UCLA

Number theory seminar

Local zeta functions for some Shimura

varieties with tame bad reduction

Thomas Haines University of Maryland

April 22, 2003

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 det (1 − N𝔭^{−s}Φ_{𝔭}; H_{c}^{i}(X ×_{E} ℚ̄_{p}, ℚ̄_{ℓ})^{Γ_{𝔭}^{0}} )^{(−1)^{i+1}} ,

where

• ℓ ≠ p,

• Φ_{𝔭} = geom. Frobenius ,

• inertia subgroup = Γ_{𝔭}^{0} ⊂ Γ_{𝔭} := Gal(ℚ̄_{p}/E)

Good reduction implies: can forget (·)^{Γ^{0}} and pass to counting rational points over fields 𝔽_{q}...

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 (·)^{Γ^{0}} 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.

Local L-functions

π_{p} = irred. admissible rep. of G(ℚ_{p}) , having local Langlands parameter ϕ_{π_{p}} : W_{ℚ_{p}} × SL_{2}(ℂ) → LG := W_{ℚ_{p}} ⋉ 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}rϕ_{π_{p}}(Φ×[\begin{matrix}p^{−1/2} & 0 \\ 0 & p^{1/2} \end{matrix} ]); (ker N)^{Γ^{0}} )^{−1},

• Φ ∈ W_{ℚ_{p}} a geom. Frobenius,

• N := rϕ_{π_{p}}(1 × [\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}]), nilpotent operator on V,

• Γ^{0}-action on ker (N) ⊂ V is via rϕ_{π_{p}} restricted to Γ^{0} × id ⊂ W_{ℚ_{p}} × SL_{2}(ℂ).

Note π_{p} spherical implies: (ker N)^{Γ^{0}} = 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_{𝔭}^{0} acts unipotently on cohomology of X ×_{𝔼_{𝔭}} ℚ̄_{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_{π_{p}}(Γ_{p}^{0}) = id. (Thus, for any (r, V) we have (ker N)^{Γ^{0}} = ker N.) Conjecturally, this happens at least when π_{p} has Iwahori fixed vectors.

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).

Prototype: Y_{0}(p), attached to Iwahori (\begin{matrix}∗ & ∗ \\ p∗ & ∗ \end{matrix}) ⊂

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)...)

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} ℚ̄_{p}, ℚ̄_{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_{γ_{0}}(f) + ⋯ = _{π}^{∑} m(π)Tr π(f) + · · · .

Explanations: 1) (γ_{0}, γ, δ) ∈ G(ℚ) × G(𝔸_{f}^{p}) × 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_{pj})} ϕ(x^{−1}δσ(x))\frac{dx}{dx_{δσ}}.

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}}) (\begin{matrix}p & 0 \\ 0 & 1 \end{matrix}) GL_{2}(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.

Problems in bad reduction case

(A) non-trival inertia action on H^{i}(S_{K}×_{E}ℚ̄_{p}, ℚ̄_{ℓ}). (B) Assume S_{K}/O_{E} proper. Then H^{i}(S_{K} ×_{E} ℚ̄_{p}, ℚ̄_{ℓ}) = H^{i}(S_{K}×_{O_{E}}𝔽̄_{p}, RΨ(ℚ̄_{ℓ})), where RΨ(ℚ̄_{ℓ}) ∈ D(S_{K}×𝔽̄_{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(ℚ̄_{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_{•})^{Γ^{0}} ).

Semi-simple trace extends to give functionsheaf correspondence à la Grothendieck. In particular, there is a Leftschetz trace formula for it.

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} × 𝔽̄_{p}, RΨ)) as

_{x∈Fix(Φ^{j},S_{K}(𝔽̄_{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).

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_{𝔭}^{ss}(s, S_{K}) = _{π_{f}}^{∏} L_{p}^{ss}(s − \frac{d}{2}, π_{f}, r_{μ})^{a(π_{f})dim (π_{f}^{K})} ,

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.

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 = 𝔭𝔭̄ in 𝔼.

• K = K^{p}I_{p}, I_{p} ⊂ G(ℚ_{p}) standard Iwahori.

• (temp.) G_{ℚ_{p}} split ( ˜= GL_{n}×𝔾_{m}) (and identify μ = (1^{r}, 0^{n−r})).

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...)

Key geometric step (part 1))

Identify test function ϕ_{j} by identifying the function

x ↦ Tr^{ss}(Φ^{j}, RΨ_{x}))

on S_{K} × 𝔽̄_{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} × k̄ ≅ M^{loc} × k̄. Can embed

M^{loc} × k̄ ↪ GL_{n}(k̄((t)))/I_{k̄,t} := FL × k̄,

the affine flag variety for GL_{n}.

Any x ∈ S_{K}(k̄) gives rise to x_{0} ∈ FL(k̄); not unique, but in uniquely determined I-orbit.

So Tr^{ss}(Φ^{j}, RΨ_{x}) = Tr^{ss}(Φ^{j}, RΨ_{x_{0}}^{M^{loc}}). 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_{μ_{j}} is unique central function such that z_{μ,j}∗𝕀_{K_{j}} = 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_{ℚ_{p}} to FL_{𝔽_{p}} 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.

More about local models

There is a diagram with smooth surjective morphisms

\xymatrix{S_{K} & S̃_{K} \ar[l]_{p} \ar[r]^{q} & M^{loc}.}

[Definitions: S̃_{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

\xymatrix{Λ_{0,R} \ar[r] & Λ_{1,R} ⋯ \ar[r] \ar[r] & Λ_{n−1,R} \ar[r]^{p} & Λ_{0,R} \\ F_{0} \ar[r] \ar[u] & F_{1} ⋯ \ar[u] \ar[r] & F_{n−1} \ar[r] \ar[u] & F_{0}; \ar[u]}

Λ_{i,R} := ℤ_{p}⟨p^{−1}e_{1}, …, p^{−1}e_{i}, e_{i+1} , …, e_{n}⟩ ⊗_{ℤ_{p}} R, and F_{i} is an R-submodule, locally a direct summand of rank r.

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

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:

\xymatrix{− ∗ 𝕀_{K} : Z(H(G//I)) \ar[r] & H_{sph}(G//K)}

Definition Base change b for Z(H(G//I)) is unique map making commute:

\xymatrix{Z(H(G_{j}//I_{j})) \ar[r]|{B} \ar[d]_{b} & H(G_{j}//K_{j}) \ar[d]^{b} \\ Z(H(G//I)) \ar[r]|{B} & H(G//K),}

where b on right is usual base change for spherical Hecke algebras (defined via Satake isomorphism).

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...

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} ℚ̄_{p}, ℚ̄_{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_{p}^{I_{p}}: by the scalar

Tr (r_{μ}ϕ_{π_{p}}(Φ × [\begin{matrix}p^{−1/2} & 0 \\ 0 & p^{1/2} \end{matrix} ]); V_{μ}).

Towards true local factors

Rapoport: to recover Z from Z^{ss}, enough to know monodromy-weight conjecture: The graded gr_{k}^{M} of monodromy filtration on H^{i}(S_{K}×𝔽̄_{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_{π_{p}} : W_{ℚ_{p}} × SL_{2} → ^{L}G

satisfies: ϕ_{π_{p}}(W_{ℚ_{p}}) is bounded.

We expect this for the π_{f} which come into H^{•}(S_{K}).^{∗} ∗Note: The tempered-ness of the parameter ϕ_{π_{p}} 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 (·)^{Γ^{0}} necessary for functional equation

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.