Antoine Chambert-Loir
Département de mathématiques d’Orsay, Bâtiment 425, Faculté des sciences d’Orsay, Université Paris-Sud, F-91405 Orsay Cedex
E-mail:Antoine.Chambert-Loir@math.u-psud.fr
François Loeser
Institut de Mathématiques de Jussieu, UMR 7586 du CNRS, Université Pierre et Marie Curie, F-75252 Paris Cedex 05
E-mail:Francois.Loeser@upmc.fr
Abstract. — We consider a motivic analogue of the height zeta function for integral points of equivariant partial compactifications of affine spaces. We establish its rationality and determine its largest pole.
A major problem of Diophantine geometry is to understand the distribution of rational or integral points of algebraic varieties defined over number fields. For example, a well-studied question put forward by Manin in [1] is that of an asymptotic expansion for the number of rational/integral points of bounded height. A basic tool is the height zeta function which is a Dirichlet series.
Around 2000, E. Peyre suggested to consider the analogous problem over func- tion fields, which has then an even more geometric flavor since its translates as a problem of enumerative geometry, namely counting algebraic curves of given degree and establishing properties of the corresponding generating series. In view of the developments of motivic integration by Kontsevich, Denef–Loeser [9], etc., it is nat- ural to look at a more general generating series which not only counts thenumber of such algebraic curves, but takes into account thespace they constitute in a suitable Hilbert scheme.
A natural coefficient ring for the generating series is the Grothendieck ring of varieties KVark: ifk is a field, this ring is generated as a group by the isomorphism classes of k-schemes of finite type, the addition being subject to obvious cut-and- paste relations, and the product is induced by the product ofk-varieties. One interest of this generalization is that it also makes sense in a purely geometric context, where no counting is available.
Such a situation has been first studied in a paper by Kapranov in [13], where the analogy with the mentioned diophantine problem is not pointed out. Later, Bourqui made some progress on the motivic analogue of Manin’s problem, see [3], as well as his survey report [4].
In this article, we consider the following situation. Let k be an algebraically closed field of characteristic zero. Let C be a quasi-projective smooth curve over k and
let C be its smooth projective compactification; we let S =C\C. Let F =k(C) be the function field of C and g be its genus.
Let X be a projective irreducible k-scheme together with a non-constant flat mor- phismπ: X →C. Let G andU be Zariski open subsets of X such that G⊂U ⊂X and such that π(U)contains C. Let L be a line bundle on X; we assume that there exists an effective Q-divisor D supported on (X \U)F such that L(−D) is ample on XF.
We are interested in sectionsσ: C →X ofπwith prescribed degreen= degσ∗L and such thatσ(CF)⊂GF and σ(C)⊂U.
By Lemma 2.2.2, these conditions define a constructible set MU,n (in some k- scheme); moreover, the hypothesis on L implies that there exists n0 ∈Z such that MU,n is empty forn6n0. Considering the classes [MU,n] of these sets in KVark, we form the generating Laurent series
(1) Z(T) = X
n∈Z
[MU,n]Tn and ask about its properties.
In fact, this is not exactly a geometric/motivic analogue of Manin’s problem, but rather one for its restriction tointegral points.
Precisely, we investigate in this paper the motivic counterpart of the situation studied recently in the paper [6] by Y. Tschinkel and the first named author.
In this paper, we consider the particular case where GF is the additive group Gna,F, UF =XF andXF admits an action ofGF which extends the group action ofGF over itself. We also assume that the irreducible components of the divisor at infinity∂X = X\G are smooth and meet transversally. Finally, we restrict ourselves to the case where the restriction of L to the generic fiber XF is equal to −KXF(∂XF), the log-anticanonical line bundle.
Let L be the class of the affine line A1k in KVark and let Mk be the localized ring of KVark with respect to the multiplicative subset S generated by L and the elements La−1, for a∈ N>0. An element of KVark is said to be effective if it can be written as a sum of classes of algebraic varieties; similarly, an element of Mk is effective if its product by some element of S is the image of an effective element of KVark. For example, 1−L−a=L−a(La−1) is effective for every a >0.
Let Mk{T}and Mk{T}† be the subrings of Mk[[T]] generated byMk[T] and the inverses of the polynomials 1−LaTb, respectively forb >a>0 andb > a>0. Any element ofMk{T}† has a value at T =L−1 which is an element of Mk.
Our main theorem is the following.
Theorem 2. — The power series Z(T) belongs to Mk{T}. More precisely, there exists an integer a > 1, an element PU(T) ∈ Mk{T}† such that PU(L−1) is an effective nonzero element of Mk, and a positive integer d such that
(1−LaTa)dZ(T) =PU(T).
For any algebraic k-variety M, let d(M) = dim(M) and ν(M) be the number of irreducible components of maximal dimension.
Corollary 3. — For every integer p ∈ {0, . . . , a−1}, one of the following cases occur when n tends to infinity in the congruence class of p modulo a:
(1) Eitherd(MU,n) = o(n),
(2) Or d(MU,n)−n has a finite limit and log(ν(MU,n))/log(n) converges to some integer in {0, . . . , d−1}.
Moreover, the second case happens at least for one integer p.
The corresponding question in the arithmetic case consists in establishing the an- alytic property of the height zeta function (holomorphy for<(s)>1, meromorphy on a larger half-plane, pole of order d at s = 1) as well as showing that the num- ber of points of height 6 B grows as B(log(B))d−1. Its proof in [6] relies on the Poisson summation formula for the discrete cocompact subgroupG(F) of the adelic group G(AF). The starting point of the present work was to take advantage of the motivic Poisson formula recently established by E. Hrushovski and D. Kazhdan in [12] in order to prove new results in the geometric setting.
However, in its present form, this motivic Poisson formula suffers two limitations.
Firstly, the functions it takes as input may only depend on finitely many places of the given function field. For this reason, the question we solve in this paper is a geometric analogue of Manin’s problem for integral points, rather than for rational points. Secondly, the Poisson formula only applies to vector groups, and this is why our varieties are assumed to be equivariant compactifications of such groups.
The plan of the paper is the following.
We begin the paper by an exposition, in a self-contained geometric language, of the motivic Poisson formula of Hrushovski–Kazhdan. We then gather in Section 2 some preliminary results needed for the proof. In particular, we show in Proposition 2.1.3 that Corollary 3 is a consequence from Theorem 2. For eventual reference, we also prove there a general existence theorem for the moduli spaces which we study here, see Proposition 2.2.2. We end this Section by recalling a few notation on Clemens complexes, and on functions on arc spaces with values inMk.
In Section 3, we lay out the foundations for the proof of Theorem 2. Its main goal consists in describing the moduli spaces as adelic subsets of the groupG.
The core of the proof of Theorem 2 begins with Section 4. We first apply the motivic Poisson summation formula of Hrushovski and Kazhdan. We show that this formula gives an expression Z(T) as a “sum” (in the sense of motivic integration) over ξ ∈ G(F) of rational functions Z(T, ξ) whose denominators are products of factors of the form 1−LaTb for b > a. The point is that the term corresponding to the parameter ξ= 0 is the one which involves the largest number of such factors with a = b; intuitively, the “order of the pole of Z(T, ξ) at T = L−1” is larger for ξ = 0 than for ξ 6= 0. Admitting these facts, it is therefore a simple matter to conclude the proof of Theorem 2.
The proof of these facts are the subject of Sections 5 and 6. In fact, once rewritten as a motivic integral, the power seriesZ(T,0) is a kind of “geometric” motivic Igusa zeta function. Its analysis, using embedded resolution of singularities, would be
classical; in fact, our geometric setting is so strong that we even do not need to resolve singularities in this case. For general ξ, however, what we obtain is a sort of “motivic oscillatory integral”. Such integrals are studied in a coordinate system in Section 5. Finally, in Section 6, we establish the three propositions that we had temporarily admitted in Section 4.
Acknowledgments. — The research leading to this paper was initiated during a visit of the second author to the first author when he was visiting the Institute for Ad- vanced Study in Princeton for a year. We would like to thank that institution for its warm hospitality. The first author was supported by the Institut universitaire de France, as well as by the National Science Foundation under agreement No.
DMS-0635607. The second author was partially supported by the European Re- search Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant Agreement nr. 246903 NMNAG.
1. The motivic Poisson formula of Hrushovski–Kazhdan
For the convenience of the reader, we begin this paper with an exposition of Hrushovski-Kazhdan’s motivic Poisson summation formula. We follow closely the relevant sections from [12], but adopt a self-contained geometric language. In the rest of the paper, we will make an essential use of the formalism recalled here.
To motivate the definitions, let us discuss rapidly the dictionary with the Poisson summation formula for the adele groups of global fields. So assume that F is a global field. Let AF be the ring of adeles of F; it is the restricted product of the completions Fv at all places v of F and is endowed with a natural structure of a locally compact abelian group. The fieldF embeds diagonally in AF and its image is a discrete cocompact subgroup. Fix a Haar measure µ on AF as well as a non trivial character ψ: AF → C∗. For any Schwartz-Bruhat function ϕ on AnF, its Fourier transform is defined as the function
Fϕ(y) =
Z
AnF
ϕ(x)ψ(xy) dµ(x)
onAnF; it is again a Schwartz-Bruhat function. Moreover, the global Haar measure, additive character and Fourier transform can be written as product of similar local objects. Then, one has
X
x∈Fn
ϕ(x) =µ(AF/F)−n X
y∈Fn
Fϕ(y).
The motivic Poisson summation formula provides an analogue of this formalism, whenF is the function field of a curveC over an algebraically closed field. Integrals belong to the Grothendieck ring of varieties, more precisely, to a (suitably localized) variant “with exponentials” of this ring. They are constructed using motivic inte- gration at the “local” level of completions Fv; here Fv is identified with the field k((t)) of Laurent series, so that Fvn can be considered as an infinite dimensional k- variety, more precisely, an inductive limit of arc spacest−mk[[t]]n'L(Ank). Motivic
Schwartz-Bruhat functions are elements of relative Grothendieck rings. The possi- bility to define the “sum over Fn” of a motivic function follows from the fact that it is non-zero outside of a finite dimensional subvariety of this ind-arc space. The Poisson summation formula then appears as a reformulation of the Riemann Roch theorem for curves combined with the Serre duality theorem, as formulated in [17].
1.1. The Grothendieck ring of varieties with exponentials
1.1.1. — Letk be a field. TheGrothendieck group of varieties KVark is defined by generators and relations; generators are k-varieties X (=k-schemes of finite type);
relations are the following:
X−Y, whenever X and Y are isomorphic k-varieties;
X−Y −U,
wheneverXisk-variety,U an open subset ofXandY =X\U is the complementary (reduced) closed subset. We write [X] for the class in KVark of a k-variety X.
The Grothendieck group of varieties with exponentials KExpVark is defined by generators and relations (cf. [8, 12]). Generators are pairs (X, f), where X is a k-variety and f: X →A1k is a k-morphism. Relations are the following:
(X, f)−(Y, f◦u)
whenever X, Y are k-varieties, f: X → A1k a k morphism, and u: Y → X a k- isomorphism;
(X, f)−(Y, f|Y)−(U, f|U)
whenever X is a k-variety, g: X → A1k a morphism, U an open subset of X and Y =X\U the complementary (reduced) closed subset;
(X×A1k,pr2)
where X is a k-variety and pr2 is the second projection. We will write [X, f] to denote the class in KExpVark of a pair (X, f).
There is a morphism of Abelian groups ι: KVark → KExpVark which sends the class of X to the class [X,0].
Any pair (X, f) consisting of a constructible variety X and of a piecewise mor- phismg: X →A1k has a natural class in KExpVark.
1.1.2. — One endows KVark with a ring structure by setting [X][Y] = [X×kY]
whenverX and Y are k-varieties. The unit element is the class of the point Speck.
One endows KExpVark with a ring structure by setting [X, f][Y, g] = [X×kY,pr∗1f + pr∗2g],
whenever X and Y are k-varieties, f: X → A1k and g: Y → A1k are k-morphisms;
pr∗1f+ pr∗2g is the morphism from X×kY toA1k sending (x, y) tof(x) +g(y). The unit element for this ring structure is the class [Speck,0] =ι([Speck]).
The morphism ι: KVark→KExpVark is a morphism of rings.
One writesLfor the class ofA1k in KVark, or for the class of (A1k,0) in KExpVark. LetS be the multiplicative subset of KVarkgenerated by Land the elementsLn−1, forn>1. The localizations of the rings KVark and KExpVark with respect toS are denotedMkandExpMkrespectively. There is a morphism of ringsι: Mk →ExpMk. Lemma 1.1.3 ([8, Lemma 3.1.3]). — The two ring morphisms ι: KVark → KExpVark and ι: Mk →ExpMk are injective.
Proof. — For t∈A1(k), letjt be the map which sends a pair (X, f) to the class of [f−1(t)]. One observes that j0 −j1 defines a morphism of groups j: KExpVark → KVark. Indeed, for everyt∈A1(k),jtmaps the additivity relations in KExpVark to additivity relations in KVark. Moreover, jt(Y ×A1,pr2) = [Y] for everyk-varietyY, so thatj((Y ×A1,pr2)) = 0. This proves the existence ofj; it is obviously a section of ι, henceι is injective.
Lemma 1.1.4. — Let X be a k-variety with a Ga-action, let f: X →A1k be such thatf(t+x) =t+f(x)for every t∈Ga and every x∈X. Then, the class of(X, f) is zero in KExpVark.
Proof. — By a theorem of Rosenlicht [16], there exists a Ga-stable dense open subset U and a quotient map U → Y which is a Ga-torsor. Every such torsor is locally trivial for the Zariski topology, hence, up to shrinkingU (andY accordingly), this Ga-torsor is trivial, hence a Ga-equivariant isomorphism u: Ga×Y ' U. Let g: Y → A1k be the morphism given by y 7→ f(u(0, y)). For y ∈ Y and t ∈ Ga, one hasf(u(t, y)) =f(t+u(0, y)) =t+f(u(0, y)). This shows that the class of (Y, f◦u) equals the product of the classes of (Ga,Idk) and (Y, g). It is zero in KExpVark, so that the class of (U, f|U) is zero too. One concludes the proof by Noetherian induction.
1.1.5. Relative variant. — Let S be a k-variety. There are similar rings KVarS, KExpVarS, MS and ExpMS defined by replacing k-varieties and k-morphisms by S-varieties and S-morphisms in the above definitions, and the affine lineA1k by the line A1S.
Any morphism u: S → T of k-varieties induces morphisms u∗ and u∗ between these Grothendieck rings. The definitions are similar; let us explain the case of KExpVar.
Sending the class of (X, f) to itself,Xbeing viewed as aT-scheme viau,uinduces a morphism of groups
u∗: KExpVarS →KExpVarT. In othe other direction, it induces a morphism of rings,
u∗: KExpVarT →KExpVarS
given, at the level of generators, by (X, f) 7→ (X ×S T, f ◦ pr1). Elements of KExpVarS can be thought of asmotivic functionswith sourceS. Forf ∈KExpVarS
and a points ∈S, considered as a morphism Speck(s)→S, one writesf(s) for the elements∗f of KExpVark(s).
Lemma 1.1.6. — Let f ∈ KVarS (resp. MS, resp. KExpVarS, resp. ExpMS).
If f(s) = 0 for every s∈S, then f = 0.
As a corollary, Lemmas 1.1.3 and 1.1.4 hold in this greater generality.
Proof. — We give the proof for KExpVarS, the other three cases are similar. Let us fix a representativeM of f in Z[ExpVarS], the free Abelian group generated by pairs (X, u), where X is a S-scheme and u: X → A1k is a morphism. Let s be a generic point of S; since f(s) = 0, the object Mk(s) is a linear combination of elementary relations. By spreading out the varieties and the morphisms expressing these relations, there exists a dense open subset U of S such that the object MU in Z[ExpVarU] is a linear combination of the corresponding elementary relations.
One has [MU] = 0, hence [MT] = 0, where T = S \U. By Noetherian descending induction, it follows that [M] = 0.
Remark 1.1.7. — Let A =Z[T] and B be the localization of A with respect to the multiplicative subset generated byT and theTn−1, forn>1. The unique ring morphism fromAto KVark which sendsT toLendowes KVarkand KExpVarkwith structures ofA-algebra. Moreover,Mk'B⊗AKVarkandExpMk 'B⊗AKExpVark and
More generally, for any k-variety S, KVarS and KExpVarS are A-algebras, and one has natural isomorphisms
MS 'B ⊗AKVarS 'Mk⊗KVark KVarS and
ExpMS 'B⊗AKExpVarS 'ExpMk⊗KExpVark KExpVarS.
Thanks to this remark, we will often allow ourselves to write formulas or proofs at the level of KExpVarS, when the generalization to ExpMS follows directly by localization.
1.1.8. Exponential sums. — The class of a pair (X, f) in KExpVark can be thought of as an analogue of the exponential sum
X
x∈X(k)
ψ(f(x)),
whenk is a finite field andψ:k →C∗ is a fixed non-trivial additive character. This justifies the notation Px∈Xψ(f(x)) for the class [X, f] in KExpVark.
More generally, let θ ∈ExpMS and let u:S →A1k be a morphism. We define
(1.1.9) X
s∈S
θ(s)ψ(u(s)) = θ·[S, u],
the product being taken in ExpMS. Let us explicit this definition, assuming that θ = [X, f], where g: X → S is a S-variety and f: X → A1k a k-morphism; in this
case,
X
s∈S
θ(s)ψ(u(s)) = [X, f +u◦g].
Observe also that whenk is a finite field, one has
X
s∈S(k)
θ(s)ψ(u(s)) = X
s∈S(k)
X
x∈X(k) g(x)=s
ψ(f(x))
= X
x∈X(k)
ψ(f(x) +u(g(x)).
1.2. Local Fourier transforms
1.2.1. Schwartz-Bruhat functions and their integrals. — LetF be the field of frac- tions of a ring which is complete for a (normalized) discrete valuation ord, and has residue field k. We assume that F and k have the same characteristic; any uni- formizer,i.e., anyt∈K of valuation 1, gives rise to an isomorphismk((t))'F. Let F◦ be the valuation ring of F.
Fix such a local parameter t. For any two integers M 6 N, we can identify the quotient set {x; ord(x) > M}/{x; ord(x) > N} = tMF◦/tNF◦ of the elements x in F satisfying ord(x) > M, modulo those satisfying ord(x) > N, with the affine spaceA(M,N)k =AN−Mk , via the formula
x=
N−1
X
j=M
xiti (mod tN)7→(xM, . . . , xN−1).
In view of this, we define a motivic Schwartz-Bruhat function of level (M, N) to be a function A(M,Nk ) → ExpMk, that is, an element of ExpMA(M,N)
k
. We write S(F;M, N) for the ring of Schwartz-Bruhat functions of level (M, N). We denote by1F◦ the class of 1 inS(F; 0,0).
1.2.2. Compatibilities. — The natural injection tMF◦/tNF◦ → tM−1F◦/tNF◦ is turned into a closed immersion ι: A(M,N)k → A(M−1,N)k . This gives rise to a mor- phism of restriction ι∗: S(F;M−1, N)→S(F;M, N), as well as to a morphism ι∗: S(F;M, N)→S(F;M −1, N) (extension by zero). One has ι∗ι∗ = Id.
Similarly, the natural injection tMF◦/tNF◦ →tMF◦/tN+1F◦ induces a morphism π: A(M,Nk +1) → A(M,N)k which is a trivial fibration with fiber A1k. This gives rise to a morphismπ∗: S(F;M, N)→S(F;M, N + 1).
The space of motivic smooth functions (of arbitrary level) is then defined by
(1.2.3) D(F) = lim←−
M,ι∗
lim−→
N,π∗
S(F;M, N),
while the space of motivic Schwartz-Bruhat functions is defined by
(1.2.4) S(F) = lim−→
M,ι∗
lim−→
N,π∗
S(F;M, N).
These spaces have a ring structure, but S(F) has no unit element; the natural injectionS(F)⊂D(F) is a morphism of rings.
We define the integral of a motivic Schwartz-Bruhat function ϕ of level (M, N) by
Z
F
ϕ(x) dx=L−N X
x∈A(M,N)k
ϕ(x).
Observe that ι∗ commutes with the sum over points, while π∗ only commutes up to multiplication by L. Consequently, the integral of a Schwartz-Bruhat function does not depend on the choice of a level (M, N). This gives rise to an additive map S(F)→ExpMk.
1.2.5. — This theory is extended in a straightforward way to finitely many vari- ables, as well as to products of local fields. More precisely, for n > 1, one sets Sn(F;M, N) = Sn(M, N) =S(F;M, N)n, one extends the definition of the mor- phismsι∗ and π∗ directly, one sets
(1.2.6) S(Fn) = lim−→
M,ι∗
lim−→
N,π∗
Sn(M, N).
One defines similarly an additive map RFn: S(Fn)→ExpMk. Given finitely many fields Fs, s ∈S as above, with residue field ks and local parameter ts, and families of integers Ms6Ns, one sets
(1.2.7) Sn(Y
s∈S
Fs; (Ms, Ns)) = Y
s∈S
Sn(Fs;Ms, Ns) and, similarly,
(1.2.8) S(An;Y
s∈S
Fs) = lim−→
(Ms,ι∗)
lim−→
(Ns,π∗)
Sn(Y
s∈S
Fs; (Ms, Ns)),
using a self-explanatory extension of notation. In particular, given a family of func- tionsϕs∈S(Fsn), one obtains a function⊗s∈Sϕs∈S(An;Qs∈SFs). The definition of the integral extends to a linear map S(An;Qs∈SFs)→ExpMk.
1.2.9. The Fourier kernel. — Letr: F →k a non zerok-linear map which vanishes ont−aF◦ for some integera. We define the conductorν ofras the smallest integera such thatr vanishes on t−aF◦. In the sequel, our main source of such a linear form will be a residue: assume that F is the completion at a closed rational point of a function field in one variable over k, let ω ∈ ΩF /k be a non-zero meromorphic differential form, then set r(x) = res(xω), where res : ΩF /k → k is the residue map;
in that case, the conductor ofr is the order of the pole ofω.
Let x ∈ F, let us write x = Pnxntn, where xn = 0 for n < ord(x). One has r(x) =
−a−1
P
n=ord(x)
xnr(tn). Consequently, restricted to the subset of F consisting of elements x such that ord(x) 6 M, r can be interpreted as a linear morphism r(M,N): A(M,Nk ) →A1k, for every integer N such that N >−a−1.
The product map F ×F → F gives rise to a morphism A(M,Nk )×A(Mk 0,N0) → A(M+Mk 0,M+M0+min(N−M,N0−M0)). When N00 = M +M0 + min(N −M, N0 −M0) = min(M0+N, M +N0)>ν, we can compose withr(M+M0,N00) and obtain the kernel of the Fourier transform, as an element of D(F2).
1.2.10. Fourier transformation. — The Fourier transform of a Schwartz-Bruhat functionϕ∈S(F;M, N) is defined formally as
Fϕ(y) =
Z
F
ϕ(x)e(xy) dx,
where e(·) is a short-hand notation for ψ(r(·)). Let us make the definition explicit, assuming thatϕis of the form [U, f], where (U, u) is aA(M,N)k -variety andf: U →A1k is a morphism. Then,Fϕis represented by
L−N[U ×uA(M,Nk )×A(Mk 0,N0), f(u) +r(xy)]
in the Grothendieck group ExpMA(M0,N0)
k
, where the structural morphism U ×u
A(M,N)k ×A(Mk 0,N0) → A(Mk 0,N0) is given by the projection to the third factor. For this to make sense, we only need to takeM0 >ν−N and N0 >ν−M.
Proposition 1.2.11. — If ϕ ∈ S(F;M, N), then Fϕ∈S(F;ν−N, ν−M), where ν is the conductor of r.
Observe also that ϕ7→Fϕis ExpMk-linear.
Theorem 1.2.12 (Fourier inversion). — One has F Fϕ(x) = L−νϕ(−x).
Proof. — We may assume that ϕ is represented by [U, f] as above. To compute F Fϕ, we may set (M0, N0) = (ν −N, ν−M) and (M00, N00) = (M, N), so that F Fϕis represented by
L−N−N0[U ×A(M,Nk )×A(Mk 0,N0)×A(Mk 00,N00), f(u) +r(xy) +r(yz)].
The contribution of the part wherex+z 6= 0 is zero, because of Lemma 1.1.4. The part wherex+z = 0 is equal to
L−N−N0[U ×A(Mk 0,N0)×A(Mk 00,N00), f(u)] =L−N−M0[U ×A(Mk 00,N00), f(u)]
=L−ν[U, f],
where U is viewed as an A(Mk 00,N00)-variety via the morphism −u. This proves the theorem.
For n > 1, the definition of the Fourier Transform F extends to F: S(Fn) → S(Fn) by interpreting xy as the standard scalar product of the two elements x and y of Fn. More generally, given finitely many fields Fs, s ∈ S as above, with residue field ks, local parameter ts, and linear maps rs: Fs → ks, the definition of F extends to F: S(An;Qs∈SFs) → S(An;Qs∈SFs) by replacing r(xy) by
P
s∈Srs(xsys). Fourier inversion 1.2.12 and its proof extend to this setting.
1.3. Global Fourier Transforms
1.3.1. — LetC be a projective, connected, smooth curve overk, and let F =k(C) be its field of functions. We fix a non-zero meromorphic differential formω ∈Ω1F /k. One can interpret the field F = k(C) as the k-points of an ind-k-variety. The simplest way to do so consists maybe in considering the inductive system of all Riemann-Roch spaces H0(C,O(D)), indexed by effective divisors D onC.
Lemma 1.3.2. — For each divisor D, the subspace L(D) of rational functions on f having divisor >−D is a k-variety.
1.3.3. Global Schwartz-Bruhat functions. — Let S be a finite set of places of C.
For each s ∈ S, we fix a local parameter ts. A global Schwartz-Bruhat function on AnF is an element of S(An;Qs∈SFs) for some finite set of places S; if S0 ⊃ S, we identifyϕ∈S(An;Qs∈SFs) and ϕ⊗Ns0∈S0\S1F◦
s0 inS(An;Qs∈S0Fs).
In classical cases, simple functions are characteristic functions of a ball, or of product of balls. For any family (Ms, Ns)s∈S and any a ∈ Qs∈S(A(Mk s,Ns))n, let us define Wa = {a}, together with the obvious map to W = (Qs∈SA(Mk s,Ns))n. The motivic function associated to the pair (Wa →W,0) is called a simple function. It represents the characteristic function of the product of the balls of center as and radiusNs.
A family of simple functions parametrized by a k-variety Z is a motivic function of the form [Z −→u W,0], where W is a finite product (Qs∈SA(Mk s,Ns))n.
Lemma 1.3.4. — (1) LetZ be ak-variety, letϕbe a family of simple functions parametrized by Z and let χ: Z →ExpMk be a motivic function. Then, there exists a unique global Schwartz-Bruhat function Φ such that Φ(x) = Pz∈Zχ(z)ϕz(x) for every x∈Qs∈SAnF.
(2) Any global Schwartz-Bruhat function on AnF can be written as in (1).
Proof. — (1) We may assume that χis represented by [X −→u Z, f], whereX is a Z-variety and f: X →A1k; let ϕ= [Z −→ϕ W,0], where W = Qs∈SA(Mk s,Ms). Then, the function Pz∈Zχ(z)ϕz is represented by the pair [X −−→ϕ◦u W, f]. Uniqueness follows from Lemma 1.1.6.
(2) Conversely, let Φ be a global Schwartz-Bruhat function on AnF. Again, we may assume that Φ is represented by a pair [Z, f], where Z is a variety over W =
Q
sA(Mk s,Ns) and f: Z →A1k. Let ϕ be the family of simple functions parametrized byW given by the pair [W −Id→W,0]. One checks readily that
Φ = X
w∈W
ϕ(w)ϕw.
1.3.5. Summation over rational points. — Letϕbe a global Schwartz-Bruhat func- tion onAnF, represented by a class ϕS in ExpMk(Qs∈SA(Mk s,Ns)), for some finite set S⊂C(k).
Consider the divisor D = PMs[s] on C and the k-variety L(D). There is a natural morphism α: L(D) → Qs∈SA(Mk s,Ns), hence we can consider α∗ϕS ∈ ExpMk(L(D)) and define Px∈Fnϕ(x) to be its image in ExpMk. It does not de- pend on the choice of the set S, of the integers (Ms, Ns), nor on the choice of the divisor D.
Let us give a more explicit formula, assuming that ϕS is of the form [X, f], where X →Qs∈SA(Mk s,Ns) and f: X →A1k. In that case, one has
(1.3.6) X
x∈Fn
ϕ(x) = [L(D)×Q
A(Ms,Ns)k X, f ◦pr2].
1.3.7. Global Fourier transformation. — We fix a nonzero meromorphic differential formω∈ΩF /k. For any place s∈C(k), we definers(x) = ress(xω); its conductor is νs = ords(ω). Let ν be the divisor Pνs[s] on C. One has deg(ν) = 2g−2, where g is the genus of C.
Observe that if νs = 0, then 1Fs◦ is its own Fourier transform. Consequently, we may define the Fourier transform Fϕ of any global Schwartz-Bruhat function ϕ is defined as the Fourier transform ofϕS, where S is any finite set of places such that νs = 0 fors6∈S, andϕS ∈S(An,Qs∈SFs) is a representative ofϕ. By construction, Fϕis itself a global Schwartz-Bruhat function on the “dual” space AnF.
Theorem 1.3.8 (Fourier inversion formula). — Let ϕ be a global Schwartz- Bruhat function on AnF. Then
F Fϕ=L2g−2ϕ(−x).
Theorem 1.3.9 (Motivic Poisson formula). — Let ϕ be a global Schwartz- Bruhat function on AnF. Then,
X
x∈An(F)
ϕ(x) =L(g−1)n X
y∈An(F)
Fϕ(y).
Before proving this formula, we need to recall a few results concerning the Riemann-Roch theorem for curves. Let D be a divisor on C. Let L(D) be the set of rational functions y ∈ F such that div(y) +D > 0. Let Ω(D) be the set of meromorphic forms α ∈ ΩF /k such that div(α) > D; the meromorphic form ω being fixed, the space Ω(D) is identified with the space L(div(ω)−D) by the map y7→yω.
LetR =Q0s∈C(k)Fsbe the ring of adeles of F =k(C), that is, the restricted prod- uct of all completionsFs; we embed F diagonally inR. For any divisorD, letR(D) be the subspace ofR consisting of families (xs) such that div(xs) + ords(D)>0 for every s ∈ C(k). There is an isomorphism of k-vector spaces (see [17], Chapitre II,
§5, proposition 3)
H1(L(D))'R/(R(D) +k(C)).
According to Serre’s duality theorem ([17], Chapitre II, §8, théorème 2), the mor- phism
θ: ΩF /k →Hom(R, k), α7→ (xs)7→X
s
ress(xsα)
!
identifies Ω(D) with the orthogonal of R(D) +k(C) in Hom(R, k), i.e., with the dual of H1(L(D)).
Prof of Theorem 1.3.9. — For simplicity of notation, we assume that n = 1. We may also assume thatϕis a simple function ⊗s∈Sϕs, where ϕs is represented by the point as ∈A(Mk s,Ns). Then, for every s∈ S, Fϕs is a Schwartz-Bruhat function on Fs and
Fϕ(ys) =
e(ress(asysω))L−Ns if ords(yω) + ords(D))>0;
0 otherwise.
Let D be the divisor Ps∈SNss on C. Then Fϕ is a global Schwartz-Bruhat function on QsFsn such that
Fϕ((ys)) =
e(Ps∈Sress(asysω))L−deg(D) if div(yω) +D>0;
0 otherwise.
By Lemma 1.3.10 below, we thus have
X
y∈F
Fϕ(y) = L−deg(D) X
y∈L(div(ω)+D)
e(hθ(yω),(as)i)
=
L−deg(D)+ndimL(div(ω)+D) if (as)∈Ω(−D)⊥,
0 otherwise.
Let us now compute the left hand side of the Poisson formula. In the case where (as)∈Ω(−D)⊥=R(−D) +k(C),
there exists a∈k(C) such that ords(a−as)>Ns for all s. Then, ϕ(x) =
1 if x−a∈L(−D), 0 otherwise
so that
X
x∈Fn
ϕ(x) = X
x∈F
ϕ(x−a) = X
x∈L(−D)
1 =LdimL(−D). By the Riemann-Roch formula,
dimL(−D) = dim H1(C,−D)−deg(D) + 1−g
= dim Ω(−D)−deg(D) + 1−g
= dimL(div(ω) +D)−deg(D) + 1−g.
This shows that
X
x∈F
ϕ(x) = Lg−1 X
y∈F
Fϕ(y).
In the other case, there does not exist any a∈k(C) such that ords(a−as)>Ns for all s. Then, ϕ(x) = 0 for all x ∈F and Px∈F ϕ(x) = 0, and we have seen that the the right hand side of the Poisson formula vanishes too.
Lemma 1.3.10. — Let V be a k-vector space, let f be a linear form on V. Then,
X
x∈V
e(f(x)) =
Ldim(V) if f = 0;
0 otherwise.
Proof. — By definition, the left hand side is the class of (V, f) in KExpVark. This equals [V] = Ldim(V) if f = 0. Otherwise, let a ∈ V be such that f(a) = 1 and let us consider the action of the additive group Ga onV given by (t, v)7→v+ta. Since f(v+ta) = f(v) +t, it follows from Lemma 1.1.4 that [V, f] = 0.
Remark 1.3.11. — By Fourier inversion, we haveF Fϕ(−x) =L−ndeg(ν)ϕ(−x) = L(2g−2)nϕ(−x). Consequently, if we apply the Poisson formula to Fϕ, we obtain
X
y∈An(F)
Fϕ(y) =L(1−g)n X
x∈An(F)
F Fϕ(x) = L(g−1)n X
x∈An(F)
ϕ(x),
as expected.
2. Further preliminaries 2.1. Motivic invariants
Letkbe a field. For everym>0, let KVar6mk be the subgroup of KVarkgenerated by classes of varieties of dimension 6 m. If x ∈ KVar6mk and y ∈ KVar6nk , then xy∈KVar6m+nk . Let (Mk6m)m∈Z be the similar filtration on Mk; explicitly,Mk6m is generated by fractions [X][Y]−1 whereX is a k-variety, Y is a product of varieties of the formA1,Aa\{0}(fora >1), and dim(X)−dim(Y)6m. For any classx∈Mk, letd(x) ∈ Z∪ {−∞} be the infimum of the integers m ∈ Z such that m ∈ Mk6m. For any x, y ∈ Mk, one has d(x+y) 6 max(d(x), d(y)) and d(xy) 6 d(x) +d(y);
moreover,d(xLn) =d(x) +n for any n∈Z.
Assume that k is algebraically closed. For any k-variety X, let Hpc(X) denote its pth singular cohomology group with compact support andQ-coefficients (ifk=C), or itspth étale cohomology group with proper support andQ`-coefficients (for some fixed prime number ` distinct from the characteristic of k). There is a unique ring morphism P from KVark to the polynomial ring Z[t] such that for any variety X, P([X]) is the Poincaré polynomial of X. Its definition relies on the weight filtration on the cohomology groups with compact support of X. If k has characteristic zero (which will be the case below), the morphism P is characterized by its values on projective smooth varieties: for any such X, one has
P([X]) = PX(t) =
2 dim(X)
X
p=0
dim(Hp(X))tp.
This implies that for any variety X, the leading term of P([X]) is given by ν(X)t2 dim(X), where, for any variety X, ν(X) is the number of irreducible compo- nents ofX of dimension dim(X).
One has P(L) = P([P1])−1 = t2; for every a > 1, P(La −1) = t2a −1 = t2a(1−t−2a) is invertible in the ring Z[[t−1]][t], with inverse
X
m>1
t−2ma.
Consequently, the morphismP extends uniquely to a ring morphism fromMk to the ring Z[[t−1]][t]. For any variety X, the largest monomial of the power series P([x]) is equal toν(X)t2 dim(X).
Lemma 2.1.1. — Let A be a ring and let P1, . . . , Pr ∈ A[T] be polynomials with coefficients inA. Assume that for everyi, the leading coefficient ofPi is a unit inA, and that for every distinct i, j, the resultant of Pi and Pj is a unit in A. Then, for any polynomial P ∈A[T] and any family (n1, . . . , nr) of nonnegative integers, there exists a unique family (Qi,j) of polynomials in A[T], indexed by pairs integers (i, j) such that 1 6 i 6 r and 1 6 j 6 ni, and a unique polynomial Q ∈ A[T] such that deg(Qi,j)6deg(Pi)−1 for all i, j and
P(T) = Q(T)
r
Y
i=1
Pi(T)ni +
r
X
i=1 ni
X
j=1
Qi,j(T)Y
k6=i
Pk(T)nk
ni
X
j=1
Pi(T)ni−j.
Since the leading coefficient ofPi is a unit,Piis not a zero divisor inA[T]. Observe that the last equality is a decomposition into partial fractions
P(T)
Qr
i=1Pi(T)ni =Q(T) +
r
X
i=1 ni
X
j=1
Qi,j(T) Pi(T)j in the total ring of fractions ofA[T].
Proof. — First assume that r = 1. In this case, the desired assertion follows from considering the Euclidean divisions byP1 of the polynomial P, of its quotient, etc.
P(T) =Q1(T)P1(T) +R1(T)
=Q2(T)P1(T)2+R2(T)P1(T) +R1(T)
=. . .
=Qr(T)P1(T)r+Rr(T)P1(T)r−1+· · ·+R1(T),
whereQ1, . . . , Qr∈A[T] and R1, . . . , Rr are polynomials of degrees6deg(P1)−1.
Now assume r >2. Let i, j be distinct integers in {1, . . . , r}. By the assumption and basic properties of the resultant, there exist polynomialsU, V ∈A[T] such that 1 = U Pi +V Pj. Consequently, the ideals (Pi) and (Pj) generate the unit ideal of A[T]. By induction on n1, . . . , nr, it follows that the ideals (Qk6=iPknk), for 1 6 i6r, are comaximal inA[T]. Therefore, there exist polynomials U1, . . . , Ur ∈A[T]
