The covolume of quaternion groups on the four-dimensional hyperbolic space

LXXVII.1 (1996)

The covolume of quaternion groups on the four-dimensional hyperbolic space

by

Stefan Breulmann (Mannheim) and Volker Helmke (M¨ unster)

Introduction. In this paper we compute in two different ways the vol- ume of the fundamental domain for quaternion groups acting on the four- dimensional hyperbolic space. The first proof of our volume formula

v(F) = π

1080

Y

p|D

(p

+ 1)(p − 1),

where D denotes the discriminant of the corresponding rational quaternion algebra, is based on the Maaß–Selberg relations using Stokes’ theorem as described in [EGM] and the Fourier expansion of the Eisenstein series E

. The second proof uses the residue of the Eisenstein series E

for explicit computation of the volume. This method was already used in [Sa]. We draw the relevant Fourier expansion and the residue of the Eisenstein series from [GS].

It was kindly pointed out to the authors by F. Grunewald (D¨ usseldorf) that our volume formula can also be proved with some labour by the usual Tamagawa measure method. But in analogously defined cocompact situa- tions the evaluation of the volume formula by this method seems still very complicated due to our lack of knowledge of certain Cartan involutions.

1. Preliminaries. Let B be a definite rational quaternion algebra with discriminant D and class number h. For w = r + xi + yj + zij ∈ B let w = r − xi − yj − zij be the conjugate element. As usual we denote by n(w) = ww, tr(w) = w + w, w

:=

tr(w) and Im w :=

(w − w

) the norm, trace and the real and imaginary part, respectively, of an element w ∈ B.

For A ⊂ B let A

:= {a ∈ A : tr(a) = 0} be the subset of all quaternions in A with trace zero.

By H = B ⊗ R we denote the Hamilton quaternions. Let H = {P ∈ H : tr(P) > 0}

[9]

be the four-dimensional hyperbolic space, where we identify H = ]0, ∞[ × R

⊂ R

. We always write P = (r, x, y, z) for P ∈ H and P

:= r.

The associate volume measure resp. volume form is given by dv = dr dx dy dz

r

resp. dv = dr ∧ dx ∧ dy ∧ dz r

and the corresponding Laplace–Beltrami operator is

∆ = r

 ∂

∂r

+ ∂

∂x

+ ∂

∂y

+ ∂

∂z



− 2r ∂

∂r .

Let MSp(1; H) (in the notation of [Kr], p. 316) be the modified symplectic group of degree 1 over H, i.e.,

MSp(1; H) =



M ∈ Mat(2; H) : M

 0 1 1 0

 M =

 0 1 1 0

  . It is generated by the elements

(1)

 1 a 0 1



(a ∈ H, tr(a) = 0),

 0 1 1 0

 ,

 a 0 0



(a 6= 0).

MSp(1; H) operates on H in the usual way P 7→ M P := (aP + b)(cP + d)

(M =

∈ MSp(1; H)). The hyperbolic volume measure dv is MSp(1; H)- invariant and for f ∈ C

(H) and M ∈ MSp(1; H) we have

∆(f ◦ M ) = (∆f ) ◦ M.

Let G be the group G :=



M ∈ Mat(2; B) : M

 0 1 1 0

 M =

 0 1 1 0

  . For a maximal order R in B we define Γ

:= GL(2; R) ∩ G.

An element M ∈ G is called parabolic if there exist γ ∈ G and 0 6= b ∈ B

with γ

M γ = ±

, and an element σ ∈ H ∪ {∞} is called a cusp of Γ

if σ is fixed by some parabolic element of Γ

. For a cusp σ ∈ H ∪ {∞}

we denote by (Γ

)

the stabilizer subgroup of σ in Γ

. Since

(Γ

)

:= {M ∈ (Γ

)

: trace M = ±2} =



±

 1 b 0 1



: b ∈ R



we obtain

(2) [(Γ

)

: (Γ

)

] = |R

|/2.

It is known that the set of cusps of Γ

equals B

∪ {∞}, and from that we can give the explicit correspondence between the classes of Γ

-inequivalent cusps and the classes of left R-ideals in B:

If we denote by σ

, . . . , σ

a maximal system of representatives of Γ

-

inequivalent cusps, the ideals I

:= Ra

+ Rc

, i = 1, . . . , n (σ

= a

c

(a

, c

∈ B)) are a complete system of representatives of classes of left R- ideals, in particular we have n = h. We always choose the ideals I

such that n(I

) = Z and hence I

= I

, furthermore we denote by R

the right order of I

.

We define the Eisenstein series for the cusp σ

= γ

∞ (γ

∈ G) by the formula

(3) E

(P, s) := X

γ∈(ΓR)_σi\ΓR

(γ

γP)

= r

X

(a,c)∈(Ii×Ii)/R^×_i Ra+Rc=Ii

tr(ac)=0

n(aP + c)

,

and the modified Eisenstein series by E e

(P, s) := r

X

(0,0)6=(a,c)∈I_i×I_i tr(ac)=0

n(aP + c)

.

Both series converge for Re s > 3 uniformly on compact sets and are Γ

- invariant functions since

(4) E e

(P, s) = X

Z(I

, s)E

(P, s), Z(I

, s) = X

06=x∈Iij

n(x)

,

with I

:= I

I

= I

I

.

The ideals I

, i = 1, . . . , h, constitute a complete system of representa- tives of the classes of left R

-ideals, the ideal I

is associated with the cusp σ

:= γ

σ

and we obtain, with an appropriate choice of γ

,

(5) Γ

= γ

Γ

γ

.

We define the Eisenstein series E

(P, s) and e E

(P, s) for the cusp σ

as above, replacing (j fixed) R by R

, I

by I

and σ

by σ

.

The correspondence between these series is given by

(6) E

(P, s) = E

(γ

P, s), E e

(P, s) = e E

(γ

P, s).

Computing the Fourier coefficients of these Eisenstein series it turns out that one obtains relatively simple expressions if one forms weighted means of the Eisenstein series. The series

E e

(P, s) :=

X

1

|R

| E e

(P, s) (j = 1, . . . , h)

are called weighted Eisenstein series. To get a functional equation, we mul- tiply by an additional factor and define

(7) E

(P, s) := F (s) e E

(P, s).

Here F (s) is defined by

F (s) := π

D

Γ (s/2)Γ (s)ζ(2s − 2)ζ(s − 1)

Y

p|D

(1 + p

), where ζ denotes the Riemann zeta function.

The series E

(P, s) are obviously Γ

-invariant functions and satisfy the differential equation

(8) −∆E

(P, s) = s(3 − s)E

(P, s).

We draw from [GS] the Fourier expansion of the series E

(P, s) at infinity.

This yields the Fourier expansion of E

(P, s) at every cusp σ

(by (6)).

Theorem 1.1. The Eisenstein series E

(P, s) (j = 1, . . . , h) have the Fourier expansions

(9) E

(P, s)

= r

f

(s) + r

f

(3 − s) + 4 √

π X

u∈_R

d

u6=0j

n(2u)

X

t∈N (t,D)=1 u∈t_R

d

j

X

l| ^D q(2u) l>0

tl L



s − 1, − n(2u)D

t

l



× r

K

(2πr p

n(2u)) exp(2πitr(uP)).

Here the following notations are applied:

(10) f

(s) := (s − 1)D

Y

p|D

(1 − p

)ζ

(s)ζ

(2s − 2) with the customary abbreviation

ζ

(s) := π

Γ (s/2)ζ(s).

Moreover , d

R

:= {w ∈ B

: tr(wR

) ⊂ Z} is the dual lattice of R

and q(2u) denotes the denominator of n(2u). The function L

(s, ∆) is defined by

L

(s, ∆) :=

 π

Γ (s/2)∆

L(s, ∆) for ∆ > 0, π

Γ ((s + 1)/2)|∆|

L(s, ∆) for ∆ < 0,

where we denote by L(s, ∆) the Cohen–Zagier L-function ([Za], p. 110), and K

is the modified Bessel function

K

(w) = 1 2

∞

\

0

exp



− w

2 (t + t

)



t

dt.

Corollary 1.2. The Eisenstein series E

(P, s) (j = 1, . . . , h) have meromorphic continuations to the whole complex plane and satisfy the func- tional equation

E

(P, s) = E

(P, 3 − s).

They are holomorphic functions except for simple poles at s = 0, 3 with the residue

(11) res(E

(P, s), s = 3) = ζ(3) 2π

Y

p|D

(p

− 1).

2. The volume of the fundamental domain. In this section we compute in two different ways the volume of a fundamental domain F for the operation of Γ

on H. The first proof is based on the Maaß–Selberg relations, a method which was already described in [EGM]. The second proof follows the line of argument of [Sa].

2.1. First proof. For g ∈ C

(H) we define the differential form ω

by ω

:= g

dx ∧ dy ∧ dz

r

− g

dy ∧ dz ∧ dr

r

+ g

dz ∧ dr ∧ dx

r

− g

dr ∧ dx ∧ dy r

. Let ω = ω

, T ∈ G, P = (r, x, y, z) ∈ H, P

= T P = (r

, x

, y

, z

) and ω

be the differential form with P replaced by P

. Simple calculations by means of the generators (1) show that

(12) ω = ω

.

For g, h ∈ C

(H) let ω

:= ω

h. Trivial calculations show that dω

= ((∆g)h + r

(grad g)

grad h) dv.

Hence

(13) dω

− dω

= ((∆g)h − (∆h)g) dv.

Now consider a Poincar´e normal polyhedron F for the operation of the group Γ

on H. The boundary of F is contained in finitely many three- dimensional planes (cf. [Gr], p. 215, [Wi], p. 866 ff.) and the “sides” of F are pairwise Γ

-equivalent.

Similarly to [Ro], p. 288, we subdivide F in a relatively compact part and finitely many cusp sectors. For preparation we need an analogue of Shimizu’s lemma.

Lemma 2.1. For Y ≥ 1 two points P, P

∈ H with P

, P

> Y are

equivalent with respect to γ

Γ

γ

= Γ

if and only if they are equivalent

with respect to (Γ

)

.

P r o o f. Let H

:= {P ∈ H : P

> Y }. For M ∈ Γ

\ (Γ

)

, M =

∗ ∗ c d

, i.e. c 6= 0 and P ∈ H

we have

(M P)

= P

n(cP + d) = P

n(c)n(P + c

d) ≤ P

n(c)P

< 1 ≤ Y,

since tr(c

d) = 0 implies n(P + c

d) ≥ P

and from c ∈ R

we conclude n(c) ≥ 1. Hence H

∩ M H

= ∅.

Applying Lemma 2.1 and observing that γ

F is a fundamental domain for Γ

we conclude that for a suitable choice of F there is a fundamental domain P

for (Γ

)

such that

{P ∈ P

: P

> Y } = (]Y, ∞[ × Im H) ∩ γ

F.

Let S

be a cusp sector “at height” Y at the cusp σ

, i.e.

S

= γ

{P ∈ P

: P

> Y } ⊂ F and

F

:= F ∩

 \

j=1

γ

(]0, Y ] × Im H)



the fundamental domain “cut off at height Y ”. Similar considerations to the above lemma show that the cusp sectors are pairwise disjoint if Y ≥ 1.

We now choose

g : H → C, P 7→ E

(P, s), h : H → C, P 7→ E

(P, t)

for fixed s, t > 3, s 6= t, and let Y ≥ 1. For j ∈ {1, . . . , h}, P = γ

P

∈ S

with P

∈ P

and P

> Y the functions g and h have Fourier expansions at the cusp σ

of the form

(14) g(P) = ϕ(P

) + ψ

(P

), h(P) = σ(P

) + τ

(P

), where

(15) ϕ(P

) = (r

)

f

(s) + (r

)

f

(3 − s), σ(P

) = (r

)

f

(t) + (r

)

f

(3 − t)

and where ψ

, τ

denote the terms involving the higher Fourier coefficients in (9) with P replaced by P

.

We subtract the zeroth Fourier coefficients “in height Y ”, i.e., g

(P) :=

 g(P) for P ∈ F

,

g(P) − ϕ ◦ γ

(P) for P ∈ S

,

∆g

=

 ∆g in F

,

∆(g − ϕ ◦ γ

) in S

,

where in the last line we take the usual derivative. We conclude from (8) and simple calculations in the cusp sectors

(16) −∆g

= s(3 − s)g

, −∆h

= t(3 − t)h

. We first treat the contribution of F

.

Lemma 2.2. With the same notations as above set Q

(Y ) := {P ∈ P

: P

= Y }. Then

J(Y ) := (t(3 − t) − s(3 − s)) \

F^Y

g

h

dv (17)

= X

\

Qj(Y )

(g

h − h

g) dx

dy

dz

(r

)

. P r o o f. From (13) and (16) we have

((t(3 − t) − s(3 − s))g

h

) dv = ((∆g

)h

− (∆h

)g

) dv

= dω

− dω

,

where these differential forms are defined only on F

. Since further calcu- lations only deal with F

, we write g and h instead of g

and h

.

Denoting by ∂F

the positively oriented boundary of F

and applying Stokes’ theorem we obtain

\

F^Y

(dω

− dω

) = \

∂F^Y

(ω

− ω

).

The boundary of F

consists of pairwise equivalent parts of three-dimen- sional hyperbolic planes and the “cross-sections” γ

Q

(Y ) = γ

{P ∈ P

: P

= Y } at the cusps.

Since Γ

-equivalent pairs of bounding hyperbolic planes inherit opposite orientation, the corresponding contributions to the above integral cancel because of (12) and the Γ

-invariance of g and h. Hence we obtain

J(Y ) = X

\

γjQj(Y )

(ω

− ω

).

If we denote by ω

the differential form with respect to g and h with P replaced by P

= γ

P and similarly with g and h interchanged, (12) again implies

J(Y ) = X

\

Qj(Y )

(ω

− ω

) = X

\

Qj(Y )

(g

h − h

g) dx

dy

dz

(r

)

.

This proves the assertion.

We treat the cusp sectors applying a similar argument to the “cusp sectors cut off at height X > Y ”

S

= γ

{P ∈ P

: Y < P

≤ X}

and letting X → ∞. We only note the result.

Lemma 2.3. For an arbitrary cusp sector S

we have (18) (t(3 − t) − s(3 − s)) \

S_j

g

h

dv

= − \

Qj(Y )

 ∂

∂r

ψ

(P

)



τ

(P

) −

 ∂

∂r

τ

(P

)

 ψ

(P

)

 dx

dy

dz

(r

)

. Theorem 2.4. We have the equality

(19) (t(3 − t) − s(3 − s)) \

F

g

h

dv

= X

\

Qj(Y )

 ∂

∂r

ϕ

 σ −

 ∂

∂r

σ

 ϕ

 dx

dy

dz

(r

)

. P r o o f. We decompose F = F

+ P

j=1

S

and add (17) and (18).

Note that the integrals over the sets Q

(Y ) in (17) with the higher Fourier coefficients from (14) vanish because of their periodicity.

Now we can compute the volume of the fundamental domain explicitly.

Theorem 2.5. The volume of the fundamental domain for the group Γ

is given by the equation

v(F) = π

1080

Y

p|D

(p

+ 1)(p − 1).

P r o o f. We compute the integrand on the right hand side of (19) by means of (15). For simplification we write f instead of f

and r instead of r

. This gives

(20)

X

\

Qj(Y )

 ∂

∂r ϕ

 σ −

 ∂

∂r σ

 ϕ

 dx dy dz r

= (f (s)f (t)Y

(s − t) + f (3 − s)f (t)Y

(3 − s − t)

+ f (s)f (3 − t)Y

(s + t − 3) + f (3 − s)f (3 − t)Y

(t − s))

× X

λ

(Im Q

(Y )),

where λ

denotes the Lebesgue measure on B

⊗ R = R

. From (2) and λ

(B

⊗ R/R

) = D/2 ([GS], p. 247) we conclude that

(21) λ

(Im Q

(Y )) = D/|R

|.

For further simplification we set

(22) Θ :=

X

D

|R

| = D 24

Y

p|D

(p − 1)

(cf. [De], p. 134).

We now substitute s = 3 + 2a, t = 3 + a with a > 0, multiply both sides of (19) with 2a

/(3a

+ 3a) and conclude from (20)–(22) that

2a

\

F

g

h

dv =



f (3 + 2a)f (3 + a) 2a

3a + 3 Y

(23)

+ (−2a)f (−2a)f (3 + a)Y

− 2(−a)f (−a)f (3 + 2a)Y

+ (−2a)f (−2a)(−a)f (−a) −1

3a + 3 Y

 Θ.

The right side of this equation converges as a → 0+ to (0 + f (3)res(f ; 0) − 2f (3)res(f ; 0) + O(Y

))Θ

= −f (3)res(f ; 0)Θ + O(Y

).

For the same reason as in [EGM], p. 211, we obtain from (23), letting a → 0+,

v(F

) = −f (3)res(f ; 0)Θ

(res(E

(P, s), s = 3))

+ O(Y

) and therefore, letting Y → ∞,

v(F) = −f (3)res(f ; 0)Θ (res(E

(P, s), s = 3))

= D

π

45

Y

p|D

1 − p

p

− 1

X

1

|R

| (see (10), (11))

= π

1080

Y

p|D

(p

+ 1)(p − 1).

2.2. Second proof

Theorem 2.6. With the function f

from (10) we have for Re s > 3 E

(P, s) = f

(s)

X

2

|R

|

X

γ∈(Γ_Rj)⁰_∞\Γ_Rj

(γγ

P)

. P r o o f. Since

X

Z(I

, s)

|R

| = ζ(s)ζ(s − 1) Y

p|D

(1 − p

) (cf. [GS], p. 255), we derive from (4) and (7) that

E

(P, s) = f

(s) X

E

(P, s).

To prove the assertion, we transform the Eisenstein series E

(P, s) in (3). From (2) and (5) we obtain

E

(P, s) = 2

|R

|

X

γ∈(Γ_Rj)⁰_∞\Γ_Rj

(γγ

P)

, which gives the desired result.

Let F be a Poincar´e normal polyhedron for the action of Γ

on H.

Without loss of generality we may assume that F contains a full cusp sector for Γ

at infinity.

Γ

contains the hyperbolic rotations of the form γ

:=

∈ (Γ

)

with a ∈ R

. We now consider the images γ

F and see that the union

P := [

a∈R^×/{±1}

γ

F

yields a set which we can assume to have the following shape (note the formula

γ

P

r

= (γ

P)

for all w ∈ R

):

The imaginary coordinates of the points in P with sufficiently large real part run through a fundamental parallelepiped F for the lattice R

. The real coordinates run through a set (depending on the imaginary part P − P

of the points)

]A(P − P

), ∞[ with A(P − P

) ≥ 0.

Here we have A(P − P

) = 0 if and only if P − P

is a cusp of Γ

not equiva-

lent to ∞. Without loss of generality let us assume that the representatives

σ

of the cusps are in ∂F. Then the fundamental domains F

:= γ

F,

j = 1, . . . , h, “contain” the cusp infinity, too. Since linear-fractional trans-

formations map hyperbolic straight lines to hyperbolic straight lines, we can

assume for the computation of the volume that P

:= [

a∈R^×_j/{±1}

γ

F

has the same shape as P = P

with certain F

, R

and A

instead of F, R

and A, respectively. From now on we choose the fundamental domains F

and the integration domains P

as described above. For a ∈ R

we always denote by γ

the matrix

.

S e c o n d p r o o f o f T h e o r e m 2.5. Consider the function b(s) := \

F



E

(P, s) − f

(s) X

(γ

P)

 dv(P).

Since the residue of E

(P, s) does not depend on P, we obviously have res(b(s), s = 3) = v(F)res(E

(P, s), s = 3).

To compute the residue of b(s) in a different manner, we put s

:= 3 + 1/n and find, by Theorem 2.6,

res(b(s), s = 3) = lim

n→∞

(s

− 3)b(s

)

= lim

n→∞

1 n

\

F

f

(s

) X

2

|R

|

×

 X

γ∈(Γ_Rj)⁰_∞\Γ_Rj

(γγ

P)

− |R

|

2 (γ

P)

 dv(P)

= lim

n→∞

1 n

\

F

f

(s

) X

2

|R

|

X

γ∈(Γ_Rj)⁰_∞\Γ_Rj γ6=γ_a,a∈R^×_j

(γγ

P)

dv(P)

= lim

n→∞

1 n

X

2

|R

| f

(s

) X

γ∈(Γ_Rj)⁰_∞\Γ_Rj γ6=γ_a,a∈R^×_j

\

γFj

r

dv(P)

= lim

n→∞

1 n

X

2

|R

| f

(s

) \

F_j

A_j(x,y,z)

\

0

r

dr dx dy dz

= f

(3) X

D

|R

| ,

since λ

(F

) = D/2 ([GS], p. 247).

Putting together these equations we obtain from Corollary 1.2 and some well-known special values of Γ and ζ,

v(F) = f

(3) P

j=1

D/|R

| res(E

(P, s), s = 3) = π

45 Y

p|D

(p

+ 1) X

1

|R

|

= π

1080

Y

p|D

(p

+ 1)(p − 1) (see (22)).

By way of example, in the special case of the quaternion algebra B =

−1,−1 Q

and the Hurwitz order R = Zi+Zj +Zk +Z

we have D = 2 and obtain

v(F) = π

216 .

This special result can be also obtained by means of both methods from the calculations in [Kr].

Using the above method, we can now compute the residue of the Eisen- stein series E

(P, s) at s = 3 without knowing the Fourier expansion explic- itly. The Eisenstein series E

(P, s) have meromorphic continuations to the half plane Re s > 1 with a simple pole at s = 3 (cf. [GS], p. 245) and we have

res(E

(P, s), s = 3) = D

|R

|v(F) .

References

[De] M. D e u r i n g, Algebren, 2. Auf., Springer, Berlin, 1968.

[EGM] J. E l s t r o d t, F. G r u n e w a l d and J. M e n n i c k e, Eisenstein series on three- dimensional hyperbolic space and imaginary quadratic number fields, J. Reine Angew. Math. 360 (1985), 160–213.

[Gr] L. G r e e n b e r g, Finiteness theorems for Fuchsian and Kleinian groups, in: Dis- crete Groups and Automorphic Functions, Proc. Conf. Cambridge 1975, W. J.

Harvey (ed.), Academic Press, London, 1977, 199–257.

[GS] V. A. G r i t s e n k o and R. S c h u l z e - P i l l o t, Eisenstein series on four-dimen- sional hyperbolic space, Acta Arith. 67 (1994), 241–268.

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Fakult¨at f¨ ur Mathematik und Informatik Mathematisches Institut

Universit¨at Mannheim der Universit¨at M¨ unster

Seminargeb¨aude A 5 Einsteinstraße 62

D-68131 Mannheim, Germany D-48149 M¨ unster, Germany

E-mail: breul@hurwitz.math.uni-mannheim.de E-mail: helmke@math.uni-muenster.de

Received on 18.9.1995 (2855)