Motivic Poisson summation - IMJ-PRGzoe/GTM/HrushovskiJune08.pdf · (iii) Avoiding splitting. Let f...

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Motivic Poisson summation Ehud Hrushovski, David Kazhdan 1. Local motivic integration. 2. Deligne-Kazhdan-Vigneras; statement of problem. 3. Adeles. The quotients T (O)\T (A)/T (K ). 4. Adelic volumes (integrals of global test functions.) 5. The sum-over-rational-points functional δ K . Poisson summation formula: δ K F = δ K . 6. Motivic covolume and rational points on orbits. 7. Form-independence of Fourier transform. 1

Transcript of Motivic Poisson summation - IMJ-PRGzoe/GTM/HrushovskiJune08.pdf · (iii) Avoiding splitting. Let f...

Page 1: Motivic Poisson summation - IMJ-PRGzoe/GTM/HrushovskiJune08.pdf · (iii) Avoiding splitting. Let f be a eld, l = f a cyclic extension of order p. We will be interested in certain

Motivic Poisson summationEhud Hrushovski, David Kazhdan

1. Local motivic integration.

2. Deligne-Kazhdan-Vigneras; statement ofproblem.

3. Adeles. The quotients T (O)\T (A)/T (K).

4. Adelic volumes (integrals of global testfunctions.)

5. The sum-over-rational-points functionalδK.

Poisson summation formula: δKF = δK.

6. Motivic covolume and rational points onorbits.

7. Form-independence of Fourier transform.

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0. Grothendieck rings.

Let T be a theory in a language L. By a con-

structible set X we mean here a quantifier-free

formula φ(x) of L, viewed as a set X(M) =

φ(M) in a model M of T . If we wish to speak

of formulas with parameters from A ≤M |= T ,

we refer to TA. We assume substructures are

closed under constructible bijections. [X] is

the class of X, up to constructible bijections.

K+(T ) = {[X] : X constructible };

[X] + [Y ] = [X.∪Y ], [X · Y ] = [X × Y ]

K(T ) is the ring formed by adding formal ad-

ditive inverses.

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We will also add multiplicative inverses for cer-

tain X such that X(A) 6= ∅ for all A. We think

of an element of K(T ) as a generalized num-

ber.

If A ≤M |= T , A finite, we obtain a homomor-

phism

K(T )→ Q, [X] 7→ X(A)

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Summation.

Let X be a constructible set. If a ∈X(M),M |= T , let Ta = Th(M,a). We

have an X-parameterized family of rings

K(Ta).

One can similarly define a Grothendieck ring of

constructible sets Y over a given constructible

set X; we denote it Fn(X,K) and view it as a

ring of sections

a 7→ [Y (a)] ∈ K(Ta)

We have an additive map Fn(X,K)→ K(T ),

Y 7→∑x∈X

Y (x) := [Y ]

This should be viewed as part of the structure

of the Grothendieck ring.

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Quotients.

Let T be a universal theory extending T∀.

K(T/T ) := K(T )/({[X] : T |= (¬∃x)(x ∈ X)})

Example

ACFF=theory of algebraically closed fields

with an F -algebra structure. If char(F ) = 0,

K(ACFF ) = K(V arF ).

For the main theorem we will permit multi-

plicative inverses for the classes of all abelian

algebraic groups; details below.)

Let D be a quantifier-free piecewise-

constructible (=strict Ind-constructible)

k-algebra. For each field F we have D(F ).

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The set of F such that D(F ) has no 0-

divisors is closed under ultraproducts and

substructures, so it is the set of models of a

universal theory Div(D). We will work with

the quotient K(ACFF , /Div(D)). A posteriori

our main theorem will be valid in K(ACF )Q.

(Explained below.)

K is the result of these two operations, plus

the provision for an additive character ψ:

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(i) Motivic exponential sums∑φ(x)ψ(x).

Needed for the definition of the Fourier trans-

form.

Let K = ACF , Let H be a constructible group,

K0H = Fn(H,K), Then KH admits a convolu-

tion: f ∗ g(a) =∑b∈H f(a)g(a−1b). The char-

acteristic function χ1 is the identity element.

Let KH = KH[χc−1] where c 6= 1 is some con-

structible element. For φ ∈ Fn(H,K), define∑x∈H φ(x)ψ(x) to be the image of φ in KH.

Note: ∑x∈H

ψ(x) = 0

We will use the case H = Ga

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(ii): Localizing by group classes

In part (6) of this talk we will need to localize

by the multiplicative subset of all commutative

group varieties. Instead of constructing this lo-

calization we note and impose a consequence:

Let (Ay : y ∈ Y ) be a definable family of com-

mutative algebraic groups, (Xy), (X ′y) two fam-

ilies of definable sets, and assume:

[Ay][Xy] = [Xy]2, [Ay][X ′y] = [X ′y]2

[Xy]2 = [Xy][X ′y] = [X ′y]2

for y ∈ Y . Then∑y∈Y

[Xy] =∑y∈Y

[X ′y]

.

Let K(ACFF )g be the quotient of K(ACFF )

obtained by imposing the above relation, as

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well as the relations: a = b whenever [a][A] =

[b][A].

The relation [A][X] = [X]2 is typical of principal homo-

geneous spaces Xy.

Proof: Let ey =[Xy][Ay] , e

′y =

[X ′y][Ay]. Then ey =

e2y = eye′y = (e′y)2 = e′y. So∑y∈Y

[Xy] =∑y∈Y

ey[Ay] =∑y∈Y

e′y[Ay] =∑y∈Y

[X ′y]

Page 10: Motivic Poisson summation - IMJ-PRGzoe/GTM/HrushovskiJune08.pdf · (iii) Avoiding splitting. Let f be a eld, l = f a cyclic extension of order p. We will be interested in certain

(iii) Avoiding splitting.

Let f be a field, l = f a cyclic extension of

order p. We will be interested in certain infi-

nite dimensional division algebras D over f; our

theorem becomes trivial upon base change to

l.

Let Y be a finite variety, such that l = f(Y ).

We will work in K′ = K(ACFf)/[Y ].

Note that [Y ]2 = p[Y ], and so e = [Y ]/p is

idempotent in KQ. Since our theorem is true

(trivially) in K/(1 − e), if we prove an identity

in K′ it will be true in KQ.

Assume V ≤ V ′ are varieties, such that when-

ever V ′ \ V has a point in a field F , we have

l ≤ F . Then [V ] = [V ′] ∈ K′. It will thus suffice

to consider fields F such that DF is a division

ring. This will allow us to consider certain∨

-

definable sets as definable.8

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1. Local Integration (of very smooth func-

tions)

L((s)) = {∑

aisi : ai ∈ L, an = 0 for n << 0}

VN,M = s−NL[[s]]/sML[[s]], a finite-dimensional

L-space (with basis.)

A local test function = φ ∈ Fn(Vn,m,K);

viewed as a function on L((s)), supported

on s−nL[[s]], and locally constant modulo

sML[[s]]. Similarly for several variables.

Integration.∫φ = [k]−M

∑x∈VN,M

φ(x)

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Convolution:

(φ1 ∗ φ2)(x) =∫φ1(u)φ2(u−1v)

Fourier transform: fix a linear r : D → k van-

ishing on s−ML[[s]] for some M ; with O⊥ =

{x : (∀y ∈ O)r(xy) = 0} = s2νO.

F(φ)(x) = [k]−ν∫yφ(y)ψ(r(xy))

If φ is defined modulo sML[[s]], then F(φ) is

supported on s−ML[[s]].

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The smooth integration above applies equally

when L[[s]] is non-commutative.

Let L = (L, σ) be a difference field. Form L[s]

with sa = σ(a)s.

Example: (Manin’s quantum plane) L =

k[u], σ(u) = qu. Obtain k[u, s], su = qus

D0 = L[[s]], L((s)); a division ring. PD0 :=

D∗0/Z∗. O = L[[s]].

Center = Z = F ((t)) where F = Fix(σ), t =

sn. For each irreducible P [X] ∈ F [X] in one

variable, {a ∈ D : F (a) = 0} is either empty or

a conjugacy class of D∗.

Example: Given a non-commutative polyno-

mial g(X1, . . . , Xk) over L, obtain a power se-

ries Pg(t) =∑

[Wn]tn ∈ K(Diff.-Var)[[t]]. Wn =

[{x ∈ L[[s]]/sn : g(x) = 0 mod sn}].11

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We will work with σn = 1 on L; and in this talk

will assume n is prime.

Then L((s)) is an algebra of dimension n2 over

the center kn((sn)). Convolution of test func-

tions, and Fourier transform can be under-

stood via n2-dimensional, commutative mo-

tivic integration. The non-commutative view-

point will be used at one point in the proof (to

show that G(O)G(F ) = G(A) for G = PsD∗ via

the Euclidean algorithm in D.), but in general

we will take the n2-dimensional, commutative

view.

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2. Deligne-Kazhdan-Vigneras.

Let [L : F ] be a cyclic Galois extension, [L :

F ] = n, and let σ, σ′ be two choices of a gen-

erator. Form D,D′ = L((s)) as above. They

have center Z = F ((t)). Let Y be the set of ir-

reducible central polynomials over Z, of degree

1 or n. For y ∈ Y let Cy = {d ∈ D : y(d) = 0},similarly C′y.

Can form a character table Y ×IrrRepPD → C,

namely trρ(c) where f(c) = 0.) Similarly for

D′; we have a bijection between the columns

(conjugacy classes.)

Theorem. [DKV] p a prime power, L = Fpn.

There exists a bijection Rep(PD) → Rep(PD′)respecting the character table.

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Equivalent formulation: The identification of

conjugacy classes induces an isomorphism of

convolution algebras. I.e. let f1, f2, f3 ∈ Y ,

fix m and c with f3(c) = 0. Let χi be the

characteristic function of x : valfi(x) ≥ n}. Let

Cm(f1, f2, f3) = χ1 ∗χ2(c) = vol({x : valf1(x) ≥n, valf2(c−1x) ≥ m}). Then C(f1, f2, f3,m) is

the same for D,D′.

Nearly equivalent: table for Fourier transform,

F(χ1)(c) does not depend on choice of σ.

[DKV] obtain this by comparing both division

rings to GLn; we will not consider this here.

Analogs for other groups are known; cf. Wald-

spurger.

No local proof is known for any such result.

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Theorem 1. Let n be prime, L any field, σ, σ′ ∈Aut(L) with σn = 1. Let f = Fix(σ) = Fix(σ′).

The Fourier transform table with values in K

is the same for σ, σ′.

The proof is commutative, global and motivic.

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3a. Tloc, theory of valued fields valued fields

with a section i of res

with sort K,k,Γ for the valued field, value

group, residue field.

(K,+, ·; Γ,+, <,0,1; val : V F ∗ → Γ; res(x

y); i : res→ V

Tloc: K is an algebraically closed field, val is a

valuation, with valuation ring O and maximal

ideal M, and M⊕ k = O.

Delon - Leloup.

Tloc admits quantifier elimination. k,Γ are em-

bedded, stably embedded and strongly orthog-

onal. Note (K, i(k),+, ·) has no QE.

Example: a small neighborhood of k in K.

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3b. Theory of valued fields over a curve

We describe here a first-order theory T = T gl

convenient as the background for adelic work.

It has the following sorts.

k - an algebraically closed field with a distin-

guished field of constants F . k is endowed with

the language of F -algebras.

C(k), where C is a smooth, complete curve

over F .

Γ - an ordered Abelian group, with distin-

guished element 1 > 0.

V F . This sort comes with a map V F → C(k);

the fibers are denoted Kx. Each Kx comes with

valuation ring Ox, a surjective homomorphism

resx : Ox → k, and a ring embedding ix : k →Ox, such that resx ◦ ix = Idk. Also, a map

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vx : Kx \ {0} → Γ, denoting a valuation with

valuation ring Ox.

We identify k with its image ix(k). .

As a final element of structure, we have a func-

tion c : C(k) → V F , such that c(x) ∈ C(Kx);

and for any f ∈ k(C), valf(c(x)) = ordx(f) · 1.

So for any limited subset S of k(C), an image

of S in Kx is definable.

T admits quantifier-elimination. k and Γ are

embedded and stably embedded.

All TC are interpreted in TP1. We will work

in TP1, but will need other TC when analyzing

some structures definable there.

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Over TC, we form:

K = k(C) as an piecewise definable field

We view K = (K,+, ·,k, res(dx)) as Ind-

definable in T. For C = P1, the pieces are the

rational functions of degree ≤ d.

Consider the integral k-adeles: O =∏x∈C(k) Ox.

Adeles A, a subring of∏x∈C(k)Kx.

We have a diagonal embedding K → A.

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3. T (O)\T (A)/T (K)

Let X be a definable set. By a definable func-

tion f : X → V (∏c∈CKc) we mean a definable

function f on X×C, such that f(x, c) ∈ V (Kc).

f : X → V (A) means: For all x, for all but

finitely many c, f(x, c) ∈ V (Oc).

Let E be an Ind-definable-in-definable families-

equivalence relation on V (A). a representative

constructible set for V/E to be a definable set

Y , such that for some definable X and f : X →V (A) and surjective g : X → Y , every element

of V (A) is E-equivalent to some element g(x),

and g(x) = g(x′) iff f(x)Ef(x′).

If Y, Y ′ are representative constructible sets for

V (A)/E, there exists a constructible bijection

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Y → Y ′: g(x) 7→ g′(x′). Define [V (A)/E] = [Y ].

Let T be a torus over F (C), and assume given

T (O). For each v ∈ C(k) we have hv : T →X∗(T )⊗Γ. Let h =

∑v hv, and T (A)0 = ker(h).

We assume T (Ov) = ker(hv) for almost all v.

Lemma 2. There exists a representative set for

T (O)\T (A)/T (K). It admits a definable map

to Γ; the kernel, T (O)\T (A)0/T (K), is an al-

gebraic group.

The classes [T (O)\T (A)0/T (K) will play an es-

sential role.

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Note: the Adeles are not pro-definable or *-

definable in the usual sense.

The construction we need is not Pro(Def),

but (Def-Pro)(Def). For instance consider

J = Gm(O)\Gm(A)0/Gm(K). If we take

the Ind/Pro interpretation, every bijection

becomes an isomorphism.

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4a. Semi-local volumes.

Semi-local test functions: Consider∏v∈S Av.

Let S ⊂ C(k) be finite, definable. Let tv be a

parameter for Kv. It is natural to define semi-

local test functions as definable functions,∏v tMv Ov-invariant for some M , and supported

on∏v t−Mv Ov. Iterated integration.

In particular, for a product X =∏v∈SXv, Xv ⊆

Omv .

vol(X) =∏v∈S

vol(Xv)

Objections:

i) vol(Xv) belong to different rings!

cf. Weil reduction of scalars.21

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ii) Assume S is a Galois orbit. Classically, one

takes only one copy of Av. Then expect:

vol(X.∪Y ) = vol(X) + vol(Y ).

cf. Frobenius.

Let ρ = ρF : K → K(ACFf, Th∀(f)) ,

[X] 7→ X(f). (If f is a finite field, this is the

“counting rational points” map.)

ρ∏v∈S

(av + bv) = ρ∏v∈S

av + ρ∏v∈S

bv

(A separable descent analogue of (a+b)p = ap+bp?)

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4b. By a global test function φ we mean a

semi-local test function φS at a finite definable

subset S ⊆ C, extended by 1Ov to all v /∈ S. I.e.

φ(a) = φS(a) if av ∈ Ov for v /∈ S, otherwise

φ(a) = 0.

So for test functions, global integration re-

duces to the semi-local case.

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5. Rational points and Poisson summation.

Let φ be a global test function.

The set of points of K = k(C) in the supportof φ is a limited set, i.e. contained in one ofthe definable approximations Kn to K.

Define δK(φ) =∑a∈Kn φ(a).

• δK(ψ(r(ax))φ(x)) = δK(φ) for a ∈ K.

• δK(φ(ax) = δK(φ) for a ∈ K.

• δK(1O) = [k]

Poisson summation formula:

(1) δKF = δK

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6. Rational points on orbits and motivic co-

volume.

Define integers Rv of Dv, uniformly in v ∈ C.

For almost all places Rv is a maximal sta-

bly dominated subring of Dv. At the ramified

places 0,∞ we take R0 = D0, nothing that

modulo the center D∗0 is Div(D)-equivalent to

a stably dominated group.

R =∏vRv. When C = P1 we have:

PD(R)PD(K) = PD(A).

For T ≤ D∗, set T (O) = T ∩R∗v.

Let c ∈ D(F ). Let O be the orbit of c under

R∗-conjugation, interpreted geometrically, and

let T = CD∗(c). Then

(2) δK(1O) = n[T (O)\T (A)/T (K)][L∗/Gm]

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Here L is is viewed as a definable ring, with

group of units L∗; so [L∗/Gm] is directly the

class of a finite-dimensional variety. This is the

generic case; when O meets the centralizer of

an element normalizing L, this multiplier needs

to be modified slightly.

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Proof of Theorem 1.

We work over P1; we define Rv for v 6= 0,1

in such a way that in ACV FFv, (Dv, Rv) is a

form of (Mn,Mn(O); they are isomorphic with

parameters.

Given any c ∈ D(F ), we show existence of

c′ ∈ D′(F ) such that (Dv, Rv, c) and (D′v, Rv, c′)

are isomorphic with parameters. The family of

isomorphisms is a torsor for T (Rv), which acts

trivially on T . They induce a unique, hence de-

finable, isomorphism T (c) → T (c′), preserving

T (O).

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Let φ be a local test function at 0. Let O be

the conjugacy class of c. Let Nv be a stan-

dard neighborhood of c at v, with N1 small

compared to others. N =∏v 6=0Nv.

“ K is discrete in A ”

(3) φ(c) =δK(φ_1N)

δK(1N1O)

Writing this for F(φ) and applying Poisson

summation we obtain:

Fφ(c) =δK(φ_F(1N))

δK(1NF−11O)

But by (2) and the isomorphism T → T ′, the

right hand side is independent of the form.

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Problems.

1) (Noncommutative integration.)

Recall ξn : K(DiffvarF ) → K(V arF ), [X] 7→[X(L⊗Fk)].

a) Assume σn(x) = 1 on L, F = Fix(σ) ∩ L.

Then ξn depends on the difference field struc-

ture of L.

Let ρ : K(V arF ) → K(V arF a) be the natu-

ral homomorphism. Then ρ ◦ ξn([X]) = [X ∩Fix(σ)]. In particular, ρ ◦ ξn(X) does not de-

pend on the difference field structure of L.

Theorem 1 can be stated as saying that ρ(X)

does not depend on L, where X is a class in

the Fourier table of D. Give a more general

criterion for X to be absolute in this sense.

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b) Given a non-commutative polynomial

g(X1, . . . , Xk) over L, obtain a power se-

ries Pg(t) =∑

[Wn]tn ∈ K(Diff.-Var)[[t]].

Wn = [{x ∈ L[[s]]/sn : g(x) = 0 mod sn}]. For

any n, ξn(Pg) is rational. A statement about

Pg implying this? In quantum plane picture,

variation of q?

c) Find local, non-commutative (difference va-

riety) proof by studying variation of a finite di-

mensional qunantity depending on a transfor-

mally transcendental q, specializing at qn = 1

to V ∩ σn = 1. (E.g. V ∩ σn(s) = qns? )

d) Change of variable.

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2) Local motivic integration on valued fields

with a field of representatives.

Every formula of VF-dimension N has bound-

ary of dimension < N . Kontsevich-style inte-

gration into completion is possible if one takes

Γ = Z. The values∑anL−n obtained as vol-

umes are represented by rational functions.

Every formula has normal form g(X∗) where

X ⊆ VFn+m × Γl is an ACV FA- definable set,

and g an ACV FA- definable function on X, and

X∗ = X ∩ (VFn × km)× Γl, such that:

(N) g : X∗ → X is bijective.

Question: in char. 0, what about Cluckers-

Loeser? Or H. - Kazhdan?

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3) Further results of DKV:

a) Convolution.

b) GLn.

4) Adeles. The Tamagawa number can be ex-

pressed motivically (volG(O)[G(O)\G(A)1/G(K)])

Motivic Weil; for G = T , Ono, Oesterle.

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5) Global integration: uniformity in Γ.

The contribution of logic to p-adic integra-

tion consists of uniformity in k and in Γ. The

present global theory is uniform in k only, and

it is known that the induced structure on Γ

cannot be linear; the number of rational curve

on a Calabi-Yau would fit into this framework

(Givental, Kontsevich). Dimension growth?

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