V ) := k Tr(Φ, (gr k V ) Γ0 ). Semi-simple trace extends to give functionsheaf correspondence à la Grothendieck. In particular, there is a Leftschetz trace formula for it. 10

11 We can define Z ss (s, X) and L ss (s, π p, r) using Tr ss in place of Tr. To express Z ss in terms of various L ss (s, π p, r), for simple Shimura varieties, we can imitate Langlands. 1) Via LTF, write Tr ss (Φ j, H (S K F p, RΨ)) as x Fix(Φ j,s K ( F p )) Tr ss (Φ j, RΨ x ). Via Honda-Tate theory, write latter in form γ 0 γ,δ (±vol)o γ (f p )TO δσ (φ j ), where φ j is a suitable function in H(G(Q p j)//i j ). 2) Fundamental lemma: STO δσ (φ j ) = SO Nδ (bφ j ), where bφ j is base-change of φ j, 3) Use Arthur-Selberg trace formula (simple since G/A G anisotropic...) The main difficulties are 1) and 2) (the stabilization leading to 3) being just like Kottwitz s work in good reduction case). 11

12 Theorem 1 (H., B.C. Ngô) Let S K = Sh(G, µ, K) be a simple Shimura variety with Iwahori (more generally, parahoric) level structure at p. Let r µ : L G Aut(V µ ) be the irreducible representation of L G with highest weight µ. Let d = dim(s K ). Then Z ss p (s, S K) = π f L ss p (s d 2, π f, r µ ) a(π f)dim(π K f ), where π f ranges over irred. adm. reps. of G(A f ), the integer number a(π f ) is given by a(π f ) = π Π m(π f π )Tr π (f ), where m(π f π ) is the multiplicity in L 2 (G(Q)A G (R) 0 \G(A)), and where Π is the set of adm. reps. of G(R) having trivial central and infin. character. [Aside: f = ( 1) d pseudo. coeff. for some π 0 Π.] Taking K p = hyperspec. max. cpt., we recover Kottwitz s theorem. 12

13 Simple Shimura varieties E/E 0 /Q CM field, (D, ) central div. alg./e with positive invol. of second type, G the group defined by G(R) = {x D Q R xx R }. X := G(R) h: in case E 0 = Q, fix isom. D Q R = M n (C), and 1 r n 1; we can take h(z) = diag(z r, z n r ) and µ(z) = h C (z, 1) : C G C. G(R) =GU(r, n r) Assume: p inert in E 0, p = p p in E. K = K p I p, I p G(Q p ) standard Iwahori. (temp.) G Qp split ( =GL n G m ) (and identify µ = (1 r, 0 n r )). 13

14 Remarks Let E = E p. Then S K is proper over O E, smooth over E. For r = 1, used by Harris-Taylor in their proof of the local Langlands correspondence for GL n. For general 1 r n 1, studied by E. Mantovan and also L. Fargues (with arbitrary level structure at p, but less precise results...). [Aside: deeper level structure always occurs at some prime (if S K generically smooth), but it is interesting to note that after finite base change, expect tame reduction at such a prime ( potential semi-stable reduction ) in this sense Iwahori level structure is the most geometric kind.] No endoscopy problems (special feature of group G observed by Rapoport-Zink and Kottwitz...) 14

15 Key geometric step (part 1)) Identify test function φ j by identifying the function x Tr ss (Φ j, RΨ x )) on S K F p. We relate to nearby cycles on affine flag variety for G = GL n via Rapoport- Zink local models. Let k = F p j. Assume K p = I p = Iwahori. Theory of Rapoport-Zink: Étale locally, get S K k = M loc k. Can embed M loc k GL n ( k((t)))/i k,t := FL k, the affine flag variety for GL n. Any x S K ( k) gives rise to x 0 FL( k); not unique, but in uniquely determined I-orbit. 15

16 So Tr ss (Φ j, RΨ x ) = Tr ss (Φ j, RΨ M loc x 0 ). Latter is function in Iwahori-Hecke algebra for G. In fact we have the Kottwitz conjecture : Theorem 2 (H. and B.C. Ngô; D. Gaitsgory) Tr ss (Φ j, RΨ M loc ) = q d/2 z µ,j, where z µ,j Z(H(G(Q p j)//i j )) is the Bernstein function attached to the minuscule dominant cocharacter µ of G. Here z µj is unique central function such that z µ,j I Kj = char(k j µ(p j )K j ) H sph (G(Q p j)). In particular, we know how it acts on unramified principle series... We proved a more general theorem: for GL of GSp, there is a deformation of affine Grassmannian Gr Qp to FL Fp such that for S P K (Gr), nearby cycles RΨ(S) are central in P I (FL). Via sheaf-function dictionary, Tr ss (Φ, RΨ(S)) is central function in Iwahori-Hecke algebra. 16

17 More about local models There is a diagram with smooth surjective morphisms S K p S K q M loc. [Definitions: S K consists of points A together with a rigidification of their de Rham cohomology. M loc consists of chains of O E lattices in certain µ-admissible relative positions with respect to the standard lattice chain. The morphism q takes the image of the Hodge filtration under the rigidification...] Definition of M loc for GL n, µ = (1 r, 0 n r ): M loc (R) consists of diagrams Λ 0,R Λ 1,R Λ n 1,R p Λ 0,R F 0 F 1 F n 1 F 0 ; Λ i,r := Z p p 1 e 1,..., p 1 e i, e i+1,..., e n Zp R, and F i is an R-submodule, locally a direct summand of rank r. 17

18 [Aside (Warning): r, n not the same as before, and really need partial flags version...]. 18

19 Fundamental lemma (part 2)) Base change homomorphism for central elements in Iwahori-Hecke algebras: Assume G unramified over Q p ; let I j (resp. I) be the Iwahori subgroups over Q p j (resp. Q p ). Let K j (resp. K) denote corresponding hyperspec. max. cpt. s.g. s. Have Bernstein isomorphism B: I K : Z(H(G//I)) H sph (G//K) Definition Base change b for Z(H(G//I)) is unique map making commute: Z(H(G j //I j )) B H(G j //K j ) b Z(H(G//I)) B b H(G//K), where b on right is usual base change for spherical Hecke algebras (defined via Satake isomorphism). 19

20 Theorem 3 (H., Ngô) Let φ j be central in the Iwahori-Hecke algebra over Q p j. Let δ G(Q p j) be σ-semi-simple. Then STO δσ (φ j ) = SO Nδ (bφ j ). Also, SO γ (bφ j ) = 0 if γ not of form Nδ. Remarks In case of spherical Hecke algebras, proved by Clozel and Labesse. Also get parahoric version, which is much harder than Iwahori case. Strategy same as in Labesse. One neat new thing: naive constant term f f (P ) actually preserves central elements (!), and so can use in descent step... 20

21 Putting steps 1)-3) together, we proved Theorem 4 (H.,Ngô) For S K simple as above, Tr ss (Φ j, H (S K E Q p, Q l )) is equal to Tr(b(z µ,j ) I K p f ; L 2 (G(Q)A G (R) 0 \G(A))). This implies Theorem 1, since we know how b(z µ,j ) acts on π I p p : by the scalar [ ] p 1/2 0 Tr(r µ φ πp (Φ 0 p 1/2 ); V µ ). 21

22 Towards true local factors Rapoport: to recover Z from Z ss, enough to know monodromy-weight conjecture: The graded grk M of monodromy filtration on Hi (S K F p, RΨ) (defined using inertia action) is pure of weight i + k. Nothing known in higher dimensions for situation at hand. However, I can prove cases of local weight-monodromy conjecture for the perverse sheaf RΨ. Automorphic analogue: can recover L(s, π p, r) from L ss (s, π p, r) provided π p is tempered, i.e. local parameter φ πp : W Qp SL 2 L G satisfies: φ πp (W Qp ) is bounded. We expect this for the π f which come into H (S K ). 22

23 Note: The tempered-ness of the parameter φ πp is sufficient for L ss (s, π, r) to determine L(s, π, r). However, as pointed out by Don Blasius, it is not necessary. Moreover, we actually should not expect all π p which appear in the cohomology to be tempered. [In closing, if time: explain 1)heuristic that ( ) Γ0 equation necessary for functional 2) action of Γ 0 on cohomology of Shimura variety with Iwahori level structure is unipotent (from Gaitsgory s result that action on nearby cycles is unipotent)...] THE END. 23