JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 10, Number 2, April 1997, Pages 371–392 S 0894-0347(97)00229-4

LATTICE POINTS IN SIMPLE

MICHEL BRION AND MICHELE` VERGNE

1. Introduction

Consider a convex n-dimensional P in Rn with all vertices in the lattice Zn. In this article, we give a formula for the number of lattice points in P ,inthe case where P is simple, that is, if there are exactly n edges through each of P . More generally, for any polynomial function φ on Rn, we express φ(m) m n P ∈XZ ∩

in terms of P (h) φ(x)dx where the polytope P(h) is obtained from P by independent parallel motions of all facets. This extends to simple lattice polytopes the Euler- R Maclaurin summation formula of Khovanskii and Pukhlikov [8] (valid for lattice polytopes such that the primitive vectors on edges through each vertex of P form a basis of the lattice). As a corollary, we recover results of Pommersheim [9] and Kantor-Khovanskii [6] on the coefficients of the Ehrhart polynomial of P . Our proof is elementary. In a subsequent article, we will show how to adapt it to compute the equivariant Todd class of any complete toric variety with quotient singularities. The Euler-Maclaurin summation formula for simple lattice polytopes has been obtained independently by Ginzburg-Guillemin-Karshon [4]. They used the dictio- nary between convex polytopes and projective toric varieties with an ample divisor class, in combination with the Riemann-Roch-Kawasaki formula ([1], [7]) for com- plex manifolds with quotient singularities. A counting formula for lattice points in lattice simplices has been announced by Cappell and Shaneson [2], as a consequence of their computation of the Todd class of toric varieties with quotient singularities.

2. Euler-Maclaurin formula for polytopes Let V be a real vector space of dimension n.LetMbe a lattice in V .Points of M will be called integral points. The vector space V has a canonical Lebesgue measure dx giving volume 1 to a fundamental domain for M. More precisely, let e1,e2,...,en be a basis of V such that

1 2 n M = Ze Ze Ze . ⊕ ⊕···⊕ 1 2 n If x = x e + x e + +xne is a point in V ,thendx = dx dx dxn. 1 2 ··· 1 2 ···

Received by the editors October 3, 1996 and, in revised form, October 16, 1996. 1991 Mathematics Subject Classification. Primary 52B20; Secondary 11P21. Key words and phrases. Simple lattice polytope, Euler-Maclaurin summation formula.

c 1997 American Mathematical Society

371

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We denote by V ∗ the dual vector space to V .IfLis a lattice in a vector space W ,wedenotebyL∗ its dual lattice in the dual vector space W ∗:

L∗ = y W ∗, (x, y) Z, for all x L . { ∈ ∈ ∈ } We will denote M ∗ by N .ThenNis a lattice in V ∗. Let P be a contained in V with nonempty interior P 0.We denote by vol(P )thevolumeofP with respect to the measure dx on V . We denote by the set of closed faces of P .Wehave F n = (k), F F k [=0 where (k) is the set of faces of dimension k.Wehave (n)= P . By definition, the setF (0) of extremal points of P is the set of verticesF of P .Theset{ } (1) is the set of edgesF of P . A face of codimension 1 is called a facet. A facet F F(n 1) is ∈F − the intersection of P with an affine hyperplane y;(uF,y)+λF =0 . We choose the { } normal vector uF V ∗ to the facet F such that P is contained in y,(uF ,y)+λF ∈ { ≥ 0 . In other words, we choose the inward-pointing normal vector uF . This normal } vector is determined modulo multiplication by an element of R+. If f is a face of P , we define (2.1) f = F (n 1); f F . F { ∈F − ⊂ } We denote by f the vector space generated by elements p q where p f and q f.Iffis in h(ki), then f is of dimension k. − ∈ ∈ F h i Definition 2.1. Define Cf to be the convex cone generated by elements p q with − p P and q f.WesaythatCf is the tangent cone to P at its face f. ∈ ∈ The cone Cf is also known as the barrier cone. It contains f as its largest linear subspace. h i

Definition 2.2. Define σf to be the polar cone to Cf :

σf = y V ∗ (x, y) 0 for all x Cf . { ∈ | ≥ ∈ } The cone σf is also known as the normal cone. We have + σf = R uF . F f X∈F If f = P ,thenσf = 0 . { } { } Definition 2.3. A convex polytope P is said to be simple if there are exactly n edges through each vertex.

For example, in R3, a , a pyramid with triangular basis, and a are simple. Definition 2.4. A convex polytope P is a lattice polytope if all vertices of P are in the lattice M. Consider a convex lattice polytope P . We can then choose for each facet F the normal vector uF in the dual lattice N. We normalize uF in order that uF be a primitive element of N,thatis,iftuF N,thentis an integer. ∈

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LetusnumberfacetsofP as F ,F ,...,Fd. Wedenotebyui Nthe normalized 1 2 ∈ normal vector uFi to Fi.Letλi=λFi.ThusPis the intersection of d half-spaces:

P = x V,(ui,x)+λi 0,1 i d . { ∈ ≥ ≤ ≤ } d d Consider h R , h =(h1,h2,...,hd). For h R , define ∈ ∈

(2.2) P (h)= x V,(ui,x)+λi +hi 0,1 i d . { ∈ ≥ ≤ ≤ } Then P (h) is a convex polytope. Moreover, for small h, P (h)andP have the same directions of faces. In particular, P (h) is simple if P is simple and h is small enough. Let C be a closed convex cone in a vector space W with a lattice L.Wedenote by C the vector space spanned by C. The dimension of C is defined to be equal to theh i dimension of the vector space C .WesaythatCis acute (or pointed) if C does not contain any nonzero linearh subspace.i A cone C is said to be polyhedral (respectively, rational polyhedral) if C is generated by a finite number of elements of W (respectively, of L). An acute polyhedral cone C of dimension k is said to be simplicial if C has exactly k edges. If P is a convex lattice polytope, then cones Cf associated to faces f of P , and their polar cones σf are rational and polyhedral. The cone Cf is acute if and only if f is a vertex of P . The cones σf are acute for all f . ∈F A finite collection Σ of rational polyhedral acute cones in V ∗ is called a fan if 1) for any face τ of an element σ Σ, we have τ Σ; 2) for σ, τ Σ, we have σ τ Σ.∈ ∈ ∈ ∩ ∈ The fan is complete if σ Σ σ = V ∗. We denote by Σ(k) the set∈ of cones in the fan Σ of dimension k. S If P is a convex lattice polytope, the collection ΣP = σf ,f is a complete fan, called the normal fan of P ([11]). If f (n k) is{ a face∈F} of codimension k ∈F − of P ,thenσf has dimension k.ThefanΣP depends only on the directions of the faces of P . In particular, the homothetic polytope qP (q a positive integer) has the same fan as P . Definition 2.5. A fan Σ is said to be simplicial if each cone σ Σ is simplicial. ∈ A lattice polytope is simple if and only if its fan is simplicial. Let Σ be a simplicial fan. Let d be the cardinal of Σ(1). We denote elements of Σ(1) as ` ,` ,...,`d.Letui,1 i d, be the primitive integral vector (with 1 2 ≤ ≤ respect to N) on the half-line `i. Definition 2.6. Let σ Σ. We denote by (σ) the subset of the set 1, 2,...,d ∈ E { } consisting of those i such that the half-line `i is an of σ.

The elements ui,i (σ) are linearly independent. Let { ∈E } (2.3) U(σ)= Zui i (σ) ∈EM and T (σ)= σ /U(σ). h i Let k be the dimension of σ.ThenU(σ)=Zk is a lattice in σ = Rk,and k k h i T(σ)=R /Z is a k-dimensional torus.

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Consider the lattice N(σ)=N σ of σ .ThenU(σ) is a sublattice of N(σ), which is usually different from N(σ∩h). Definei h i (2.4) G(σ)=N(σ)/U(σ). Then G(σ) is a finite subgroup of T (σ). The order of G(σ) is called the multiplicity of σ in toric geometry. If τ is a face of σ,wehaveU(τ)= τ U(σ)since (τ) (σ)andthe hi∩ E ⊂E elements ui,i (σ),are linearly independent. Thus we have a natural inclusion T (τ) T (σ).∈E This induces a natural inclusion of the finite group G(τ)inG(σ). ⊂ By the definition of ui as a primitive vector, the group G(σ) is trivial if σ Σ(1). ∈ Definition 2.7. Let Σ be a simplicial fan. We denote by TΣ the set obtained from the disjoint union of the tori T (σ)(σ Σ) by identifying the subsets T (σ τ)of T(σ)andT(τ) for all (σ, τ ) Σ Σ. ∈ ∩ ∈ × We write TΣ = σ Σ T (σ). In TΣ,wehaveT(σ) T(τ)=T(σ τ). A visual ∈ ∩ ∩ way to represent the set TΣ associated to a rational fan Σ is the following. We denote by Q(σ) theS subset

Q(σ)= [0, 1[ui i (σ) ∈EX of σ . It is clear that the map Q(σ) T (σ) (restriction of the quotient map σ h i σ/U(σ)) is an isomorphism.7→ Furthermore, Q(σ) Q(τ)=Q(σ τ). h i→hi ∩ ∩ Consider the subset QΣ of V ∗ defined by

(2.5) QΣ = Q(σ). σ Σ [∈ Then the set QΣ is isomorphic to the set TΣ. Consider the finite subgroup G(σ) T (σ). ⊂ Definition 2.8. The subset ΓΣ of TΣ is defined to be

ΓΣ = G(σ). σ Σ [∈ Thus we can think of ΓΣ as the union of all finite groups G(σ)(σ Σ) with equivalence relations given by G(σ τ)=G(σ) G(τ) T . In particular,∈ the ∩ ∩ ⊂ Σ neutral elements of all the groups G(σ) are identified to a unique element of ΓΣ, denoted by 1. In the identification of TΣ with the subset QΣ of V ∗, the subset ΓΣ of T is identified with Q N. Σ Σ ∩ Example 2.9. Let a, b, c be pairwise coprime integers, and let P (a, b, c)bethe in R3 with vertices O =(0,0,0),A=(a, 0, 0),B=(0,b,0),C=(0,0,c).

The rational fan ΣP associated to P has edges + + + + `1 = R e1,`2=Re2,`3=Re3,`0=R(bce1 cae2 abe3), − − − 3 where (e1,e2,e3) is the canonical basis of (R )∗. Let us list the nontrivial abelian groups G(σ)forσ ΣP.DenotebyG(j,...,k) the group associated to a cone in ∈ ΣP generated by (`j,...,`k). We have G(0, 2, 3) = Z/bcZ,G(0, 3, 1) = Z/caZ,G(0, 1, 2) = Z/abZ

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and G(0, 1) = Z/aZ,G(0, 2) = Z/bZ,G(0, 3) = Z/cZ.

Our set ΓΣ is equal to Γ=(Z/bcZ) (Z/acZ) (Z/abZ), ∪ ∪ where we identify the common subsets Z/aZ, Z/bZ, Z/cZ. Definition 2.10. A simple lattice polytope P is called a Delzant polytope if each cone σ ΣP is spanned by a part of a basis of N, i.e., if G(σ)= 1 for each ∈ {} element σ ΣP . ∈ Equivalently, P is a Delzant polytope if the set ΓΣ constructed from the complete fan ΣP of P is reduced to 1 . This is a very strong hypothesis. As shown by the example above, many lattice{ } simplices are not Delzant. As another example, consider the lattice simplex P (a)inR3 with vertices (0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1,a), where a 2 is an integer. Then P (a) is not Delzant, and the only lattice points in P (a) are≥ its vertices. It follows that P (a) is not a union of Delzant polytopes. Remark 2.11. In the dictionary (that we do not use here) between convex polytopes and toric varieties, a projective toric variety with quotient singularities is associated to a simple lattice polytope P . This toric variety is nonsingular if and only if the polytope P is a Delzant polytope (see [3]). We now define for k 1,2,...,d functions ak on T associated to the d ∈{ } Σ elements `k of Σ(1). The torus T (σ)= σ/U(σ) comes equipped with a basis h i k 2iπyk of its lattice of characters: for each k (σ), we define χσ(g)=e σ if y = j ∈E y uj is an element of σ representing g. j (σ) σ h i The∈E following lemma is obvious. P Lemma 2.12. For any k 1,2,... ,d , there exists a unique function ak : T ∈{ } Σ → C∗ such that k 1) if k/ (σ),thena(g)=1for all g T (σ) TΣ; 2) if k ∈E(σ),thenak(g)=χk(g)if g ∈ T (σ) ⊂ T . ∈E σ ∈ ⊂ Σ k Observe that there exists a unique continuous function ξ on V ∗ which is linear k k on each cone of Σ, and such that ξ (uk)=1andξ (uj) = 0 for all j = k.Letus 6 identify TΣ with the subset QΣ of V ∗.Thenifg TΣis represented by the element k ∈ y Q ,wehaveak(g)=e2iπξ (y). ∈ Σ We can characterize the subset G(σ)ofΓΣ as follows. Lemma 2.13. Let σ Σ. We have ∈ G(σ)= γ Γ ,ak(γ) = 1 for all k/ (σ) . { ∈ Σ ∈E } Now we turn to the definition of Todd operators. Consider the analytic function

∞ z 1 k 1 Bk 2k Todd(z)= =1+ z+ ( 1) − z , 1 exp( z) 2 − (2k)! k − − X=1 where Bk are the Bernoulli numbers. Let a be a complex number. Consider more generally the function z Todd(a, z)= . 1 aexp( z) − −

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This function is analytic in a neighborhood of 0. Consider its Taylor expansion

∞ Todd(a, z)= c(a, k)zk k X=0 for z small. Let h be a real variable. For any a C, consider the operator ∈ ∞ Todd(a,∂/∂h)= c(a, k)(∂/∂h)k. k X=0 We have

1 ∞ k 1 Bk 2k (2.6) Todd(1,∂/∂h)=1+ ∂/∂h + ( 1) − (∂/∂h) , 2 − (2k)! k X=1 while for a =1, 6 1 ∞ k (2.7) Todd(a,∂/∂h)=(1 a)− ∂/∂h + c(a, k)(∂/∂h) . − k X=2 We denote Todd(1,∂/∂h) simply by Todd(∂/∂h). If φ is a polynomial function of h, then Todd(a,∂/∂h)φ(h) is well defined, as (∂/∂h)kφ = 0 for large k. Definition 2.14. Let Σ be a complete simplicial fan. For g T , define ∈ Σ d k Todd(g,∂/∂h)= Todd(a (g),∂/∂hk). k Y=1 Define Todd(Σ,∂/∂h)= Todd(γ,∂/∂h). γ Γ X∈ Σ Recall a version of the Euler-Maclaurin formula. If φ(x) is a polynomial function on R and s t are integers, then ≤ t t+h2 (2.8) φ(k) = Todd(∂/∂h1)Todd(∂/∂h2) φ(x)dx . s h1 h =h =0 k=s Z − ! 1 2 X We will generalize this formula to simple lattice polytopes.

Let P be a simple lattice polytope with d facets. Let M P be the number of lattice points in P ,and M P0 the number of lattice| points∩ | in the interior P 0 of P .LetP(h) be the deformed| ∩ polytope| obtained from P after d independent parallel motions of its facets (formula (2.2)). The main theorem of this article is Theorem 2.15. Let V be a vector space with a lattice M .LetPbe a simple lattice polytope in V , and let Σ be its associated fan. Then, for small h, the volume vol(P (h)) of the deformed polytope P (h) is a polynomial function of h, and we have

M P = Todd(Σ,∂/∂h)vol(P(h)) h , | ∩ | | =0 while 0 M P = Todd(Σ, ∂/∂h)vol(P(h)) h . | ∩ | − | =0 More generally, if φ is a polynomial function on V ,then

I(φ)(h)= φ(x)dx ZP(h)

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is a polynomial function of h for small h,and

φ(m) = Todd(Σ,∂/∂h)I(φ)(h) h , | =0 m M P ∈X∩ while

φ(m) = Todd(Σ, ∂/∂h)I(φ)(h) h . − | =0 m M P 0 ∈X∩ Remark 2.16. If, moreover, P is a Delzant polytope, then the corresponding set ΓΣ is reduced to 1 ,Todd(Σ,∂/∂h) is the usual Todd operator considered by Khovanskii and Pukhlikov{ } [8] and Theorem 2.15 is due to them in this case. We will prove Theorem 2.15 in the next section.

3. Integral formulas As an example of our method, let us first prove identity (2.8). It will be conve- nient to extend the action of Todd operators to exponential functions h ehz,for za small complex number. Indeed, for z small, the series 7→

hz hz 1 ∞ k 1 Bk 2k Todd(∂/∂h)e = e 1+ z+ ( 1) − z 2 − (2k)! k ! X=1 is convergent and equal to Todd(z)ehz. Let [s, t] be an interval. Then we have t etz esz (3.1) ezxdx = . z − z Zs Assume t and s are integers; then

t (t s+1)z kz sz z (t s)z sz 1 e − e = e (1 + e + +e − )=e − , ··· 1 ez k s X= − that is,

t etz esz (3.2) ekz = + . 1 e z 1 ez k s − X= − − On the other hand,

t+h2 tz sz zx h z e h z e (3.3) e dx = e 2 e− 1 . s h z − z Z − 1 Therefore

t+h2 zx Todd(∂/∂h1)Todd(∂/∂h2) e dx s h1 h =h =0 Z − ! 1 2

etz esz = Todd(z) Todd( z) . z − − z Comparing with formula (3.2), we obtain

t+h2 t zx kz Todd(∂/∂h1)Todd(∂/∂h2) e dx = e . s h1 h =h =0 Z − ! 1 2 k=s X

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If we take the Taylor expansion at the origin of this identity in z,weobtain formula (2.8). Our proof of Theorem 2.15 for an n-dimensional lattice polytope P will be based on the same approach. Let y V , and let P V be a polytope (not necessarily C∗ a lattice polytope). Define ∈ ⊂

E(P )(y)= e(x,y)dx. ZP Then the volume of P is the value of E(P )aty=0. If, moreover, P is a lattice polytope, define D(P )(y)= e(m,y) m M P ∈X∩ and D(P 0)(y)= e(m,y). m M P0 ∈X∩ Then the number M P of lattice points in P is the value of D(P )aty= 0. Although E(P )(y),| D∩(P )(|y)andD(P0)(y) are analytic functions of y,“sim- ple” expressions (similar to formulae (3.1) and (3.2)) for E(P )(y), D(P )(y)and D(P0)(y) will be given only when P is simple and y is “generic”. On this for- mula for E(P (h))(y), it will be easy to analyze the action of the Todd operator Todd(Σ,∂/∂h) and to compare it with D(P )(y). Recall that Cf denotes the tangent cone to P at its face f.Choosev f.Set 0∈ + CP(f)=v0+Cf. + As Cf is invariant by translation by vectors in f , the affine cone C (f)doesnot h i P depend of the choice of v0 f. We call it the inward pointing affine cone tangent + ∈ + to P at f.ThusCP(f) contains P and P = f CP (f). Let ∈F T C−(f)=v Cf P 0− be the outward pointing affine cone at f. If E is a subset of V ,wedenotebyχE its characteristic function. Proposition 3.1. Let P be a convex polytope with non empty interior P 0.Then we have the identities (1)

dim f χP = ( 1) χC+(f), − P f X∈F (2)

n dim f ( 1) χ 0 = ( 1) χ , P C−(f) − − P f X∈F (3)

dim f χ 0 = ( 1) χCf . { } − f X∈F

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Proof. A version of these identities can be found in [5]. We give another proof, based on the Euler identities

(3.4) ( 1)dim f =1 − f X∈F and, for any point m in the boundary of P ,

(3.5) ( 1)dim f =0. − f ,m f ∈FX∈ Let m be an arbitrary point of V .Wehavetoprovetherelations (1) If m P ,then ∈ ( 1)dim f =1. − f ,m C+(f) ∈F X∈ P (2) If m/P,then ∈ ( 1)dim f =0. − f ,m C+(f) ∈F X∈ P (3) If m P 0,then ∈ ( 1)dim f =( 1)n. − − f ,m C−(f) ∈F X∈ P (4) If m/P0,then ∈ ( 1)dim f =0. − f ,m C−(f) ∈F X∈ P (5) ( 1)dim f =1. − f ,0 C ∈FX∈ f (6) If m =0,then 6 ( 1)dim f =0. − f ,m C ∈FX∈ f + First observe that m P if and only if m CP (f) for all f . So assertion (1) is just the Euler identity∈ (3.4), and the same∈ holds for (5). For∈F (3), if m P 0, ∈ then the unique face f such that m CP−(f)isf=P. Let us prove (4). Let m/P0. Consider∈ the convex hull H of P and m.Let ∈ (H)bethesetoffacesofH.Let m(H) be the set of faces of H containing m. F F Write (H) as the disjoint union m(H) n(H). Using relations (3.4), (3.5) for the polytopeF H and its boundaryF point m∪F,weobtain ( 1)dim g =1.It g n(H) − is easy to see that the faces g in (H) are faces of P and∈F that these are all the n P faces f of P such that m/C (fF). Thus we have ( 1)dim f =1. P− f ,m/C−(f) ∈ P − Subtracting the Euler identity for P , we obtain (4). ∈F ∈ P Let us prove (2). Let m/P. Consider the convex hull H of P and m.LetR be the closure of H P .Theset∈ Ris not convex in general; however, it can be contracted to m. Therefore,\ the Euler identities holds for R.Let (R)betheset F

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of faces of R.Let m(R) be the set of faces of R containing m.Write (R)asthe F F disjoint union m(R) n(R). Using relations (3.4), (3.5) for the polytope R and its vertex m,weobtainF ∪F ( 1)dim g = 1. It is easy to see that the faces g in g n(R) ∈F − + n(R) are faces of P and that these are all the faces f of P such that m/CP(f). F P dim f ∈ Thus we have f ,m/C+(f)( 1) = 1. Subtracting the Euler identity for P , P − we obtain relation∈F (2). ∈ P Finally, let us prove (6). We may assume that 0 is an interior point of P .Let + m= 0. Choose a small positive number t. Recall that CtP (tf) denotes the inward pointing6 affine cone for the face tf of tP ,wheretis a small positive number. Then + there exists t sufficiently small such that m Cf if and only if m CtP (tf). Thus the last relation is deduced from relation (2)∈ by considering the polytope∈ tP for t sufficiently small.

To a point m of V , we associate its δ-measure δ(m), defined as follows. For any continuous function φ on V ,wehave(δ(m),φ)=φ(m). If S is a discrete subset of

V ,wedenotebyδ(S)= s Sδ(s)itsδ-measure. The following proposition∈ follows immediately from Proposition 3.1. P Proposition 3.2. Let P be a convex lattice polytope. We have the equalities (1)

δ(M P )= ( 1)dim f δ(M C+(f)), ∩ − ∩ P f X∈F (2)

n 0 dim f ( 1) δ(M P )= ( 1) δ(M C−(f)), − ∩ − ∩ P f X∈F (3)

dim f δ( 0 )= ( 1) δ(M Cf ). { } − ∩ f X∈F We will consider Fourier transforms of the measures δ(M Cf ). They make sense in the framework of generalized functions. We will use the∩ function notation Θ(y) for a generalized function Θ on V ∗, although the value of Θ at a particular point y may not have a meaning. We denote by Θ(y)φ(y)dy the value of Θ on a test V ∗ density φ(y)dy. We will say that Θ is smooth on an open subset U of V ∗ if there exists a smooth function θ(y)onUsuch thatR Θ(y)φ(y)dy = θ(y)φ(y)dy for V ∗ V ∗ all test functions φ with compact support contained in U. Then the value of Θ at R R y U is defined to be θ(y). If there exist two smooth functions f,g on V ∗, with g ∈not identically 0, such that the equation g(y)Θ(y)=f(y) holds in the space of generalized functions on V ∗, then Θ is smooth on the open set U = y,g(y) =0 and Θ(y)=f(y)/g(y)onU. { 6 } Consider for example V = . Consider the discrete measure δ( )= δ(n). R Z n Z We denote its Fourier transform by Θ(y)= eiky . This means that the∈ gener- k Z alized function Θ(y) is the limit in the space of∈ generalized functions ofP the smooth iky P functions k K e . We have thus for a smooth test function φ on R | |≤ P Θ(y)φ(y)dy = eiky φ(y)dy. R k R Z X∈Z Z

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Clearly (1 eiy)( eiky )=0sothatΘ(y) is supported on 2π .Infact, k Z Z Poisson summation− formula∈ is P 1 (2π)− Θ(y)φ(y)dy = φ(2πk). R k Z X∈Z Let + Z = 0, 1, 2, 3,... . { } Let a be a complex number of modulus 1. Consider the discrete measure h(a)= anδ(n) and its Fourier transform Θ+(y)= aneiny.Wehavethe n Z+ a n Z+ equality∈ ∈ P P

(3.6) (1 aeiy)Θ+(y)=1. − a + Thus the generalized function Θa (y) is smooth outside the set iLoga +2πZ,and for y/iLoga +2πZ, ∈ 1 (3.7) Θ+(y)= . a 1 aeiy − + iy In particular, Θa (y) is a rational function of e . We will generalize this formula to higher dimensions. y We call a meromorphic function Φ(y)onV∗ a rational function of e if for some n C basis (e1,... ,en)ofN,writingy= i=1 yi∗ei,Φ(y1∗,... ,yn∗) is a rational function y y of e 1∗ ,... ,e n∗. This does not depend on the choice of the basis of N. P Definition 3.3. Let C be a rational polyhedral convex cone in V .Denoteby Θ(C)(y) the Fourier transform of δ(M C): ∩ Θ(C)(y)= ei(m,y). m M C ∈X∩ Proposition 3.4. Let C be a rational polyhedral convex cone in V ; let W be the largest linear subspace contained in C. (1) The generalized function Θ(C) is supported on a discrete union of translates of W ⊥. (2) If C is acute (i.e. W =0), then there exists a meromorphic function φ on V such that for y outside a union of a discrete set of affine hyperplanes C∗ Θ(C)(y)=φ(iy). The order at 0 of the function φ is at least n. Moreover, φ(y) is a rational function of ey. − Proof. Observe that W is a rational subspace of V . Moreover, for any m M W , 0 ∈ ∩ (1 ei(m0,y))Θ(C)(y)=0asM Cis invariant under translation by elements of M − W . Hence the support of Θ∩ is contained in the set ∩ y V ∗ (y,m0) 2πZ for all m0 M W { ∈ | ∈ ∈ ∩ } and this set is a discrete union of translates of W ⊥. It is enough to prove (2) when C is simplicial. Indeed, we can always subdivide C by simplicial cones Cj ,andthenΘ(C) is a sum (with signs) of the generalized functions Θ(Cj ). Now consider a simplicial cone C V , the intersection of n ⊂ k distinct hyperplanes (wk,x) 0 with wk in the lattice N of V ∗.Letw Vbe the ≥ ∈

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k dual basis of wk.Anelementx Vis written as x = ηkw , with ηk =(wk,x). Thus ∈ P + k C = R w . k M n k k Let L = k=1 Zw be the lattice of V spanned by w .Wehave + k L L C= Z w. ∩ k M Let χ be a multiplicative character of L. Consider the discrete measure h(C, χ, L)= χ(m)δ(m) m L C ∈X∩ and its Fourier transform Θ(C, χ, L)(y)= χ(m)ei(m,y). m L C ∈X∩ Formula (3.6) gives n k (1 χ(wk)ei(w ,y))Θ(C, χ, L)(y)=1. − k Y=1 This equation implies that Θ(C, χ, L) is smooth outside the zeroes of the analytic n k i(wk,y) function g(y)= k=1(1 χ(w )e ). This zero set is a union of a discrete set of hyperplanes. Thus we− obtain Q Lemma 3.5. For y outside a union of a discrete set of hyperplanes, 1 Θ(C, χ, L)(y)= . n (1 χ(wk)ei(wk,y)) k=1 − With the notation as above, a basisQ of the dual lattice L∗ to L consists of w1,... ,wn.LetT=V∗/L∗ = V ∗/( k Zwk). Then T is an n-dimensional torus. Characters of L are parametrized by T :anelementg Tgives a character χg by 2iπ(x,y) L ∈ writing χg(x)=e if x L and if y V ∗ represents g V ∗/L∗. ∈ ∈ ∈ Consider the finite subgroup G = N/( k Zwk) T. Recall that N is the dual lattice to M.Thus,forx L, ⊂ ∈ L

χg(x)=0, if x/M, ∈ g G X∈ χg(x)= G, if x M. || ∈ g G X∈ We obtain 1 δ(M C )= G− h(C, χg,L). ∩ | | g G X∈ From Lemma 3.5, we obtain Lemma 3.6. Let C V be a rational simplicial cone. Then, for y outside a union of a discrete set of hyperplanes,⊂ we have

1 1 Θ(C)(y)= G− . | | n (1 χ (wk)ei(wk,y)) g G k=1 g X∈ − Q

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This explicit formula for simplicial cones implies Proposition 3.4. To check that φ(y) is a rational function of ey, let us state another formula for Θ(C)(y). Let α1,... ,αn be the primitive vectors of M on the edges R+w1,... ,R+wn of C. Then the set n k S(C)=M tkα 0 tk < 1 ∩ | ≤ (k ) X=1 is finite and n k Θ(C)(y) (1 ei(α ,y))= ei(m,y) . · − k=1 m S(C) Y ∈X

In the sequel, we will say that a property holds for generic y V ∗ if there exists some nonzero analytic function g such that the property holds∈ for all y with g(y) =0. Let6 P be a lattice polytope. Consider the Fourier transform of identities (1), (2), and (3) of Proposition 3.2. By definition, the Fourier transform of δ(M P )is ∩ the function y D(P )(iy). We choose an integral element v0 M on each face f 7→ + ∈ of P .ThenM CP(f)=v0+(M Cf), while M CP−(f)=v0 (M Cf). We obtain the following∩ proposition. ∩ ∩ − ∩

Proposition 3.7. We have the equality of generalized functions: (1)

dim f i(v0,y) D(P )(iy)= ( 1) e Θ(Cf )(y), − f X∈F (2)

n 0 dim f i(v0,y) ( 1) D(P )(iy)= ( 1) e Θ(Cf )( y), − − − f X∈F (3)

dim f 1= ( 1) Θ(Cf )(y). − f X∈F For each vertex s, consider the acute cone Cs. Proposition 3.4 shows that there exists a meromorphic function φ(C )onV (in fact, a rational function of ey)such s C∗ that Θ(Cs)(y)=φ(Cs)(iy)forygeneric. Proposition 3.8. We have the equalities of meromorphic functions on V : C∗ (1)

(s,y) D(P )(y)= e φ(Cs)(y), s (0) ∈FX (2)

0 n (s,y) D(P )(y)=( 1) e φ(Cs)( y). − − s (0) ∈FX

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Furthermore (3)

φ(Cs)(y)=1. s (0) ∈FX Proof. Consider formulae (1) and (2) of Proposition 3.7. When f is a face of ∈F strictly positive dimension, the cone Cf contains the nonzero linear vector space f ,andΘ(Cf) is supported on a union of affine spaces of dimension strictly less h i than n. Thus we obtain the identities above for y iV ∗ generic. The last identity is obtained from relation (3). ∈ Consider a simple lattice polytope P with associated fan Σ. In this case, we have explicit expressions for the meromorphic functions φ(Cs). Indeed Cs is the s intersection of the n half-spaces (uF ,y) 0,F . The elements uF belong to N. ≥ ∈F Let σs Σ be the polar cone to Cs. Recall the definitions of U(σ)andofG(σ)by ∈ s formulae (2.3) and (2.4). The lattice U(σs) is the lattice with Z-basis (uF ,F ) F s ∈F and the group G(σs) is the group N/U(σs). Let (ms ,F ) be the dual basis to s ∈F (uF ,F ). Applying Lemma 3.6, we obtain ∈F 1 1 φ(C )(y)= . s F (mF ,y) G(σs) s (1 χg(m )e s ) g G(σ ) F s | | ∈Xs ∈F − This leads to the following explicit formulaeQ for D(P )andD(P0)asasumof meromorphic functions attached to each vertex s of P . These formulae are the generalization of formula (3.2) in the 1-dimensional case.

Proposition 3.9. For y V ∗ generic, we have ∈ C e(s,y) 1 e(m,y) = , F (mF ,y) G(σs) s (1 χg(m )e s ) m M P s (0) g G(σ ) F s ∈X∩ ∈FX | | ∈X s ∈F − while Q e(s,y) 1 e(m,y) == ( 1)n . F (mF ,y) − G(σs) s (1 χg(m )e s ) m M P 0 s (0) g G(σ ) F s − ∈X∩ ∈FX | | ∈X s ∈F − Let P be a simple polytope in V . We now analyzeQ the continuous version

E(P )(y)= e(x,y)dx ZP of D(P ). We do not assume that P is a lattice polytope, as we will have to consider deformed polytopes P(h). Let s be a vertex of P. We choose inward pointing s F s normal vectors (uF ,F ) and dual elements (ms ,F ). Then the volume ∈F F s ∈F F of the parallelepiped constructed on (ms ,F )isequalto det(ms ) F s .The following formula expresses the analytic function∈F E(P ) as a sum| over all| vertices∈F of meromorphic functions attached to each vertex s of P . This is the n-dimensional analogue of formula (3.1). Proposition 3.10. Let P be a simple polytope. Let y V be such that (mF ,y)= C∗ s 0 for all vertices s and all F s.Then ∈ 6 ∈F (x,y) n (s,y) F 1 s e dx =( 1) e ( det(ms ) F ) F . P − | | ∈F s (m ,y) Z s (0) F s ∈FX ∈F Q

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Proof. It is possible to give a direct argument for this proposition using Proposi- tion 3.1 and explicit formulas for Fourier transforms of characteristic functions of simplicial cones. However, we can also deduce the value of E(P ) from the value of D(P ) by a limit argument using Riemann sums to evaluate an integral. Indeed, it is sufficient to prove this formula for lattice polytopes (choosing lattice M with smaller and smaller fundamental domain). We have

n (m,y) E(P )(y) = lim q− e q →∞ m (M/q) P ∈ X ∩ when q becomes a large integer. We replace M by M/q in the formula of Proposition 3.9. We obtain

(s,y) n (m,y) e 1 q− e = . F (mF ,y/q) G(σs) s q(1 χg(m )e s ) m (M/q) P s (0) g G(σ ) F s ∈ X ∩ ∈FX | | ∈X s ∈F − Q We see that only the trivial term g = 1 in each group G(σs) will contribute to 1 the limit at q = , and we obtain our proposition, as we observe that G(σs) − is ∞ F | | the absolute value of det(ms )F s . ∈F

In particular, we have vol(P ) = limt 0 E(P )(ty), and we obtain that for any generic y, → n n ( 1) F (s, y) s (3.8) vol(P )= − ( det(ms ) F ) F . n! | | ∈F s (m ,y) s (0) F s ∈FX ∈F Q Let P be a simple lattice polytope with facets F1,F2,...,Fd.Leth=(h1,...,hd) be a small parameter of deformation.

Lemma 3.11. Let φ be a polynomial function on V .Forh Rdsmall, the function ∈ I(φ)(h)= P(h)φ(x)dx is polynomial in h. Proof. ConsiderR

E(y)(h)=E(P(h))(y)= e(x,y)dx. ZP(h) We compute E(y)(h) using Proposition 3.10. Let s be a vertex of P.Letσsbe the polar cone to Cs.Wehave

+ σs= R uj. j,F s Xj ∈F s The subset of 1, 2,...,d consisting of those j with Fj is the set (σs)of { } j ∈F E Definition 2.6. We denote by (ms,j (σs)) the dual basis to (uj,j (σs)). When h is small, the point s(h)givenby∈E ∈E

j s(h)=s hjm − s j (σ ) ∈EXs is a vertex of P (h). Thus, for generic y,

(3.9) E(y)(h)= E(s, y)(h), s (0) ∈FX

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where j n (s hj m ,y) ( 1) e − j (σs ) s E(s, y)(h)= − ∈E . G(σ ) P j s j (σ )(ms,y) | | ∈E s The function E(y)(h) is analytic in y. ConsideringQ the Taylor expansion of t E(ty)(h)att= 0 , we obtain for every k and generic y, 7→ j n+k n ((s, y) hj(m ,y)) 1 k ( 1) j (σs) s − ∈E (3.10) (x, y) dx = − j . k! P (h) (n + k)! G(σs) (ms,y) Z s (0) P j (σs) ∈FX | | ∈E k Q The polynomial behaviour in h of P (h)(x, y) dx is apparent from this formula. As this result holds for any generic y and any k, we obtain our lemma. R Remark that the Todd operator Todd(a,∂/∂h) is well defined on functions h ehz provided z is sufficiently small. We rewrite also formula (3.9) as 7→ (3.11) E(y)(h)= E(s, y)(h) s (0) ∈FX with

(s,y) j e 1 n hj(ms,y) E(s, y)(h)=( 1) e− j . −   G(σs) (m ,y) j (σ ) j (σ ) s ∈EY s | | ∈EY s This shows that Todd(Σ,∂/∂h) is well defined on

E(y)(h)= e(x,y)dx ZP(h) provided that y is sufficiently small. Theorem 3.12. Let P be a simple lattice polytope, and let Σ be the associated fan. If y V ∗ is small, then ∈ C Todd(Σ,∂/∂h) e(x,y)dx = e(m,y), ZP(h) ! h=0 m M P ∈X∩

while

Todd(Σ, ∂/∂h) e(x,y)dx = e(m,y). − ZP(h) ! h=0 m M P 0 ∈X∩

Consider the Taylor expansion of both members of the first equality above at y =0.Weobtain

Todd(Σ,∂/∂h) (x, y)kdx = (m, y)k ZP(h) ! h=0 m M P ∈X∩

for all y V ∗ and k N. Thus Theorem 3.12 implies Theorem 2.15. ∈ C ∈ Proof. Consider formula (3.11) for the function E(y)(h). For s avertexofP,the function E(s, y)(h) depends only of the variables hj such that j (σs). Let k be k ∈E such that k/ (σs). From formula (2.7), we see that Todd(a ,∂/∂hk)E(s, y)(h)= k ∈E k 0ifa = 1, while if a =1,wehaveTodd(1,∂/∂hk)E(s, y)(h)=E(s, y)(h). By 6

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k Lemma 2.13, if γ Γ is not in G(σs), then there is k/ (σs) such that a (γ)is ∈ Σ ∈E not 1. Thus Todd(γ,∂/∂h)E(s, y)(h)=0ifγ/G(σs). We obtain ∈ Todd(Σ,∂/∂h)E(s, y)(h)= Todd(γ,∂/∂h)E(s, y)(h), γ G(σ ) ∈Xs and for γ G(σs), ∈

j Todd(γ,∂/∂h)E(s, y)(h)= Todd(a (γ),∂/∂hj) E(s, h).   j (σ ) ∈EYs   We have

uh u Todd(a,∂/∂h)e h = Todd(a, u)= . | =0 1 ae u − − We obtain, for γ G(σs), ∈ Todd(γ,∂/∂h)E(s, y)(h) h | =0 j (s,y) n (ms,y) e 1 =( 1) − j j − (1 aj(γ)e(ms,y)) G(σs) (m ,y) j (σ ) j (σ ) s ∈EY s − | | ∈EY s e(s,y) = j . j (ms,y) G(σs) j (σ )(1 a (γ)e ) | | ∈E s − j j j By the definition of a Q, a (γ)=χγ(ms). Comparing with the first formula of Proposition 3.9, we obtain the first formula of our theorem. By a similar proof, we obtain the second formula.

4. The coefficients of the Ehrhart polynomial Let P be a convex lattice polytope in V with nonempty interior P 0. Consider, for q a positive integer, the polytope qP.Letφbe a function on V .Let

i(φ, P )(q)= φ(m) m M (qP) ∈X∩ and i(φ, P 0)(q)= φ(m). m M (qP 0) ∈ X∩ As a consequence of Proposition 3.8, let us prove the following generalization of a well-known theorem of Ehrhart.

Proposition 4.1. If φ is a homogeneous polynomial function of degree k,then the functions q i(φ, P )(q) and q i(φ, P 0)(q) are polynomial of degree n + k. Moreover, we have7→ 7→ i(φ, P 0)(q)=( 1)n+ki(φ, P )( q) − − and i(φ, P )(0) = φ(0).

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Proof. Observe that the vertices of qP are the qs (s avertexofP) and that the tangent cone at qs to qP is Cs. Therefore, using Proposition 3.8, we obtain for generic y:

(m,y) (qs,y) e = e φ(Cs)(y) m M (qP) s (0) ∈ X∩ ∈FX and (m,y) n (qs,y) e =( 1) e φ(Cs)( y). − − m M (qP 0) s (0) ∈ X∩ ∈FX Now replace y by ty for small nonzero t and consider the expansion into Laurent j s series in t.Asφ(Cs)(y) is of order at least n,wehaveφ(Cs)(ty)= t a (y), − j n j where as(y) are homogeneous rational functions of degree j.Wethusseethat≥− j P k+n 1 k 1 j j s (m, y) = q (s, y) ak j (y) k! j! − m M (qP) s (0) j=0 ∈X∩ ∈FX X and that k+n 1 k n 1 j j k j s (m, y) =( 1) q (s, y) ( 1) − ak j (y). k! − j! − − m M (qP 0) s (0) j=0 ∈ X∩ ∈FX X We thus obtain the polynomial behaviour in q of i(φ, P )(q)andofi(φ, P 0)(q)for the polynomial function φ(x)=(x, y)k and the first identity as well. As this result holds for all y and k, it holds for all polynomial functions on V . We also obtain, for q = 0 and for φ(x)=(x, y)k,that tk i(φ, P )(0) k! X is equal to the Laurent series expansion of s (0) φ(Cs)(ty). Thus we obtain the second identity from formula (3) of Proposition∈F 3.8 as we have identically P s (0) φ(Cs)=1. ∈F P For φ = 1, the polynomial i(φ, P )(q) is called the Ehrhart polynomial. We denote it simply by i(P ). We write n k i(P )(q)= M (qP) = q ak(P ). | ∩ | k X=0 It follows from Proposition 4.1 that the term a0(P ) is equal to 1. Let us give, for example, the values of ak(P )forthesimplexP(a, b, c) considered in Example 2.9. Let p and q 1 be two coprime integers. Let s(p, q)bethe Dedekind sum, defined by ≥ q i pi (4.1) s(p, q)= , q q i=1 X    1 where ((x)) = 0 if x is integral and ((x)) = x [x] 2 otherwise. We have − −

a3 (P )=abc/6,a2(P)=(ab + bc + ca +1)/4,a0(P)=1,

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while a1(P )isequalto 1 ab bc ca 1 + + + +(a+b+c)/4 s(bc, a) s(ca, b) s(ab, c)+3/4. 12 c a b abc − − −   This formula, originally due to Mordell, has been generalized recently by Pom- mersheim [9] and Kantor-Khovanskii [6]: more generally, they computed the coeffi- cient an 2(P ) of the Ehrhart polynomial. Let us show how to deduce their results from Theorem− 2.15, which gives (in principle) an explicit formula for the Ehrhart polynomial of any simple polytope. Let P be a simple lattice polytope, and let Σ be its fan. Consider the Todd operator Todd(Σ,∂/∂h). We write it as the sum of its homogeneous components

∞ Todd(Σ,∂/∂h)= Ak(∂/∂h). k X=0 Lemma 4.2. We have

an k(P )=Ak(∂/∂h)vol(P(h)) h=0. − | Proof. Let q be a positive integer. We have

M (qP) = T(Σ,∂/∂h)vol((qP)(h)) h . | ∩ | | =0 Formula (3.8) shows that for any generic y,

n j n ( 1) ((qs, y) j (σ ) hj(ms,y)) − ∈E s (4.2) vol((qP)(h)) = − j . n! G(σs) (ms,y) s (0) P j (σs) ∈FX | | ∈E Thus the lemma follows. Q Let us write Todd(Σ,∂/∂h) = Todd(∂/∂h)+R(∂/∂h) with d Todd(∂/∂h)= Todd(∂/∂hj) j=1 Y and R(∂/∂h)= Todd(γ,∂/∂h). γ Γ ,γ=1 ∈XΣ 6 We write

∞ Todd(∂/∂h)= Tk(∂/∂h) k X=0 and

∞ R(∂/∂h)= Rk(∂/∂h), k X=0 where Tk,Rk are homogeneous polynomials of degree k. We have thus

an k(P )=mn k(P)+rd k(P) − − −

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with

mn k(P )=Tk(∂/∂h) vol(∆(h)) h=0 − · | and

rn k(P )=Rk(∂/∂h) vol(∆(h)) h=0. − · | Lemma 4.3. We have

1 d T (∂/∂h)=I, T (∂/∂h)= ∂/∂h , 0 1 2 j j=1 X while

R0(∂/∂h)=0,R1(∂/∂h)=0. Proof. The first two equalities follow readily from formula (2.6). The groups G(σ) are trivial for σ Σ(1). Thus there is no element γ of Γ with ∈ Σ ak(γ) = 1 for all k but one, and the last equalities follow from formula (2.7).

More generally, by the same argument, we obtain the following Lemma 4.4. Assume G(σ)= 1 for all cones σ Σ of dimension at most K. {} ∈ Then Rk(∂/∂h)=0and hence an k(P )=mn k(P)for all k K. − − ≤ Let f be a face of P. Consider the vector space f and its lattice M f . We denote∈F by vol(f) the volume of the face f with respecth i to the Lebesgue measure∩h i on f determined by this lattice. Leth i f be a face of codimension 2. Then f is the intersection of two facets. To simplify notation, we assume that f = F F .Thenσf (the polar cone of Cf ) 1 ∩ 2 is generated by two normal vectors u1 and u2 to F1 and F2. The elements u1, u2 generate a sublattice U(σf )ofN σf .Wehave ∩h i G(σf)=(N σf )/U(σf ). ∩h i As u1 is primitive, we can always choose a Z-basis n1,n2 of N σf such that ∩h i u1 = n1 and u2 = pn1 + qn2 with 1 p q. The integers (p, q) are coprime. We have q = G(σ) . Recall formula (4.1)≤ for≤s(p, q). | | Definition 4.5. Let f be a face of codimension 2. Using notation above, define 1 1 µ(f)= +s(p, q). 4−4q Proposition 4.6 ([9], [6]). We have

mn(P )=vol(P),rn(P)=0,

1 mn 1(P)= vol F, rn 1(P)=0, − 2 − F (n 1) ∈FX−

rn 2(P)= µ(f)vol(f). − f (n 2) ∈FX−

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Proof. Let σ be an element of the fan Σ. Define

e(σ,∂/∂h)= ∂/∂hj. j (σ) ∈EY From formula (4.2), we readily obtain

Lemma 4.7. Let f be a face of P ,andσf Σthe corresponding cone. Then we have ∈ 1 e(σf ,∂/∂h)vol(P(h)) h = G(σf ) − vol(f ). | =0 | | The values of an(P ), an 1(P ) are well known and easily obtained from Lemmas − 4.3 and 4.7. It remains to obtain the value of rn 2(P ). Consider the subset Γ2 of − ΓΣ defined by

Γ2 = G(σ). σ Σ(2) ∈[ Let Γ0 =Γ 1 .Wehave 2 2−{ } R2(∂/∂h)= Todd(γ,∂/∂h). γ Γ X∈ 20 For σ Σ(2), let G(σ)0 = G(σ) 1 .Then Γ0 is the disjoint union of the sets ∈ −{ } 2 G(σ)0 when σ varies in Σ(2). We study

R2(σ,∂/∂h)= Todd(γ,∂/∂h) γ G(σ) ∈X 0 + + for σ Σ(2). To simplify notation, we write σ = R u1 + R u2, and as before we ∈ choose a Z-basis n1,n2 of σ N such that u1 = n1,u2 = pn1 + qn2. Elements h i∩ of G(σ)=(σ N)/U(σ) are represented by elements jn2 with 0 j

q 1 − 2πijp/q 1 2iπj/q 1 R (σ,∂/∂h)= (1 e− )− (1 e )− (∂/∂h )(∂/∂h ). 2  − −  1 2 j=1 X   If f isthefaceofP such that σ = σf ,wehavebyLemma4.7, 1 (∂/∂h )(∂/∂h )vol(P(h)) h = q− vol(f ). 1 2 | =0 By formula (18a) of [10], we have q 1 2iπjp/q 1 2iπj/q 1 q− (1 e )− (1 e− )− − − j=1 X q 1 1 1 = s( p, q)+ − =s(p, q) + . − − 4q − 4q 4 Thus we obtain

R2(σf )vol(P(h))h=0 = µ(f)vol(f). Summing over all faces of codimension 2, we obtain the desired formula for rn 2(P ). −

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References

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