Uniform estimates for orbital integrals and their Fourier ...pi.math.cornell.edu/~templier/geometric/16elmau_Gordon.pdf · Introduction An old result (2012) New results Model theory

IntroductionAn old result (2012)

New resultsModel theory results

Harmonic analysis

Uniform results fororbital integrals

How it worksLanguages

Constructible motivicfunctions andspecialization

Uniform estimates for orbital integrals andtheir Fourier transforms

Raf Cluckers, Julia Gordon and Immanuel Halupczok

University of Lille, UBC, and University of Düsseldorf, respectively.

April 2016, Simons Symposium, SchloßElmau



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Set-up

• G is a connected reductive group; g – its Lie algebra;

• F - a local field (often, F = kv where k is a global field;k can be a number field or function field of sufficientlylarge characteristic .)

• {fλ} be a family of functions, in C∞c (G(F )) or in

C∞c (g(F )), indexed by some parameter λ (very flexible,

and can stand for a tuple of parameters).

• Oγ(fλ), OX (fλ) – orbital integrals, with respect to thecanonical measure.

• DG(X ) :=∏

α∈Φα(X) 6=0

|α(X )|. (an extension of

Harish-Chandra’s definition; it’s no longer zero atnon-regular elements).

• Φ(X , fλ) = DG(X )1/2OX (fλ) – normalized orbital integral(similarly, on the group).



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A question

• For a fixed f ∈ C∞c (g(F )), the function X 7→ Φ(X , f ) is

bounded on g(F ). (Harish-Chandra: bounded ong(F )rss, Kottwitz: all g(F )).

Questions:

• How does this bound depend on the field as F = Fv

runs through the set of finite places of a global field F ?• How does this bound depend on the parameters λ in a

family of test functions fλ? (e.g. when fλ is thecharacteristic function of KaλK , so we can think of λ as atuple of integers).



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An answer

There exist constants a, b (that depend only on the rootdatum of G), such that

|Φ(X , fλ)| ≤ qa+b‖λ‖v .

Note: precise a and b are unknown; if geometric methodsgave such a and b in the function field case, our result alsoimplies that the same a, b work in characteristic zero (forlarge enough p).



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The strategy:

• Use model theory.

• Prove that Φ(X , fλ) belongs to the class of the so-calledconstructible functions

• Prove that if a constructible function on S × Zn is

bounded, its bound has to have the form qa+b‖λ‖. (HereS is any definable set).

• We emphasize that we use the fact that Φ is boundedto obtain the fact that its bound has this nice form.



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Limits

Theorem (Cluckers-Halupczok)Let H be a constructible (exponential) function on X × Z

n,such that for all F with sufficiently large residuecharacteristic, for all x ∈ XF , lim‖λ‖→∞ |HF (x , λ)|C = 0.Then in fact ∃r < 0, α : X → Z, such that

HF (x , λ)| < qr‖λ‖F

for all λ ∈ Zn with λ > αF (x).



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Limits and Fourier transform

Theorem (Cluckers-Halupczok-G.)

• The class of constructible exponential functions isclosed under taking pointwise limits, Lp-limits, andFourier transforms.

• If a constructible exponential function H is bounded forevery value of a parameter λ ∈ Z

n, then it is boundedby the expression of the form qa+b‖λ‖.



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Transfer principles

The properties of being bounded, in L1, in L2, identicallyvanishing, having a limit, converging to zero, etc. transfer (ineither direction) between fields of characteristic zero andfields of positive characteristic with isomorphic residuefields, for large enough residue characteristic.Remark on norms: We can prove: the L2-norm for a motivicexponential function HF for a field F of positivecharacteristic is estimated by qN

F ‖HF ′‖, where F ′ has theresidue field isomorphic to that of F , and N is a(non-effective) constant depending on the formulas definingthe function H. There might be a better estimate.



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Transfer principles

The properties of being bounded, in L1, in L2, identicallyvanishing, having a limit, converging to zero, etc. transfer (ineither direction) between fields of characteristic zero andfields of positive characteristic with isomorphic residuefields, for large enough residue characteristic.Remark on norms: We can prove: the L2-norm for a motivicexponential function HF for a field F of positivecharacteristic is estimated by qN

F ‖HF ′‖, where F ′ has theresidue field isomorphic to that of F , and N is a(non-effective) constant depending on the formulas definingthe function H. There might be a better estimate.



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A list of constructible things

• Moy-Prasad filtration subgroups (and lattices gx,r ) aredefinable.

• Orbital integrals: Oγ(fλ) = H(γ, λ), (on a group or Liealgebra), for a definable family of test functions {fλ}.

• Fourier transforms of orbital integrals on the Liealgebra: OX (f̂ ) =

∫

g(F ) µ̂X (Y )f (Y )dY , where µ̂X (Y ) isa locally integrable function on g

reg(F ). This function isconstructible exponential; moreover, (X ,Y ) 7→ µ̂X (Y ) isconstructible exponential.

• Shalika germs (with T.C. Hales); some extraparameters needed.



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Should be on the list?

• (In progress with Tom Hales and Loren Spice):characters of supercusipdal representations obtainedby J-K Yu’s construction (up to multiplicative characters)

• Not yet done: weighted orbital integrals (?)



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Uniform boundedness ofFourier transforms

• (Harish-Chandra, Herb). Let ω be a compact subset ofg(F ). Then

supω

|D(X )|1/2|D(Y )|1/2|µ̂X (Y )| <∞.

• It follows from Theorem 1: If ωm is a family of definablecompact sets, then

supωm

|D(X )|1/2|D(Y )|1/2|µ̂X (Y )| ≤ qa+b‖m‖F .



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Uniformity of asymptoticexpansions

Suppose H1, H2 are constructible (exponential) functions,say, on an affine space, and for every p-adic field F ,(H1)F = (H2)F on a neighbourhood of 0. Then thisneighbourhood is of the same radius for all F of sufficientlylarge residue characteristic, and this statement can betransferred to F of positive characteristic.In particular, this applies to Shalika germ expansions.



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Presburger language for Z

The symbols are:

• ’0’, ’1’ – constants;

• ’+’ – a binary operation;

• ≡n, n ≥ 1 – binary relations;

• and symbols for the variables.

(Note: no multiplication!)A set is called definable if it can be defined by ’φ(x) is true’,where φ is a formula and x – a tuple of variables.A function is called definable if its graph is a definable set.Definable functions are linear combinations ofpiecewise-linear and periodic functions.



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Language of rings

The language of rings has:

• 0, 1 – symbols for constants;

• +, × – symbols for binary operations;

• countably many symbols for variables.

The formulas are built from these symbols, the standardlogical operations, and quantifiers. Any ring is a structure forthis language.

ExampleA formula: ’∃y , f (y , x1, . . . , xn) = 0’, where f ∈ Z[x0, . . . , xn].

If we choose a structure that happens to be an algebraicallyclosed field, then definable sets are the constructible sets inthe sense of algebraic geometry (thanks to quantifierelimination for algebraically closed fields).



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Denef-Pas Language (for thevalued field)

Formulas are allowed to have variables of three sorts:

• valued field sort, (+, ×, ’0’,’1’, ac(·), ord(·) )

• value sort (Z), (+, ’0’, ’1’, ≡n, n ≥ 1)

• residue field sort, (language of rings: +, ×, ’0’, ’1’)

Formulas are built from arithmetic operations, quantifiers,and symbols ord(·) and ac(·). Example:φ(y) = ’∃x , y = x2’, or, equivalently,

φ(y) = ’ord(y) ≡ 0 mod 2 ∧ ∃x : ac(y) = x2’.



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Constructible Motivic Functions

Generally, we’ll be able to consider functions of the formf (x1, . . . xm, y1, . . . , yn, z1, . . . , zr ), where x1, . . . , xm areallowed to range over the valued field, y1, . . . , yn over theresidue field, and z1, . . . , zr – over Z (i.e., residue field andvalue group parameters are allowed).

The ring of constructible functions is built from definablefunctions, and functions of the form L

f , where f is adefinable function and L is a formal symbol. Constructiblemotivic functions take values in

Z

[

L,L−1, (1 − L−i)−1, i ≥ 1

]

.



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Specialization

(F ,̟) gives a structure for Denef-Pas language: we allowthe variables of the corresponding sorts to range over F andthe residue field of F , respectively.

• The function ord(·) specializes to the usual valuation;

• the function ac(·) specializes to the first nonzerocoefficient of the ̟-adic expansion.

• The symbol L specializes to q = #k .

• Any formula in the language of rings with n freevariables ranging over the residue field specializes tothe number of the points in kn that satisfy it.



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Integration

(due to R. Cluckers and F. Loeser)When we integrate f (x1, . . . , xm, y , z), say, with respect tox1, the answer is the same type of functionf (x2, . . . , xm, y , z), but the formulas defining it might havemore bound residue field variables.

∫

K mf (x1, . . . , xm, y , z)dx1 . . . dxm = F (y , z)

where F is built from formulas in the language of rings overthe residue field. If all these formulas were quantifier-free, itwould have been a constructible set over the residue field.



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Constructible functions

A constructible function on a definable subset of F n has theform:

fF (x) =N∑

i=1

qαiF (x)F #(p−1

iF (x))(

N′∏

j=1

βijF (x))(

N′′∏

ℓ=1

11 − qaiℓ

F

)

,

where:

• aiℓ are negative integers;

• αi : X → Z are Z-valued definable functions;

• Yi are definable sets such that YiF ⊂ k riF × XF for some

ri ∈ Z, and pi : Yi → X is the coordinate projection.



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Constructible exponentialfunctions

Roughly, these are of the form

H(x) =N∑

i=1

ai(x)ψ(hi (x)),

where ai are constructible, and hi are definable functions onF n, and ψ : F → C

∗ is an additive character of level 0.We are interested in questions such as:

• when is H in L1(F n)?

• what can we say about its bounds?

It turns out that the answer entirely depends on ai(x) (andintegrability cannot happen due to delicate cancellations).

Documents

Uniform estimates for orbital integrals and their Fourier ...pi.math.cornell.edu/~templier/geometric/16elmau_Gordon.pdf · Introduction An old result (2012) New results Model theory