Logarithms of theta functions on the upper half space

Abstract

Let K be an imaginary quadratic field whose discriminant is congruent to one modulo 8 and $\mathcal{O}$ be the ring of integers of K. Let Γ denote the group $SL(2,\mathcal{O})$ which acts discontinuously on the upper half space H. In this paper, we study a homomorphism $\phi :\mathrm{\Gamma }\to \mathbf{Z}$ obtained from a branch of the logarithm of a theta function on H which is automorphic with respect to Γ and does not vanish on H. In particular, we determine explicitly the decomposition $\phi ={\phi }_c+{\phi }_e$ of φ into the cusp part ${\phi }_c$ and the Eisenstein part ${\phi }_e$, and prove a congruence conjectured by Sczech [14] between φ and ${\phi }_e$ modulo 8 under an assumption on the 2-divisibility of a certain L-value.

Introduction

Let H be the upper half space which consists of all quaternion numbers $\tau =z+jv$ ($j^2=-1,ij=-ji$) with $z\in \mathbf{C}$ and $v>0$. Every element $A=\left(\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill \end{array}\right)$ of $SL(2,\mathbf{C})$ acts on H by

$$A\tau =(a\tau +b){(c\tau +d)}^{-1},$$

where the right hand side is taken in the skew-field of quaternions. Let K be an imaginary quadratic field embedded in the complex number field C, D its discriminant and $\mathcal{O}$ its ring of integers. The group $\mathrm{\Gamma }=SL(2,\mathcal{O})$ acts discontinuously on H. We shall consider the following theta function on H:

$$\vartheta (\tau )=\sqrt{v}\sum _{\mu \in \mathcal{O}+1/2}\exp [-\frac{2\pi |\mu {|}^{2}v}{|\sqrt{D}|}+\pi i\mathrm{tr}\left(\frac{{\mu }^{2}z+\mu }{\sqrt{D}}\right)].$$

Assume $D\equiv 1(\mathrm{mod}8)$ for simplicity. Then, it is known that

$$\vartheta (A\tau )=\rho \vartheta (\tau )({\rho }^8=1)$$

for every $A\in \mathrm{\Gamma }$, cf. Sczech [13], Theorem 8.

Sczech [14] has studied the logarithm of $\vartheta (\tau )$ and stated several theorems and conjectures. He first states, as a theorem with a brief outline of proof, that $\vartheta (\tau )$ does not vanish on H. This allows him to define a continuous branch $\log \vartheta (\tau )$ of the logarithm of $\vartheta (\tau )$ and, setting

$$\phi (A)=\frac{4}{\pi i}\{\log \phi (A\tau )-\log \phi (\tau )\}(A\in \mathrm{\Gamma }),$$

he gets a homomorphism φ from Γ to Z, namely an element of ${\mathrm{H}}^1(\mathrm{\Gamma },\mathbf{Z})$. By a general theory on ${\mathrm{H}}^1(\mathrm{\Gamma },\mathbf{C})$ due to Harder, the homomorphism φ is decomposed into the cusp part ${\phi }_c$ and the Eisenstein part ${\phi }_e$ as $\phi ={\phi }_c+{\phi }_e$, where ${\phi }_c$ and ${\phi }_e$ are in general Q-valued. Sczech obtained an expression (which is not explicitly described in [14]) for ${\phi }_e$ in terms of his Dedekind sums ([12]), calculated some examples of values of φ and ${\phi }_e$, and conjectured that the congruence

$$\phi (A)\equiv {\phi }_e(A)(\mathrm{mod}8{\mathbf{Z}}_2)$$

hold for every A in Γ, where ${\mathbf{Z}}_2$ is the ring of 2-adic integers. The purpose of the present paper is to supply detailed accounts for the above mentioned two results due to Sczech (the non-vanishing of $\vartheta (\tau )$ on H and the expression of ${\phi }_e$ in terms of Dedekind sums), together with an explicit formula for the latter result, and to prove his conjecture under a hypothesis concerning a value of an L-function. Some of our previous results in [8] will be used.

Now we shall state our results more explicitly. A general theory due to Harder gives a decomposition

$${\mathrm{H}}^1(\mathrm{\Gamma },\mathbf{C})={\mathrm{H}}_{\mathrm{cusp}}^1(\mathrm{\Gamma },\mathbf{C})\oplus {\mathrm{H}}_{\mathrm{eis}}^1(\mathrm{\Gamma },\mathbf{C}).$$

Here, ${\mathrm{H}}_{\mathrm{cusp}}^1(\mathrm{\Gamma },\mathbf{C})$ consists of $\chi \in {\mathrm{H}}^1(\mathrm{\Gamma },\mathbf{C})$ such that $\chi (A)=0$ for every parabolic element A in Γ and ${\mathrm{H}}_{\mathrm{eis}}^1(\mathrm{\Gamma },\mathbf{C})$ is the subspace of ${\mathrm{H}}^1(\mathrm{\Gamma },\mathbf{C})$ defined as follows. Let L be a lattice in C with complex multiplication by $\mathcal{O}$. Let, for $z\in \mathbf{C}$ and $n\in \mathbf{Z}(n\ge 0)$,

$${E_n(z)=E_n(z,L)=\sum _{w\in L,w+z\ne 0}{(w+z)}^{-n}|w+z{|}^{-s}|}_{s=0},$$

where the value at $s=0$ is to be considered in the sense of analytic continuation. Setting $I(z)=z-\bar{z}$, define ${\mathrm{\Phi }}_L:\mathrm{\Gamma }\to \mathbf{C}$ by the following:

$${\mathrm{\Phi }}_L\left(\begin{array}{cc}\hfill a& \hfill b\\ \hfill c& \hfill d\end{array}\right)=\{\begin{array}{cc}{E}_2(0)I\left(\frac{a+d}{c}\right)-\frac{1}{c}\sum _{r\in L/cL}{E}_1\left(\frac{ar}{c}\right)E_1\left(\frac{r}{c}\right),\hfill & c\ne 0,\hfill \\ E_2(0)I\left(\frac{b}{d}\right),\hfill & c=0.\hfill \end{array}$$

Then, by Sczech [12], ${\mathrm{\Phi }}_L$ is an element of ${\mathrm{H}}^1(\mathrm{\Gamma },\mathbf{C})$. The space ${\mathrm{H}}_{\mathrm{eis}}^1(\mathrm{\Gamma },\mathbf{C})$ is defined as the subspace of ${\mathrm{H}}^1(\mathrm{\Gamma },\mathbf{C})$ generated by all ${\mathrm{\Phi }}_L$ with lattices L having complex multiplication by $\mathcal{O}$. It is known that

$$\dim {\mathrm{H}}^1(\mathrm{\Gamma },\mathbf{C})=h(\mathrm{the\hspace{0.17em}\hspace{0.17em}class\hspace{0.17em}\hspace{0.17em}number\hspace{0.17em}\hspace{0.17em}of}K).$$

Now, we can write

$$\phi ={\phi }_c+{\phi }_e$$

according to the decomposition (1.1). First, we shall write ${\phi }_e$ as a linear combination of ${\mathrm{\Phi }}_L$ by comparing, as is indicated in [14], the both sides of the above equation on parabolic elements of Γ. Denote by $\eta (z)$ the Dedekind η-function and set $L=\mathcal{O}\omega$ with

$$\omega =\frac{\pi }{2\sqrt{3}|D{|}^{1/4}}\cdot \frac{\eta {((1+\sqrt{D})/2)}^4}{\eta {(1+\sqrt{D})}^2}.$$

The image of the homomorphism ${\mathrm{\Phi }}_L$ is contained in the absolute class field H of K and the set $\{{{\mathrm{\Phi }}_L}^{\sigma }|\sigma \in Gal(H/K)\}$ is a basis of ${\mathrm{H}}_{\mathrm{eis}}^1(\mathrm{\Gamma },\mathbf{C})$ over C (Section 4). Let $G=Gal(H/K)=\{{\sigma }_1,\cdots ,{\sigma }_h\}$ and write

$${\phi }_e=\sum _{\sigma \in G}{d}_{\sigma }{{\mathrm{\Phi }}_L}^{\sigma }(d_{\sigma }\in \mathbf{C}).$$

Denote by $C_1,\cdots ,C_h$ the ideal classes of K. We choose, for each $j(1\le j\le h)$, an integral ideal ${\mathfrak{a}}_j\in C_j$ prime to 2 whose norm is minimum among the integral ideals prime to 2 belonging to $C_j$ and put

$$T={\left(E_2{(0,{\mathfrak{a}}_j^{-1}L)}^{{\sigma }_i}\right)}_{1\le i,j\le h},$$

cf. also Lemma 13. As we shall see in Section 5, detT is non-zero.

Theorem 1

We have, for $d_{\sigma }(\sigma \in G)$ in (1.3), that

$$\sqrt{D}(d_{{\sigma }_1},\cdots ,d_{{\sigma }_h})T=(N{\mathfrak{a}}_1^2,\cdots ,N{\mathfrak{a}}_h^2).$$

Next, by the use of the above theorem, we shall study the conjecture in Sczech [14] on a congruence relation between φ and ${\phi }_e$. Take a grössencharacter ψ of K with conductor $\mathcal{O}$ satisfying

$$\psi ((\gamma ))={\bar{\gamma }}^2(\gamma \in K^{\times })$$

and put

$$L_H(\psi \circ N_{H/K},s)=\sum _{\mathfrak{a}}\psi (N_{H/K}(\mathfrak{a}))\cdot N{\mathfrak{a}}^{-s}(\mathrm{Re}(s)>2).$$

Here, $\mathfrak{a}$ runs over all integral ideals of H. We will see that the main part of $L_H(\psi \circ N_{H/K},2)$ is detT and prove the following theorem (Section 5) utilizing results in a previous work [8].

Theorem 2

The number $8{(2\omega )}^{-2h}{L}_H(\psi \circ N_{H/K},2)$ is an integer in H and if it is prime to 2, the congruence

$$\phi (A)\equiv {\phi }_e(A)(\mathrm{mod}8{\mathbf{Z}}_2)$$

holds for every A in Γ.

The congruence (1.4) can be seen as an analogue of a well-known congruence between the classical Dedekind sum and the quadratic residue symbol of Q (cf., for example, [8], §3). We do not have any value of D for which the number $8{(2\omega )}^{-2h}{L}_H(\psi \circ N_{H/K},2)$ is not prime to 2. Since Theorem 3 (the non-vanishing of $\vartheta (\tau )$) holds also in the case $D\equiv 5(\mathrm{mod}8)$, it will not be difficult to extend our results here to this case applying the results in [9].

Although we will be concerned in this paper mainly with congruence relations modulo 8 between φ and ${\phi }_e$, in view of works related to the Eisenstein cohomology (cf., for example, Berger [1]), congruence relations between the cusp part of φ and the Eisenstein part of φ will also deserve to be studied. The homomorphisms ${\phi }_c$ and ${\phi }_e$ are Q-valued (Lemma 11) and we may take the smallest positive integer l such that $\frac{1}{l}\mathbf{Z}$ contains the values of ${\phi }_c$ and ${\phi }_e$. If $l>1$, we have a non-trivial congruence $l{\phi }_c(A)\equiv l{\phi }_e(A)(\mathrm{mod}l)$ for every $A\in \mathrm{\Gamma }$ (Sczech [14] contains a list of values of D for which $l>1$). Various number theoretic consequences are expected from this kind of congruences, cf. [1]. The integer l is a divisor of $\sqrt{D}{({\mathfrak{a}}_1\cdots {\mathfrak{a}}_h)}^2{2}^h{\omega }^{-2h}{L}_H(\psi \circ N_{H/K},2)$ as is seen from Theorem 1 and Lemma 14, and it is also a divisor of the index $[{\mathrm{\Phi }}_L(\mathrm{\Gamma }):{\mathrm{\Phi }}_L({\mathrm{\Gamma }}_{\mathrm{unip}})]$ as can be seen from the proof of Lemma 11, where ${\mathrm{\Gamma }}_{unip}$ is the subgroup of Γ generated by unipotent elements of Γ.

The discussion of this paper proceeds as follows. In §2, we consider an embedding of H into the Siegel upper half plane ${\mathfrak{H}}_2$ of degree 2 and see that the theta function $\vartheta (\tau )$ is the pull-back of a well-studied theta function on ${\mathfrak{H}}_2$. Then, utilizing a known fact on the zeros of the theta function on ${\mathfrak{H}}_2$, we show that $\vartheta (\tau )$ does not vanish on H (Theorem 3). As has been explained, this enables us to define the homomorphism $\phi :\mathrm{\Gamma }\to \mathbf{Z}$. Next, in §3, as a preparation for §4, we calculate the Fourier expansion of $\vartheta (\tau )$ at each cusp of Γ. In §4, by the use of this expansion, we determine the value of φ on parabolic elements of Γ and prove Theorem 1. Finally, in §5, applying a result in [8] to an expression for ${\phi }_e$ obtained from Theorem 1, we prove Theorem 2. As has been mentioned, Theorem 1 and Theorem 3 are essentially due to Sczech [14].