1. Introduction. Let Q be a positive definite n×n matrix with integral entries and even diagonal entries. It is well known that the theta function

XCII.1 (2000)

Theta functions of quadratic forms over imaginary quadratic fields

by

Olav K. Richter (Santa Cruz, CA)

1. Introduction. Let Q be a positive definite n×n matrix with integral entries and even diagonal entries. It is well known that the theta function

ϑ

(z) := X

g∈Zⁿ

exp{πi

gQgz}, Im z > 0,

is a modular form of weight n/2 on Γ

(N ), where N is the level of Q, i.e.

N Q

is integral and N Q

has even diagonal entries. This was proved by Schoeneberg [5] for even n and by Pfetzer [3] for odd n. Shimura [6] uses the Poisson summation formula to generalize their results for arbitrary n and he also computes the theta multiplier explicitly. Stark [8] gives a different proof by converting ϑ

(z) into a symplectic theta function and then using the transformation formula for the symplectic theta function. In [4], we apply Stark’s method and use theta functions of indefinite quadratic forms to construct modular forms over totally real number fields. In this paper, we define theta functions attached to quadratic forms over imaginary quadratic fields. We show that these theta functions are modular forms of weight n/2 on some Γ

groups by regarding them as symplectic theta functions and then applying well known results for symplectic theta functions. In particular, the main result of [8] allows us to compute the theta multiplier for our theta functions in a very elegant way.

2. Symplectic theta functions. The symplectic group, Sp

(R), con- sists of those 2n × 2n real matrices

M =

 A B

C D



1991 Mathematics Subject Classification: Primary 11F41.

[1]

(each entry is n × n) such that

t

M JM = J :=

 0 −I

I

0

 ,

where I

is the n × n identity matrix. The corresponding symmetric space is the Siegel upper half plane H

which consists of n × n symmetric complex matrices Z with Im Z > 0 (positive definite). The action of M on Z is given by

M ◦ Z = (AZ + B)(CZ + D)

.

Let Γ

= Sp

(Z). The theta subgroup Γ

of Γ

is the set of all

in Γ

such that both A

B and C

D have even diagonal entries. The subgroup acts on the symplectic theta function,

ϑ

 Z,

 u v



= X

m∈Zⁿ

exp{πi[

(m + v)Z(m + v) − 2

mu −

vu]}, where u and v are column vectors in C

. It is well known (see Eichler [1], for example) that for

 A B

C D



in Γ

, we have

(1) ϑ



M ◦ Z, M

 u v



= χ(M )[det(CZ + D)]

ϑ

 Z,

 u v



,

where χ(M ) is an eighth root of unity which depends upon the chosen square root of det(CZ + D), but which is otherwise independent of Z, u, and v. It is also known that χ(M ) can be expressed in terms of Gaussian sums. Stark [8] determined χ(M ) in the important special case where pD

is integral for some odd prime p. The main result in [8] is

Theorem 1. Suppose M =

is in Γ

where C

and D

ex- ist. Suppose further that for some odd prime p, pD

is integral. Then (mod p), the symmetric matrix pD

C has rank h where det(D) = ±p

. Let (pD

C)

be a nonsingular (mod p) h × h principal submatrix of pD

C and s be the signature (the number of positive eigenvalues minus the number of negative eigenvalues) of C

D. Then

χ(M )[det(CZ + D)]

= ε

 2

det[(pD

C)

] p



e

|det(C)|

{det[−iC

(CZ + D)]}

, where ε

= 1 for p ≡ 1 mod 4, ε

= i for p ≡ 3 mod 4,

is the Legendre

symbol, |det(C)|

is positive and {det[−iC

(CZ + D)]}

is given by

analytic continuation from the principal value when Z = −C

D + iY .

Alternatively, if just C

exists and pC

is integral, det(C) = ±p

, then pC

D (mod p) has rank h and

χ(M )[det(CZ + D)]

= ε

 −2 p





det[(pC

D)

] p



|det(C)|

{det[−iC

(CZ + D)]}

.

3. Theta functions as modular forms. Let K = Q( √

d) be the imagi- nary quadratic field with discriminant d < 0. Let O

be the ring of integers of K and δ

be the different of K. The algebraic conjugate of an alge- braic number α is identical with its complex conjugate and denoted by α.

Furthermore, let Γ = SL

(O

) and, as usual, for an integral ideal N, let Γ

(N) :=

 M =

 α β γ δ

  

  M ∈ Γ and γ ∈ N

 .

Our upper half space H := {x + yk | x ∈ C, y ∈ R

} is the quaternionic upper half plane consisting of quaternions with no j-component and positive k-component. The matrix

M =

 α β γ δ



∈ SL

(K)

acts on H by M ◦ z := (αz + β)(γz + δ)

. Note that M ◦ z ∈ H. For γ and δ in K and z in H, we define

N (γz + δ) := kγz + δk

= |γx + δ|

+ |γ|

y

.

Let Q be a symmetric n × n matrix with entries in O

defining the quadratic form Q[x] :=

xQx, where x ∈ C

. Furthermore, let Q{x} :=

t

xQx and Q[x] :=

xQx. If, in addition, Q has diagonal entries which are divisible by 2, we say that Q is of level N (N ∈ O

) whenever the following two conditions are satisfied:

(a) The matrix N Q

has entries in O

, and 2 divides the diagonal entries of N Q

.

(b) For any M ∈ O

, N divides M whenever M Q

has entries in O

and 2 divides the diagonal entries of M Q

.

For the vector λ =

(λ

, . . . , λ

), we define λ :=

(λ

, . . . , λ

), where

λ

, . . . , λ

are in K. We define the theta function Θ

for a quadratic form by

Definition 1. Let Q be a symmetric n × n matrix with entries in

O

such that 2 divides the diagonal entries of Q and such that Q is of

level N . Since Q is symmetric, Q =

LL for an upper triangular complex

matrix L = (l

)

(l

= 0 for s > r). For an ideal I ⊂ O

and

z = x + yk ∈ H, set Θ

(z) := X

λ∈Iⁿ

exp n

πi h

(Q[λ]x + Q[λ]x) + 2i

 X

s=1

X

l

λ

 y

io ,

where λ =

(λ

, . . . , λ

).

Remarks. (a) For R :=

LL, we have X

s=1

X

l

λ

=

λRλ = R{λ}.

Furthermore, observe that

(2) RQ

R = Q and

R = R > 0.

Hence the matrix R is a majorant of the matrix Q (in the terminology of Siegel [7]).

(b) For any algebraic integer t ∈ K, Q[λ]t + Q[λ]t = tr(Q[λ]t) is an even rational integer, and thus Θ

(z) is invariant under linear transforma- tions, i.e.

(3) Θ

(z + t) = Θ

(z).

The first task toward showing that Θ

is a modular form is to convert Θ

into a symplectic theta function ϑ. Let us introduce some helpful notation.

For α ∈ C, define

diag(α) :=

 α 0 0 α



and the 2n × 2n matrix

diag

(α) :=



 

diag(α) . ..

diag(α)



  .

For z = x + yk ∈ H, let

Z

:=

 x iy iy x



and furthermore, define the 2n × 2n matrix

Z

:=



 Z

. ..

Z



 .

Let Λ :=

(λ

, λ

, . . . , λ

, λ

). Some computation gives

(4)

Λ

LZ

LΛ = n

(Q[λ]x + Q[λ]x) + 2i

 X

s=1

X

l

λ

 y

o , where

L :=



 

diag(l

) . . . diag(l

) . .. .. .

diag(l

)



  .

Let {ω

, ω

} be an integral basis of the ideal I ⊂ O

. The entries of the vector Λ are integers in I and can be written in terms of the basis {ω

, ω

}.

Hence, we can define a vector P =

(m

, . . . , m

) with rational integers m

, . . . , m

such that Λ = W P , where

(5) W :=



 W

. ..

W





and

(6) W

:=

 ω

ω

ω

ω

 . Furthermore,

(7) W

=

 ν

ν

ν

ν

 , where {ν

, ν

} is an integral basis for I

δ

.

With T := LW and Z :=

T Z

T , we have (8) Θ

(z) = ϑ

 Z,

 0 0

 

= X

m∈Z²ⁿ

exp{πi[

mZm]}.

To see that Z is actually in H

, we observe that Z is symmetric and that Z =

T S

Z

T , where

S :=

 0 1 1 0



and S

:=



 S . ..

S



 .

Also,

(S

Z

−

S

Z

) = yI

> 0 and a corollary of Sylvester’s theorem implies that Im Z > 0.

For

∈ Γ = SL

(O

), set

(9) M

:=

 A

B

C

D

 :=

 diag

(α) diag

(β) diag

(γ) diag

(δ)



.

It is easy to check that the diagram z

 α β γ δ



◦ z

Z

 A

B

C

D



◦ Z

²² // ²²

//

is commutative. Hence

z 7→

 α β γ δ



◦ z in H corresponds to

Z 7→

 A B

C D



◦ Z in H

, where

(10)

 A B

C D



=



T A

T

T B

T T

C

T

T

D

T

 .

When is the matrix in (10) in the theta subgroup? To answer this ques- tion, let us introduce some more notation. Assume that S = (s

)

and R = (r

)

are matrices with entries in K (not necessarily in O

).

We define the matrix

R  S := ((tr(r

s

)

))

.

Note that the entries of R  S are rational numbers. Computation shows that A = I

 A

, B = Q  B

, C = Q

 C

, D = I

 D

, and A

, B

, C

and D

are given by

A

=

 ω

ν

α ω

ν

α ω

ν

α ω

ν

α

 ,

B

=

 ω

ω

β ω

ω

β ω

ω

β ω

ω

β

 ,

C

=

 ν

ν

γ ν

ν

γ ν

ν

γ ν

ν

γ

 ,

D

=

 ω

ν

δ ω

ν

δ ω

ν

δ ω

ν

δ

 .

From the definition of A, B, C and D, we see that

AC =

CA,

BD =

DB, and

DA −

BC = I

(as αδ − βγ = 1) and hence

∈ Sp

(R). In addition, A

B =

T A

B

T = Q  (αB

) and C

D = T

C

D

T

= Q

 (δC

). Thus, if γ is in the ideal I

δ

N (N the level of Q), then the entries of A, B, C, and D are traces of algebraic integers and hence rational integers, and A

B and C

D have even diagonal entries. Hence for

 α β γ δ



∈ Γ

(I

δ

N ),

we have

M =

 A B

C D



∈ Γ

. It is easy to verify that

det(CZ + D) = det(C

Z

+ D

) = N (γz + δ)

, and therefore by (1) and (8),

(11) Θ

  α β γ δ



◦ z



= χ

  α β γ δ

 , Q



N (γz + δ)

Θ

(z), where χ

, Q

is an eighth root of unity depending on

and Q.

Thus, Θ

(z) is a modular form on Γ

(I

δ

N ) of weight n/2.

4. The eighth root of unity. It remains to determine χ

, Q
explicitly. Assume that δ is a first degree prime in O

of norm p (meaning that p is a positive odd prime in Z). In this case, pD

is integral. Note that det(D) = det(D

) = p

, and thus by Theorem 1, pD

C has rank n (mod p). Hence for Q

= (r

)

, we find that

(pD

C)

=



 

tr(r

ν

ν

pδ

γ) . . . tr(r

ν

ν

pδ

γ)

.. . .. .

tr(r

ν

ν

pδ

γ) . . . tr(r

ν

ν

pδ

γ)



 

and

det(pD

C)

≡ (pδ

γ)

(ν

ν

)

det(Q)

(mod δ).

Some computation shows that

|det(C)|

{det[−iC

(CZ + D)]}

e

= N (γz + δ)

. Hence

(12) χ

  α β γ δ

 , Q



= ε

 (pδ

2γ)

det(Q) δ



and in the special case where n is even,

(13) χ

  α β γ δ

 , Q



=

 (−1)

det(Q) δ

 . We have proved

Theorem 2. Suppose that

∈ Γ

(I

δ

N ), where δ is a first degree prime in O

of norm p. For z ∈ H, we have

(14) Θ

  α β γ δ



◦ z



= ε

 (pδ

2γ)

det(Q) δ



N (γz + δ)

Θ

(z),

where ε

= 1 for p ≡ 1 mod 4 and ε

= i for p ≡ 3 mod 4.

Actually, we have determined the eighth root of unity more explicitly than it seems. In (3), we showed that for all algebraic integers t, Θ

(z +t) = Θ

(z). It follows from (11) that for

∈ Γ

(I

δ

N ) and for all algebraic integers t,

(15) χ

  α β γ δ

  1 t 0 1

 , Q



= χ

  α β γ δ

 , Q

 .

Furthermore, Hecke [2] gives a proof of Dirichlet’s primes in progres- sion theorem for number fields. Hence for algebraic integers γ and δ with (γ, δ) = 1, the arithmetic progression {γt + δ}

contains infinitely many first degree primes (in a general number field, the progression contains in- finitely many totally positive first degree primes), and the theta multiplier is determined explicitly after locating a first degree prime with odd norm in the arithmetic progression {γt + δ}

.

There is a special case which should also be mentioned. Let δ be a prime in O

with N (δ) = p

, where p is an odd prime in Z. As before, we observe that pD

has rational integers as entries. Also, det(D) = det(D

) = p

and by Theorem 1, pD

C has rank 2n (mod p). Thus,

det(pD

C) = (N (det(Q)))

(d(N (I)

))

(N (γ

)).

Hence χ

  α β γ δ

 , Q



= ε

 2

d

N (γ

)N (det(Q)) p



= (−1)

 (−1)

N (γ

det(Q)) p

 . In the special case where n is even, we see that

(16) χ

  α β γ δ

 , Q



=

 N ((−1)

det(Q)) p

 .

The result from (13) matches the result from (16) since an element a is a square in F

(the field of p

elements) iff N

(a) is a square in F

(the field of p elements). This can be seen by observing that the mapping N : F

→ F

given by a → N (a) := N

(a) is an epimorphism.

References

[1] M. E i c h l e r, Introduction to the Theory of Algebraic Numbers and Functions, Aca- demic Press, New York, 1966.

[2] E. H e c k e, ¨ Uber die L-Funktionen und den Dirichletschen Primzahlsatz f¨ ur einen beliebigen Zahlk¨orper, Nachr. Ges. Wiss. G¨ottingen Math.-phys. Kl. 1917, 299–318.

[3] W. P f e t z e r, Die Wirkung der Modulsubstitutionen auf mehrfache Thetareihen zu quadratischen Formen ungerader Variablenzahl, Arch. Math. (Basel) 4 (1953), 448–

454.

[4] O. R i c h t e r, Theta functions of indefinite quadratic forms over real number fields, Proc. Amer. Math. Soc., to appear.

[5] B. S c h o e n e b e r g, Das Verhalten von mehrfachen Thetareihen bei Modulsubstitutio- nen, Math. Ann. 116 (1939), 511–523.

[6] G. S h i m u r a, On modular forms of half integral weight, Ann. of Math. 97 (1973), 440–481.

[7] C. S i e g e l, Indefinite quadratische Formen und Funktionentheorie II , Math. Ann.

124 (1952), 364–387.

[8] H. S t a r k, On the transformation formula for the symplectic theta function and ap- plications, J. Fac. Sci. Univ. Tokyo Sect. 1A 29 (1982), 1–12.

Department of Mathematics

University of California at Santa Cruz Santa Cruz, CA 95064, U.S.A.

E-mail: Richter@math.ucsc.edu

Received on 15.2.1998

and in revised form on 6.9.1999 (3333)