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Regularized determinants of the Laplacian for cofinite Kleinian groups with finite-dimensional unitary representations

For cofinite Kleinian groups (or equivalently, finite-volume three-dimensional hyperbolic orbifolds) with finite-dimensional unitary representations, we evaluate the regularized determinant of the Laplacian using W. Muller's regularization. We give an explicit formula relating the determinant to the Selberg zeta-function.


INTRODUCTION
The regularized determinant of the Laplacian has been well studied on Riemann surfaces. In the case of compact Riemann surfaces, D'Hoker and Phong [DP86], and Sarnak [Sar87] related the regularized determinant to the Selberg zeta-function.
For non-cocompact cofinite Fuchsian groups (or equivalently, finite-area non-compact Riemann surfaces with elliptic fixed points) Venkov, Kalinin, and Faddeev [VKF73] defined a regularized determinant for the Laplacian ∆ and related the determinant to the Selberg zeta-function. They regularized the trace of the resolvent kernel using the theory of Krein's spectral shift function [Kre53,BK62,Yaf92].
Efrat [Efr88,Efr91] defined a regularized determinant for cofinite torsion-free Fuchsian groups with singular characters, and related it to the Selberg zeta-function. His regularization was essentially based on the Selberg trace formula. Efrat's paper gave rise to an interesting question: Can the regularized determinant be defined cleanly in terms of general operator theory? In the compact case, the answer is yes. Here zeta-regularization is defined in terms of the heat kernel, which is of trace-class. In the non-compact case the heat kernel is not even Hilbert-Schmidt.
Müller's regularization can be used for elliptic operators on non-compact manifolds. In [Mül92], Müller evaluates his determinant for the case of the Laplacian for finite-area surfaces with hyperbolic ends (a class of surfaces that includes Riemann surfaces), and relates the determinant to Efrat's regularization (and hence to the Selberg zeta-function).
In [Par05] J. Park studies a closely related problem. He studies eta-invarients of Dirac operators, and relates the regularized determinant of the Dirac Laplacian to the Selberg zeta-function for odd-dimensional hyperbolic manifolds with cusps. Park also uses the regularized determinant to extract information about Selberg zeta-function.
Regularized determinants have also been evaluated in the case of infinite volume Riemann surfaces, by Borthwick, Judge, and Perry [BJP].
In this paper, we evaluate Müller's relative determinant of ∆ for the case of finite-volume threedimensional hyperbolic orbifolds with finite-dimensional unitary representations. Or in other words the Laplacian acting on the Hilbert space of χ−automorphic (χ is a finite-dimensional unitary representation) functions on hyperbolic three-space. We relate the determinant to the Selberg zeta-function using the appropriate version of the Selberg trace formula (proved previously in [Fri05a,Fri05b]).
Main Results. Next we define some of the basic notions needed to state our main results. A Kleinian group is a discrete subgroup of PSL(2, C) = SL(2, C)/ ± I. Each element of PSL(2, C) is identified with a Möbius transformation, and has a well-known action on hyperbolic three-space H 3 and on its boundary at infinity -the Riemann sphere P 1 (see [EGM98, Section 1.1]) . A Kleinian group is cofinite iff it has a fundamental domain F ⊂ H 3 of finite hyperbolic volume.
We use the following coordinate system for hyperbolic three-space, H 3 ≡ {(x, y, r) ∈ R 3 | r > 0} ≡ {(z, r) |z ∈ C, r > 0} ≡ {z + rj ∈ R 3 | r > 0}, with the hyperbolic metric ds 2 ≡ dx 2 + dy 2 + dr 2 r 2 , and volume form dv ≡ dx dy dz r 3 . The Laplace-Beltrami operator is defined by and it acts on the space of smooth functions f : Suppose that Γ is a cofinite Kleinian group and χ ∈ Rep(Γ, V ) (Rep(Γ, V ) is the space of finitedimensional unitary representations of Γ in V ). Then the Hilbert space of χ−automorphic measurable functions is defined by Here F is a fundamental domain for Γ in H 3 , and , V is the inner product on V. Finally, let ∆ = ∆(Γ, χ) be the corresponding positive self-adjoint Laplace-Beltrami operator on H(Γ, χ) .
Next we briefly describe the motivation for the functional regularized determinant. Let f (s) = ′ m∈D λ −s m be a sum over the non-zero eigenvalues of ∆ . Then formally and With this formal calculation in mind, one can think of e −f ′ (0) as the regularized determinant. Now, typically, f (0) does not even converge, but f (s) does converge for Re(s) sufficiently large. Analytic continuation gives a possible value for f ′ (0).
The formal argument above works well when the orbifold in question is compact. In the noncompact case we compare ∆ with another self adjoint operator, ∆ 0 , the self-adjoint extension of the operator 3 for the definitions of the notations used above. Define the projection (onto the constant Fourier coefficient) Once again, see §3.3 for the definitions of the notations used above.
The analogue of f (s) for non-compact spaces, following Müller, is the relative zeta-function Here Re(s) > 2. Note that in order for the integral above to converge, we need to know the asymptotics of tr e − ∆ t − e − ∆0 t p 0 − dim ker ∆ at both t = 0 and t = ∞. These asymptotics are given in Lemma 4.3.
Our main article of interest is the regularized characteristic polynomial, det ∆ −(1 − s 2 ) , which we call the regularized determinant. For Re(s) > 2 define and following [Sar87], we define Our main results are: Theorem. Let Γ be a cofinite Kleinian group with one cusp at infinity, and let χ ∈ Rep(Γ, V ) . Then there exists constants C 2 , C 3 , D 1 , depending on Γ and χ (they are explicitly determined in §5) such that Here Z(s, Γ, χ) is the Selberg zeta-function (see §5.1), Ω(s) is a meromorphic function (see Equation 5.3).
The constant Y comes about from the decomposition of F = F Y ∪ F Y into a compact set F Y and a noncompact cusp sector F Y (see §3.1 for more details).
The rest of the notation is defined in §5.
Corollary. Let Γ be a cofinite torsion-free Kleinian group with one cusp at infinity, and let χ ∈ Rep(Γ, V ) be a regular character. Then The constant L(Λ ∞ , ψ) comes about from regularity at a cusp, and its value is computed using Kroneckers second limit formula. It can be realized explicitly using the Siegel function g −v,u (τ ) , namely See §3.1.1 for more details.
The constant C E is related to the non-cuspidal elliptic elements of Γ. See §3.1.2 for more details.
I would like to thank Professor Leon Takhtajan for originally suggesting this problem to me, for reading over this paper, and for useful discussions. I would also like to thank Professor Werner Müller for answering some questions of mine related to the writing of this paper.
I would also like to thank the anonymous referee, for pointing out some important errors, and for suggesting very useful comments.

GENERAL DEFINITION OF THE RELATIVE ZETA-FUNCTION
In this section (following [Mül98]) we state some basic facts concerning Krein's spectral shift function, and show how they lead to the definition of the general relative zeta-function. Later, in §4, we specialize to ∆ on H(Γ, χ) . For more details on the spectral shift function see [Kre53,BK62,Yaf92]. We first establish some notation. For B, a self-adjoint operator on H, σ(B) and σ ess (B) are the spectrum and essential spectrum of B respectively.
Let A and A 0 , be bounded self-adjoint operators in H.
exists for a.e. λ ∈ R, is real-valued, and belongs to L 1 (R). The determinant in (2.1) is the Fredholm determinant. In addition, where · 1 is the trace norm.
The theory of the spectral shift function is reminiscent of the Selberg trace formula. Let Then for every φ ∈ G, φ(A) − φ(A 0 ) is a trace class operator and Part 3 of the next lemma will allow us to define the relative zeta function for H, H 0 .  3.1. Notation. Before we can define ∆ 0 we must establish some notation (see [Fri05a,Fri05b] for more details). In order to simplify our notation (and to make our paper more readable) we present our results under the assumption:
Let P ⊂ C be a fundamental domain 1 for the action of Γ ∞ on C, and let P ′ be the fundamental parallelogram with base point at the origin for the lattice Then for Y sufficiently large, there exists a compact set Define the singular space by and the almost singular space The set P is a euclidean polygon.
The series E i (P, s) converges uniformly and absolutely on compact subsets of {Re(s) > 1} × H 3 to a χ−automorphic function that satisfies and admits a meromorphic continuation to the whole complex plane [Fri05a].
It follows that δ(P, P ′ ) = cosh(d(P, P ′ )), where d denotes the hyperbolic distance in H 3 . Next, for k ∈ S ≡ S([1, ∞)) a Schwartz-class 2 function, define K(P, Q) by The function K(P, Q) is called a point-pair invariant. Set The decay properties of the function k guarantee that the series above converges absolutely and uniformly on compact subsets of The function k leads to two other useful function: h, the Selberg-Harish-Chandra transform of k; and g, the Fourier transform of h. Explicitly: and for r ∈ R set where |P| denotes the euclidian area of P ⊂ C. The sum and integrals converge absolutely and uniformly on compact subsets of H 3 × H 3 .
We conclude this section with some notation that will be needed to state our main result. 2 The space of smooth functions k : [1, ∞) → C that satisfy limx→∞ x n k (m) (x) = 0 for all n, m ∈ N ≥0 .

Regular representations and Siegel's theta function. Recall that
It follows that χ restricted to (the abelian group) Λ ∞ diagonalizes into characters 3 ψ l for l = 1 . . . l ∞ , and the identity character for l = l ∞ + 1 . . . dim V. For each ψ l , u l , v l ∈ R are not both integers, satisfying ψ l (1) = e 2πiu l and ψ l (τ ) = e 2πiv l . We define where g a1,a2 is the Siegel function, .
For a fixed representative R, N (T 0 ) is the minimal norm of a hyperbolic or loxodromic element of the centralizer C(R). The element R is understood to be a k−th power of a primitive non cuspidal elliptic element R 0 ∈ C(R) describing a hyperbolic rotation around the fixed axis of R with minimal rotation angle 2π m(R) . Further, E(R) is the maximal finite subgroup contained in C(R).

Cuspidal elliptic elements.
Denote by CE set of elements of Γ which are Γ-conjugate to an element of Γ ∞ \ Γ ′ ∞ = {γ ∈ Γ ∞ | γ is not parabolic nor the identity element }. We fix representatives of conjugacy classes of CE, g 1 , . . . , g d 4 that have the form Let C(g) denote the centralizer in Γ of an element g ∈ CE . In addition, let {p i , ∞} be the set of fixed points in P of the element g i . Since g i is a cuspidal elliptic element it follows that p i is a cusp of Γ (see [EGM98] page 52). Hence by Assumption 3.1 there is an element 3.2. The Heat Kernel of ∆ as a Poincaré Series. The heat kernel for 5 ∆ on H 3 is a function 3 One-dimensional unitary representations. 4 There are only finitely many distinct conjugacy classes of elliptic elements in a cofinite Kleinian group. 5 We abuse notation and allow ∆ to represent both the self-adjoint operator on H(Γ, χ) and the standard differential operator on smooth functions of H 3 .
for all f in the domain of the self-adjoint operator e − ∆ t . It is a classical result [Dav89] that In order to apply the theory of point pair invariants, we need to find a Schwartz-class function k t so that k t (δ(P, Q)) = k t (cosh(d(P, Q)) = u(P, Q, t). Using the formula: Observe 6 that as x → 1 + , k t (x) → e −t (4πt) 3/2 , and that k t ∈ S. We have: Then K Γ (P, Q, t, χ) is the heat kernel for ∆ on the Hilbert space H(Γ, χ) .

It follows from Equation 3.3 and Lemma 3.2 that
By applying Lemma 3.2 we obtain the spectral expansion of K Γ : 3.3. The Operator ∆ 0 and its Heat Kernel. For F ≡ F Y ∪ F Y , let ∆ 0 be the self-adjoint extension of the operator It is understood that ∆ 0 acts componentwise with respect to the basis for V ∞ fixed in §3.1. The operator ∆ 0 can be thought of as a Laplacian operator in its own right. In fact ∆ 0 is closely related to the restriction of ∆ to F Y . Define The limit follows from applying l'Hôpital's rule to either (3.6) or (3.7). The fact that the singularity cancels out is one advantage to working the the heat kernel instead of the resolvent kernel, which has a singularity and must be iterated. by Lemma 3.4. For all f in the domain of ∆, Proof. The proof follows from [EGM98, Pages 236-237] and the definition of p 0 .
The classical theory of the Heat Equation, for the half line, tells us that k is the heat kernel of ∆ 0 . In other words: 3.4. The Regularized Trace. In this section we will prove that the regularized heat kernel is of trace-class. Then using ideas from the standard proof of the Selberg trace formula, we will evaluate the trace explicitly. But before we can proceed, we need Theorem 3.6. Its proof is based on the classical Poisson Summation Formula, and it is proved in Appendix A.
Theorem 3.6. For P = z + rj, Next write e − ∆ t − e − ∆0 t p 0 as Let m f be the multiplication operator by f. Now rewrite (3.10) once again as The idea is to borrow r −1/4 from e − ∆ τ − e − ∆0 τ p 0 and lend it to e − ∆ τ and e − ∆0 τ . It follows from Theorem 3.6 and Equation 3.9 that each of the operators Now we can apply standard Selberg theory to explicitly evaluate the integral trace F (tr V K Γ (P, P, t, χ) − tr V k(P, P, t)) dv(P ).
Let S(s) be the scattering matrix of ∆ . That is the matrix formed from the constant terms of the Fourier coefficients of the Eisenstein Series. Let φ(s) be the determinant of the scattering matrix (see [Fri05a,Fri05b] for more details).
A standard application of the Maaß-Selberg relations ( see [Fri05a], [ A straightforward calculations shows that FA tr V k(P, P, t) dv(P ) = k(Γ, χ) The term e −t 4 k(Γ, χ) is obtained using a simple u−substitution,

THE RELATIVE ZETA-FUNCTION AND RELATIVE DETERMINANT
Now that we have an explicit formula for the trace tr e − ∆ t − e − ∆0 t p 0 (Theorem 3.8), we can proceed and give an explicit evaluation of the relative zeta-function.
Since σ(∆ 0 ) = σ ess (∆) = [1, ∞) when the representation χ is singular, Note that in order for the integral above to converge, we need to know the asymptotics of tr e − ∆ t − e − ∆0 t p 0 − q χ at both t = 0 and t = ∞. These asymptotics are given in Lemma 4.3.
Since vol(F ) < ∞, it follows that [EGM98, Theorem 3.6.4] Proof. The proof follows from the standard properties of the Mellin transform, Theorem 3.8, and Lemma 4.3. Note that q χ cancels out with any of the terms coming from the zero eigenvalues of ∆ .

Asymptotics of the Heat Kernel.
The main tool that allows us to study the asymptotic behavior (near t = 0 and t = ∞) of the regularized heat kernel is the Selberg trace formula for the case of a cofinite Kleinian group with finite-dimensional unitary representations [Fri05a,Fri05b].
Theorem 4.2. (Selberg trace formula) Let Γ be a cofinite Kleinian group with one cusp at infinity, χ ∈ Rep(Γ, V ), h be a holomorphic function on {s ∈ C | | Im(s)| < 2+δ} for some δ > 0, satisfying h(1+z 2 ) = O(1 + |z| 2 ) 3/2−ǫ ) as |z| → ∞, and let Here {λ m } m∈D are the eigenvalues of ∆ counted with multiplicity. The summation with respect to {R} nce extends over the finitely many Γ−conjugacy classes of the non cuspidal elliptic elements (elliptic elements that do not fix a cusp) R ∈ Γ, and for such a class N (T 0 ) is the minimal norm of a hyperbolic or loxodromic element of the centralizer C(R). The element R is understood to be a k−th power of a primitive non cuspidal elliptic element R 0 ∈ C(R) describing a hyperbolic rotation around the fixed axis of R with minimal rotation angle 2π m(R) . Further, E(R) is the maximal finite subgroup contained in C(R). The summation with respect to {T } lox extends over the Γ−conjugacy classes of hyperbolic or loxodromic elements of Γ, T 0 denotes a primitive hyperbolic or loxodromic element associated with T. The element T is conjugate in PSL(2, C) to the transformation described by the diagonal matrix with diagonal entries a(T ), a(T ) −1 with |a(T )| > 1, and N (T ) = |a(T )| 2 . For s ∈ C, S(s) is a k(Γ, χ) × k(Γ, χ) matrix-valued meromorphic function, called the scattering matrix of ∆, and φ(s) = det S(s). The elements g i are complete representatives for the conjugacy classes of {γ ∈ Γ ∞ | γ is not parabolic nor the identity element }, |C(g i )| is the order of the centralizer in Γ of the element g i . The numbers c i ∈ C are constants depending on the g i respectively (see §3.1.3). Finally γ is Euler's constant, and η ∞ is the analogue of Euler's constant for the lattice Λ ∞ ⊂ R 2 . The term L(Λ ∞ , ψ l ) is defined in §3.1.1. See [Fri05a,Fri05b,EGM98] for more details.
Next, using the Selberg trace formula, we study the regularized heat kernel. We have Lemma 4.3. Let θ(t) ≡ tr e − ∆ t − e − ∆0 t p 0 . Then there exists constants a, b, c, d so that and there exists a positive constant c > 0 so that Proof. Our argument is analogous to [Efr88, Efr91, Proposition 1].
The asymptotics as t → ∞ follows immediately from the spectral trace formula (Theorem 3.8).
An application of the Selberg trace formula for the pair of functions yields (on applying the left side of the Selberg trace formula) the nontrivial terms from the trace of the heat kernel in Theorem 3.8, namely It remains for us to estimate each term on the right as t → 0 + .
We start with the loxodromic sum Since N (T ) > 1 for all loxodromic T, the sum decays to zero as t → 0 + .
Next note that g(0) = exp(−t) √ 4πt and h(1) = e −t . The finite sum involving the non-cuspidal elliptic terms is easily estimated, and so are all the other terms with only g(0) or h(1).
Next we must estimate dx.
An elementary u−substitution with u = x 2 √ 2 shows that the integral above decays to zero (exponentially fast) as t → 0 + . The next integral can be calculated explicitly: requires some work. To study the integral near t = 0 we follow [Efr88, Efr91, Proposition 1]. After performing an integration by parts, the integral becomes Using Sterling's formula See [Efr88, Efr91, Proposition 1] for more details.

The Regularized Determinant.
In this section we define det(∆ −(1 − s 2 )). For Re(s) > 2 define By applying the Mellin Transform to Theorem 3.8 we obtain, for Re(s) > 2, In order to define det(∆ −(1 − s 2 )), we will need to know that H(w, s) is regular at w = 0.
Lemma 4.4. The following hold: (1) For fixed s > 2, H(w, s) is regular at w = 0. (2) where γ is Euler's constant, and the constants a, b, c, d are from Lemma 4.3.
Proof. The proof is a standard exercise using the Mellin transform, Lemma 4.3, and the following formulas [GR65]: Here Ψ(z) is the logarithmic derivative of Γ(z). Regularity follows from the fact that 1 Γ(w) vanishes at w = 0. See [Efr88, Efr91, Prop 2 and 3] for more details.
Our main result, Theorem 5.2, will give the meromorphic continuation to Re(s) ≤ 2.

SELBERG'S ZETA-FUNCTION AND THE REGULARIZED DETERMINANT
5.1. The Definition of the Selberg zeta-function. In this section we define the Selberg zeta-function for cofinite Kleinian groups with finite-dimensional unitary representations. For more details see [Fri05b]. Suppose T ∈ Γ is loxodromic (we consider hyperbolic elements as loxodromic elements). Then T is conjugate in PSL(2, C) to a unique element of the form and let by C(T ) denote the centralizer of T in Γ. There exists a (primitive) loxodromic element T 0 , and a finite cyclic elliptic subgroup E(T ) of order m(T ), generated by an element E T such that Here T 0 = { T n 0 | n ∈ Z }. Next, Let t 1 , . . . , t n , and t ′ 1 , . . . , t ′ n denote the eigenvalues of χ(T 0 ) and χ(E T ) respectively. The elliptic element E T is conjugate in PSL(2, C) to an element of the form where here ζ(T 0 ) is a primitive 2m(T )-th root of unity.
For Re(s) > 1 the Selberg zeta-function Z(s, Γ, χ) is defined by Here the product with respect to T 0 extends over a maximal reduced system R of Γ-conjugacy classes of primitive loxodromic elements of Γ. The system R is called reduced if no two of its elements have representatives with the same centralizer 7 . The function c(T, j, l, k) is defined by

The Relationship Between the Selberg Zeta-Function and the Regularized Determinant.
One way to study the Selberg zeta-function is to apply the Selberg trace formula to the pair of functions, where 1 < Re(s) < Re(B). Let Z(s) ≡ Z(s, Γ, χ) be the Selberg zeta-function under Assumption 3.1. We have [Fri05b]: Lemma 5.1.
Recall that Applying the following elementary equations: to H(w, s), we see that Caution. Differentiation through the sum and integral is justified by regularity at w = 0 (Lemma 4.4).
Instead of differentiating first with respect to w at w = 0, we switch the order of differentiation, differentiate with respect to s, and restrict w so that w > 2 (where the sum and integral converge uniformly). Finally we differentiate with respect to w, and using analytic continuation (and uniqueness of analytic continuation), we obtain (5.2).
For Re(s) > 0 set (see §3.1.3 for the definition of ǫ i and g i ). Using the equation it follows that Simplifying, and recalling the definition of det(∆ −(1 − s 2 )) we arrive at d ds 1 2s where C 1 , C 2 , C 3 are easily read off from (5.4). Next, integrating twice, we obtain: where D 1 , D 2 are constants of integration. They can be determined by letting s → ∞, and applying and applying Sterling's formula we obtain: We have proved and Corollary 5.3. Let Γ be torsion-free with one cusp at ∞, and let χ be a regular character (a one-dimensional unitary representation). Then Corollary 5.4. Let Γ be cocompact, and let χ be a regular character. Then . APPENDIX A. PROOF OF THEOREM 3.6 In this section we prove Theorem 3.6, That is for P = z + rj, P ′ = z ′ + r ′ j ∈ F Y , we show that As usual we are under Assumption 3.1.
The subgroup Γ ∞ is not an abelian group. So in general we can not diagonalize the unitary representation, χ restricted to Γ ∞ , into unitary characters. However, we have the following lemma which is almost as good as diagonalizing χ (see [Fri05b, Lemma 2.4]).
Lemma A.1. Let ∞ be the one cusp of Γ (Assumption 3.1). Then there exist E, R, S ∈ Γ ∞ with the following properties: Here R, S are parabolic elements with R(P ) = P + 1 and S(P ) = P + τ (here Λ ∞ = Z ⊕ Zτ ) for all P ∈ H 3 , and E is elliptic of order m.
The elements R and S commute, but the group Γ ∞ is not abelian when m > 1.
(4) If in addition, m > 1, then The notation used in the lemma above is explained in §3.1.
Next we split f ∞ into two sums,

Lemma A.2 (Poisson Summation Formula
). Let f : R 2 → C be a Schwartz-class function, and let Λ be a two-dimensional lattice in R 2 = C. Then Here Λ 0 is the dual lattice to Λ, and f is the Fourier transform of f, where ·, · is the standard real inner product on C = R 2 .
Using Part 1 of Lemma A.1, we can rewrite the above sum as The last equality follows because δ is a point-pair invariant. Once again applying Lemma A.1, and the definition of δ we can write the above sum as where f k (u) = k t |z − z(E k P ′ ) + u| 2 + r 2 + r ′2 2rr ′ .
A straightforward computation [EGM98, Lemma 3.5.5] shows 9 that f k (0) = rr ′ g log r r ′ Where exp − x 2 4t . 8 We abuse notation here and let r, z represent both coordinates in H 3 and coordinate functions. 9 [EGM98, Lemma 6.4.2] seems to have an extra factor of "2." Noting thatf k (0) is independent of k, and that we recover the leading term of the lemma. To conclude we show that The lemma now follows.
In order to proceed, we need to understand I V − P ∞ as projection operator on the subspace V ⊥ ∞ .
By Lemma A.1 the unitary representation χ restricted to Γ ∞ can be diagonalized in a block matrix form with respect to the decomposition V = V ∞ ⊕V ′ ∞ ⊕V b ∞ . Hence we can write I V −P ∞ = P a +P b , where P a is the orthogonal projection onto V ′ ∞ and P b is the orthogonal projection onto V b ∞ . Lemma A.4. γ∈Γ∞ (I V − P ∞ ) χ(γ)k t (δ(P, γP ′ )) = O(1).
The proof is almost identical to the proof of Lemma A.3 except that the term corresponding to ω = 0, is zero. Indeed, P a m k=0 i,j∈Z 2 χ(E k R i S j )k t (δ(P, E k R i S j P ′ )) = m k=0 χ(E) k P a i,j∈Z 2 k t (δ(P, E k R i S j P ′ )), where χ(E) is a diagonal matrix with each element on the diagonal a finite root of unity not equal to one. Since the order of each root of unity divides m we must have m k=0 χ(E) k = 0.
Hence the "constant term" (the term corresponding to ω = 0 cancels out).
The latter sum decays, while the first term would not. The effect of multiplying exp(2πi(wθ R +vθ S )) is to shift the sum away from the integers. That is l,n∈Z 2 f 1 (l, n) = l,n∈Z 2 f 2 (l + α, n + β),