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