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.

1

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.

2

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)

3

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.

4

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:

(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

6

(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]

(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

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)

9

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

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

17

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.

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.

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

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?)

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]

24

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.

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).

25

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.

26

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.

27

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.

28

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?

29

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.

30

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?

31