数理解析研究所講究録 1281 巻 2002 年 101-113

101

On certain modular functions derived from modular forms ofweight 1/2(重さ 1/2 のモジュラー形式から得られるある保型関数について)Taro Horie (堀江太郎) and Naruo Kanou (加納或男)
similar to the Dedekind eta function
Abstract: In this article, we study certain modular functions
and report some results on them. These functions are given by infinite products of Borcherds’ type.
have good properties relating to some areas, it turns out that the functions
As in the case ofof number-theory. Especially, we investigate their automorphic properties in detail.
Keywords: Dedekind eta function, Dirichlet characters, Borcherds products, Kronecker’s limit formulae,
twisted Dedekind sums, reciprocity formula, elliptic functions, Klein forms$\eta_{\psi}(z)$$\eta(z)$$\eta\psi(z)$$\eta(z)$1Introduction
Let be aDirichlet character. We assume thatIn this paper we study the function$\psi$$\psi$

is even and primitive.

$(q=\exp(2\pi iz))$$\eta\psi(z):=q^{-1_{L(-1,\psi)}}2\prod_{n=1}^{\infty}(1-q^{n})^{\psi(n)}$,where $L(s, \psi)$ denotes the Dirichlet -function attached to . We mainly consider the case of quadratic
characters. In this case the definition is unambiguous, and afew examples already appeared in Ramanujan’s Notebooks ([Be3], [Ra3]). We here take asystematic approach and show that the functions
have good properties similar to those of the Dedekind eta function$L$$\psi$$\eta\psi(z)$r)(z) $= \eta_{1}(z)=q\prod_{n=1}^{\infty}+_{4}(1-q^{n})$($1=\mathrm{t}\mathrm{h}\mathrm{e}$trivial character).

for
There exists amajor difference: the Dedekind eta function has weight 1/2, but the functions.
when
is not aproduct nor aquotient of
have weight 0. We also remark thatliterature,
cf.
There have been various character analogues of the Dedekind eta function defined in the
also
are
Our
functionsBemdt [Bel], [Be2], Goldstein and Razar [G-R], Meyer [Me] and others.
one of such analogues, and from some points of views they seem to be anatural generalization of.
(see the notations) offor the congruence subgroup$\eta\psi(z)$$\psi$$\eta(z)$$\eta\psi(z)$$\neq 1$$\psi$$\neq 1$$\eta\psi(z)$$\eta(z)$$\Gamma_{0}(N)$$\mathrm{S}\mathrm{L}_{2}(\mathrm{Z})$Notations.
We use the notation $\mathrm{e}(*):=\exp(2\pi i*)$ .
We denote by $H$ $=\{z\in \mathrm{C}|\Im z>0\}$ the complex upper half plane. For $z\in H$ , we write$q:=\mathrm{e}(z)=\exp(2\pi iz)$.

102
For the logarithm logiu $(w\neq 0)$ , we choose the principal branch
For aDirichlet character , we denote by $=r(\psi)$ the conductor of$\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}-\pi<\arg w\leq\pi$$r$$\psi$$\psi$.

and by$G( \psi):=\sum_{n=1}^{r(\psi)}\psi(n)\mathrm{e}(n/r(\psi))$the Gauss sum attached to .
We define the congruence subgroup of level $N$ of$\psi$$\mathrm{S}\mathrm{L}_{2}(\mathrm{Z})$by$\Gamma_{0}(N)=\{$ $(\begin{array}{ll}a bc d\end{array})\in \mathrm{S}\mathrm{L}_{2}(\mathrm{Z})|c\equiv 0$$(\mathrm{m}\mathrm{o}\mathrm{d} N)\}$.

For each natural number , we letbe the primitive Dirichlet character associated to the quadratic(If is asquare number, we let $\chi_{t}:=1$ be the trivial character).
field
$t$

$\mathrm{Q}(\sqrt{t})$$\chi_{t}$$t$2Construction of modular functions
Let$\psi$

be an even primitive Dirichlet character. We define the theta series twisted by$\theta_{\psi}(z):=\sum_{n\in \mathrm{Z}}\psi(n)q^{n^{2}}$,$q=\exp(2\pi:z)$$\psi$:.

We know
is amodular form of weight 1/2 and character $with respect toconductor of ([Sh, Proposition 2.2], see also [S-S]).
$\theta_{\psi}$$\Gamma_{0}(4r^{2})$, where$r$is the$\psi$

Let $D>0$ be afundamental discriminant and let
Following [Bo, Theorem 14.1], for$\chi D=(^{\underline{D}})$$\theta_{XD}(z)=\sum_{n\in \mathrm{Z}}\chi_{D}(n)q^{n^{2}}$be the associated quadratic character.

, we define.

(1)$\mathrm{L}\mathrm{i}\mathrm{f}\mathrm{t}(\theta_{XD})(z)=\eta_{XD}(z):=q^{-*L(-1,\chi_{D})}\prod_{n=1}^{\infty}(1-q^{n})^{\mathrm{X}D(n)}$Theorem 1(transformation formula). Let the notations be as above. Then, for $D>1$ ,(quasi) modular function forwith the transfomation formula$\eta_{XD}(z)$

is

$a$

$\Gamma_{0}(D)$$\eta_{\chi \mathit{0}}(\frac{az+b}{cz+d})=\nu_{D}(M)\chi_{D}^{*}(d)\eta_{XD}(z)^{XD(d)}$for all$M=(\begin{array}{ll}a bc d\end{array})$$\in \mathrm{T}_{\mathrm{q}}(\mathrm{D})$,(2)

stands forwhereor 1according as $D$ is a prime or not, and $\nu_{D}(M)$ is a root of unity.
$\nu_{D}(M)$The rsrrnber
is a 5-th root of 1if $D=5$ , a 2-th root of 1 $l.fD=8$ and always equals to 1if$D>8$, respectively (see Theorems 3, 7and (22) for an explicit $fom$). In particular,,, whereand $\eta_{XD}(z)(D>8)$ are modular functions for respective$\chi_{D}^{*}$$\chi_{D}$$\eta_{\chi\epsilon}(z)^{5}$$\eta_{\chi\epsilon}(z)^{2}$$\Gamma_{0}(D, \chi_{D})$$\Gamma_{0}(D, \chi_{D}):=\{$ $(\begin{array}{ll}a bc d\end{array})\in\Gamma_{0}(D)|\chi_{D}(d)=1\}$(which is a subgroup of index 2in

The function$\eta_{\chi D}$$\Gamma\circ(D)$).

does not have zeros nor poles in the complex upper half plane $H$ .

We emphasize that our functions have atransformation foumula for. We also remark that there
exists amajor difference between the case $D=1$ (weight 1/2) and $D>1$ (weight 0).
$\Gamma_{0}(D)$Except for the determination of the root of unity, the theorem is obtained ffom one of(1) Weil’s converse theorem for weight 0,

103(2) Kronecker’s limit formulae,(3) Klein forms (This method determines the root of unity, and is discussed in detail in Sects. 4-8).

Proof. In this section, we sketch the method (1). We prepare the Weil’s converse theorem for weight 0.

This method is closely related to the work by L. J. Goldstein and M. Razar [G-R] which treats “Hecke
integrals” in ageneral setting.
The functions$f^{*}(z):= \log\eta_{XD}(z)=-L(-1, \chi_{D})\pi iz-\sum_{m,n\geq 1}\frac{\chi_{D}(m)}{n}\exp(2\pi imnz)$

and$g(z):=- \frac{G(\chi_{D})}{2}L(1,\overline{\chi}_{D})-G(\chi_{D})\sum_{m,n\geq 1}\frac{\overline{\chi}_{D}(m)}{m}\exp(2\pi imnz)$(3)$f^{*}(-1/Dz)=g(z)$ ,(4)

satisfywhich can be shown as in the proof of [Mi, Lemma 4.4.1]. Then, we can apply to $f^{*}(z)$ and $g(z)$ theWeil’s converse theorem for weight 0, which is adirect analogue of [Mi, Theorem 4.3.15] (weight $>0$ ) forthe case of weight 0. Thus we obtain the statement except for the determination of the root of unity.
$\square$

We shall determine the root of unity (automorphic factor) by two different methods in Sects. 3and 8.

and the Dedekind eta function. S. Ramanujan
Next, we mention some interesting relations between[Be3] or [Ra3]) discovered the formulas
in
Chap.
(iii)
19
of
(see Entry 10 (iii), (viii) and Entry 11$\eta_{XD}$$\frac{1}{\eta_{\chi_{\acute{0}}}(z)}-1-\eta_{\chi \mathrm{s}}(z)$$=$$\frac{1}{q^{1/5}}\prod_{n=1}^{\infty}(1-q^{n})^{-\chi_{5}(n)}-1-q^{1/5}\prod_{n=1}^{\infty}(1-q^{n})^{X\mathrm{s}(n)}$$=$$\frac{1}{q^{1/5}}\prod_{n=1}^{\infty}\frac{1-q^{n/5}}{1-q^{5n}}=\frac{\eta(z/5)}{\eta(5z)}$$\frac{1}{(\eta_{\chi \mathrm{s}}(z))^{5}}+11-(\eta_{XS}(z))^{5}$

$=$

(5)

,

$( \frac{\eta(z)}{\eta(5z)})^{6}$

The first proof of these famous formulas was given by G. N. Watson. See also B. C. Berndt’s book[Be3, pp. 265-267], where ashort proof based on other formulas of Ramanujan, is given. There areother proofs in the literature: K. G. Ramanathan [Ra2, pp. 696-697] gave another proof for (5); M. D.

Hirschhorn has given more than one proof.
In the case of other , we have similar results:$\mathrm{D}$Theorem 2. For $D=13$ , we have$\frac{1}{\eta_{X13}(z)}-3-\eta_{X13}(z)=(\frac{\eta(z)}{\eta(13z)})^{2}$

For $D=8$ , we have$\frac{1}{\eta_{\chi\epsilon}(z)^{2}}-6+\eta_{\chi_{8}}(z)^{2}=\frac{\eta(z)^{4}\eta(4z)^{2}}{\eta(2z)^{2}\eta(8z)^{4}}$.

For $D=12$ , we have.

$\frac{1}{\eta_{\chi_{12}}(z)}-4+\eta_{X12}(z)=\frac{\eta(z)^{3}\eta(4z)\eta(6z)^{2}}{\eta(2z)^{2}\eta(3z)\eta(12z)^{3}}$104, which is easily seen by the transformation formula(2) (for the left-hand side) and by [CN, pp.331-333] (for the right-hand side). We evaluate both modular
functions at the cusps, and obtain the results.

Proof. Both sides are modular functions for$\Gamma_{0}(D)$$\square$

3Automorphy and reciprocity
We shall carry out the determination of the root of unity $\nu D(M)$ in Theorem 1. Note that for
we havefundamental discriminants $D=1,5,8$ , 12, 13, 17, 21, 24, 28, 29, 33,$\ldots$$- \frac{1}{2}L(-1, \chi_{D})=\frac{1}{24}$

For $D=1,5,8$ ,As amaP,,$\frac{1}{5},$$\chi_{D}$, 1, 1, 2, 2, 3, 4, 3, 6,(cf. [Za, Sect. 7]).

$\ldots$

has anon-trivial multiplier system $\nu D(M)$ .
is aquasi multiplier system in the sense that$\eta_{XD}$$\nu_{D}(*)$

Definition. We call amap $\nu_{D}(M)$ :character$\frac{1}{2}$, if for any$\Gamma_{0}(D)arrow \mathrm{C}^{1}$$M_{1}=(\begin{array}{ll}a_{1} b_{1}c_{1} d_{1}\end{array})$,($=\mathrm{t}\mathrm{h}\mathrm{e}$unit circle) aquasi multiplier system with

$M_{2}=(\begin{array}{ll}a_{2} b_{2}c_{2} d_{2}\end{array})$$\nu_{D}(M_{1}M_{2})=\nu_{D}(M_{1})\nu_{D}(M_{2})^{\mathrm{X}D(d_{1})}$

Theorem 3. The number $\nu_{D}(M)$ in Theorem 1where,$\in\Gamma_{\phi}(D)$.

$M=(\begin{array}{ll}a bc d\end{array})$$\in\Gamma_{0}(D)$, is explicitly given as

follows. For $D=5$ , we have$\mathrm{v}\mathrm{d}(\mathrm{M})=\mathrm{e}(\frac{1}{5}\chi_{5}(d)\mathrm{W})=\mathrm{e}(\frac{1}{5}ab)$.

For $D=8$ , we have$\nu_{8}(M)=\mathrm{e}(\frac{1}{2}(\frac{a-d+bc}{8}-b))$.

For $D>8$ , we have$\nu_{D}(M)=1$.

Proof. The proof uses what is called (tthe method by Riemann and Dedekind”. First we find that acertain Dedekind’s type sums $S(\chi D;c, a)$ appear in the transformation formula ofan explicit formula for $\nu D(M)$ in terms of the components of any matrix$\log\eta_{\chi D}$. Next we find

$M=(\begin{array}{ll}a bc d\end{array})$$\in\Gamma_{0}(D)$andprove it by induction. In the case of $D=1$ , $\nu_{1}(M)$ is nothing but the multiplier system of the Dedekind
eta function
and this analysis is well-known (cf. [Ral, Chap. 9]).
$\eta(z)$$\mathrm{O}$

From the first part of the proof of Theorem 3, we derive aremarkable formula:
Theorem 4. We define the number$\phi_{D}(M)\in \mathrm{Q}$by$\log\eta_{XD}(Mz)=\chi_{D}(d)\log\eta_{\mathrm{X}D}(z)+2\pi:\phi_{D}(M)$The map$\phi_{D}$:$\Gamma_{0}(D)arrow \mathrm{Q}$,$M=(\begin{array}{ll}a bc d\end{array})$$\in\Gamma_{0}(D)$satisfies$\phi_{D}(M_{1}M_{2})=\phi D(M_{1})+\chi_{D}(d_{1})\phi_{D}(M_{2})$$(\forall M_{1}, \forall M_{2}\in \mathrm{T}0(\mathrm{D}))$

,

.

105
where

$d_{1}$

is the right lower entry of$M_{1}$. Then, for$M=(\begin{array}{ll}a bc d\end{array})$$\in\Gamma_{0}(D)$with $c\neq 0$ , we have.

$\phi_{D}(M)=-\frac{1}{2}L(-1, \chi_{D})\frac{a+\chi_{D}(d)d}{c}-\frac{1}{2}S(\chi_{D};c, a)$Here we define the “twisted Dedekind sum” by,$S( \chi_{D};c, a):=\mathrm{s}\mathrm{g}\mathrm{n}(c)\sum_{m(\mathrm{m}\mathrm{o}\mathrm{d} \mathrm{c})}\chi_{D}(m)((\frac{m}{c}))((\frac{am}{c}))$where $((x))=x-[x]-1/2$ is the (modified) fractional part of

and

$x$

(6)

$\mathrm{s}\mathrm{g}\mathrm{n}(c):=c/|c|$.

$\square$Remark.
(i) Many authors studied various character analogues of the logarithm of the Dedekind eta function,
cf. [Bel], [Be2], [G-R], [Me], [Na]. In each case, certain generalizations of Dedekind sums attached
to Dirichlet characters arise in the transformation formulas. In particular, Theoerem 4was essentiallyobtained in [Be2, (6.5)] by adifferent method. We mention that the definition of character analogues of$\log\eta(z)$ in [Be2] omitted the terms involving special values of the Dirichlet L-function.
(ii) We can regard our twisted Dedekind sums $S(\chi_{D}; c, a)$ as “periods” of Eisenstein series of weight 2.
Now, we find areciprocity formula for twisted Dedekind sums.

Theorem 5. For$M=(\begin{array}{ll}a bc d\end{array})$$\in\Gamma \mathrm{o}(D)$with $bc\neq 0$ , the following reciprocity formula holds:,$S( \chi_{D};c, a)+S^{*}(\chi_{D}; Db, d)=-L(-1, \chi_{D})\frac{a+\chi_{D}(d)d}{c}$where we define the other “twisted Dedekind sum” by$S^{*}( \chi_{D}; c, a):=\frac{G(\chi_{D})}{4c}\sum_{0\neq m(\mathrm{m}\mathrm{o}\mathrm{d} \mathrm{c})}\chi_{D}(m)\cot(\pi\frac{m}{c})\cot(\pi\frac{am}{c})$.

(7)Proof. For $g(z)=\log\eta_{XD}(-1/Dz)$ defined by (3), in asimilar way to the proof of Theorem 4(themethod by Riemann and Dedekind), we can prove$\phi_{D}^{*}(M)=-\frac{1}{2}S^{*}(\chi_{D};c, a)$

Therefore, for$M=(\begin{array}{ll}a bc d\end{array})$$\in \mathrm{T}\mathrm{o}(\mathrm{D})$.

, we have$\log\eta_{\chi_{D}}(\frac{-a/Dz+b}{-c/Dz+d})=\chi_{D}(d)\log\eta_{\chi_{D}}(-1/Dz)-\pi iL(-1, \chi_{D})\frac{a+\chi_{D}(d)d}{c}-\pi iS(\chi_{D};c, a)$(by Theotem 4) on the one hand and$\log\eta_{XD}(\frac{-a/Dz+b}{-c/Dz+d})$$=$$\log\eta_{\chi_{D}}(\frac{Dbz-a}{Ddz-c})$$=$$\log\eta_{\chi D}(-1/Dz’)$$=$$g(z’)=g( \frac{dz-c/D}{-Dbz+a})$$=$$\chi D(a)g(z)-2\pi i\frac{1}{2}S^{*}(\chi D;-Db, d)$$(z’:= \frac{dz-c/D}{-Dbz+a})$(by (4))

106
on the other hand. By comparing these two expressions, we get the result.

$\square$Remark.
(i) For $D>1$ , two twisted Dedekind sums $S(\chi D;c,a)$ and $S^{*}(\chi D;c,a)$ defined by (6) and (7) do notcoincide with each other, in general.
(ii) Related results for generalized Dedekind sums attached to Dirichlet characters were obtained by
Berndt [Bel, Theorems 4, 7], [Be2, Theorem 7] and Nagasaka [Na, Theorem 2].

4The Legendre relation
We need so called the Legendre relation (Riemann condition, CCR
commutative relation,etc.). First, we define the notion of Eisenstein summation. For some one-variable function $f(x)$ , wedefine the Eisenstein sum of by$=\mathrm{c}\mathrm{a}\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{c}\mathrm{d}$$f$$f(m)= \lim_{Marrow\infty}\sum_{m=-M}^{M}f(m)$$\sum_{m}\mathrm{e}$For the case of two variable functions, we define$\sum_{m,n}\mathrm{e}=\sum_{m}\mathrm{e}$$\sum_{n}\mathrm{e}$.

. We define the Eisenstein product by asimilar way. If we write, then it means that we remove $m=0$ or $(m,n)=(0,0)$ from the summation.
The infinite product of the sine function is written by$\sum_{\mathrm{e}}’$$\sin x=x\prod_{n=1}^{\infty}(1-\frac{x^{2}}{\pi^{2}n^{2}})=x\prod_{n}!(1-\frac{x}{\pi n})$From this formula, we deduce (for$x\not\in\pi \mathrm{Z}$

.

(8)

)

$\frac{\sin(x-t)}{\sin(x)}=\prod_{n}\mathrm{e}(1-\frac{t}{x+n\pi})$

Next, we consider Eisenstein series for aposivie integer

$k$

.

(9)

and abasis$\varpi$$=$$(\begin{array}{l}\varpi_{2}\varpi_{1}\end{array})$of alattice in C.

We define$e_{k}( \varpi)=\sum_{n}\mathrm{e}\frac{1}{n\varpi^{k}}’=\sum_{n_{1},n_{2}}\mathrm{e}’\frac{1}{(n_{1}\varpi_{2}+n_{2}\varpi_{1})^{k}}$and$E_{k}(u; \varpi)=\sum_{n}\mathrm{e}\frac{1}{(u+n\varpi)^{k}}=\sum_{n_{1}.n_{2}}\mathrm{e}\frac{1}{(u+n_{1}\varpi_{2}+n_{2}\varpi_{1})^{k}}’$,where $n=(n_{1}n_{2})$ . For larger $k(\geq 3)$ , these series are independent of the summation process. We
take another basisof some sublattice of the lattice which is generated by . Then,satisfies the
distribution relation$\varpi’$$\varpi$$e_{2k}$, , if $2k=2$ ,$e_{2}( \varpi)-\frac{2\pi\cdot\delta c}{\varpi_{1}\varpi_{1}}.$$e_{2k}( \varpi’)+\sum_{r\in R’}E_{2k}(r;\varpi’)=\{$$e_{2k}(\varpi)$

Here we write $\varpi’=\gamma\varpi=$$(\begin{array}{ll}a bc d\end{array})$

modulo the sublattice generated byorientation of , , it is equal$\varpi$$\mathrm{i}.\mathrm{e}$$\varpi$(10)

if $2k\geq 4$ .

, take $R$ aset of representatives of the lattice generated by$\varpi$

and put $R’:=R-\{0\}$ . The number $=\pm 1$ is the sign of theifis on the upper half plane ??,otherwise$\varpi’$$\mathrm{t}\mathrm{o}+1$,$\delta$$z=\varpi_{2}/\varpi_{1}$$\mathrm{a}\mathrm{n}\mathrm{d}-1$

107
Especially, for$\gamma=$$(\begin{array}{ll}0 11 0\end{array})$

and $\Im z>0$ , then$\delta=1$, $R’$ is empty and (10) becomes$e_{2}( \varpi’)-e_{2}(\varpi)=-\frac{2\pi i}{\varpi_{1}\varpi_{2}}$By comparing the Fourier development of$e_{2}$.

(11)

and the eta constant, we know$e_{2}( \varpi)=\frac{\eta_{1}(\varpi)}{\varpi_{1}}$.

Therefore, the equation (11) is nothing but the Legendre relation$\eta_{1}\varpi_{2}-\eta_{2}\varpi_{1}=2\pi i$

where$\eta_{2}:=\eta_{1}(\gamma\varpi)$

,

(12)

.

Remark.
All the facts in this section are found in the text book of Weil [We2], in particular Chap. III.

5Some properties of the sigma function
We define the Weierstrass sigma function by the infinite product,$\sigma(u;\varpi)=u\prod_{n}(\prime 1-\frac{u}{n\varpi})\exp(\frac{u}{n\varpi}+\frac{u^{2}}{2n\varpi^{2}})$

where$u\in \mathrm{C}$,$\varpi=$$(\begin{array}{l}\varpi_{2}\varpi_{1}\end{array})$

and we denote$n=(n_{1}n_{2})\in \mathrm{Z}^{2}$

(13)

.

We apply the Eisenstein product for this product (we call this operation “observation”):$\sigma(u;\varpi)$

$=$

$=$

$u \prod_{n}\mathrm{e}(\prime 1-\frac{u}{n\varpi})\exp(\frac{u^{2}}{2n\varpi^{2}})$$ue^{\}u^{2}e_{2}(\varpi)} \prod_{n}\mathrm{e}(\prime 1-\frac{u}{n\varpi})$.

Moreover, by using (8) and (9), we have$\sigma(u;\varpi)$$=$$\frac{\varpi_{1}}{\pi}\exp(\frac{1}{2}u^{2}e_{2}(\varpi))\sin(\frac{\pi u}{\varpi_{1}})\prod_{n_{1}}\frac{\sin(\frac{\pi(n_{1}\varpi_{2}-u)}{\varpi_{1}})}{\sin(\begin{array}{l}\underline{\pi n}\mapsto\varpi\varpi_{1}\end{array})}\prime \mathrm{e}$$=$$\frac{\varpi_{1}}{2\pi i}\exp(\frac{1}{2}\varpi_{1}\eta_{1}(\varpi)v^{2})(x\#$, $v=u/\varpi_{1}$ ,
where
and
In fact #11 has the triple product expansion$2=\varpi_{2}/\varpi_{1}$$q=\mathrm{e}(z)$$\theta_{11}(v;z)=-iq(:x:$x=\mathrm{e}(v)$$-x^{-:}) \prod_{n=1}^{\infty}\frac{(1-x^{-1}q^{n})(1-xq^{n})}{(1-q^{n})^{2}}$

,

(14)

. This is almost equal to the Jacobi theta function.

-x^{-\int})\prod_{n=1}^{\infty}(1-q^{n})(1-x^{-1}q^{n})(1-xq^{n})$

.

(15)

Combining (14) and (15), we have.

$\sigma(u;\varpi)=-\frac{\varpi_{1}}{2\pi}\exp(\frac{1}{2}\varpi_{1}\eta_{1}(\varpi)v^{2})\frac{\theta_{11}(vz)}{\eta(z)^{3}}$

(16)

108
This $\eta(z)=q^{1/24}\prod(1-qn)$ is the Dedekind eta function.
Next, we consider the automorphic properties of the sigma function. We restrict our consideration to, then the action
a2)(. Assume that is independent of andonly the action by$u$$\mathrm{S}\mathrm{L}_{2}(\mathrm{Z})$by$\gamma=(\begin{array}{ll}a bc d\end{array})$$\in \mathrm{S}\mathrm{L}_{2}(\mathrm{Z})$on$\varpi$$\alpha=$$\varpi$$\in \mathrm{R}^{2}$$\alpha_{1}$is as the following:$\sigma(u+\alpha\gamma\varpi;\gamma\varpi)=\sigma(u+\eta\varpi;\varpi)$.

In particular, if $\alpha=0$ , by (16), we have
$1= \frac{\sigma(u+\alpha\varpi\gamma\varpi)}{\sigma(u+\alpha\varpi\varpi)}=\frac{\varpi_{1}’}{\varpi_{1}}\frac{\theta_{11}(v’z’)}{\theta_{11}(vz)}\frac{\eta(z)^{3}}{\eta(z)^{3}},\exp(\frac{u^{2}}{2}(\frac{\eta_{1}’}{\varpi_{1}},-\frac{\eta_{1}}{\varpi_{1}}))$

where$\varpi_{1}’=c\varpi_{2}+d\varpi_{1}$,, $\eta_{1}’=O|2+d\eta_{1}$ , $v’=v/\varpi_{1}’$ and $z’=(az+b)/(cz+d)$ . Note that$\frac{\eta_{1}’}{\varpi_{1}},-\frac{\eta_{1}}{\varpi_{1}}=\frac{2\pi\dot{\kappa}}{\varpi_{1}(c\varpi_{2}+d\varpi_{1})}$,by the distribution relation (10). Now, we shall use the automorphy of the Dedekind eta function$\eta(z’)=\epsilon_{\gamma}(cz+d)^{\}}\eta(z)$If we put$\varpi=Z=(\begin{array}{l}z1\end{array})$,$|\epsilon_{\gamma}|=1$.

, then we obtain the automorphy of the Jacobi theta function

.

(17)

$\theta_{11}(\frac{v}{cz+d};\frac{az+b}{cz+d})=\epsilon_{\gamma}^{3}\mathrm{e}(\frac{\mathrm{c}v^{2}}{2(cz+d)})(cz+d)^{\}}\theta_{11}(v;z)$

Put $\beta_{11}=(1/21/2)$ . For
theta functions are defined by$/\mathit{3}=( \beta_{1} \beta_{2})$$=\alpha+\beta_{11}=(\alpha_{1}\alpha_{2})+(1/21/2)\in \mathrm{R}^{2}$, the characterized$\theta_{\beta}(v;z)=\mathrm{e}(\frac{1}{2}\alpha_{1}^{2}z+\alpha_{1}v+\alpha_{1}\ ) \theta_{11}(v+\alpha Z;z)$(hence$\theta_{\beta_{11}}=\theta_{11}$). We deduce the automorphy of these functions by (17):$\theta_{\beta}(\frac{v}{cz+d};\frac{az+b}{cz+d})=\epsilon_{\gamma}^{3}(cz+d)^{\}}\mathrm{e}(\frac{cv^{2}}{2(cz+d)})\delta_{\gamma}(\alpha)\theta_{\beta’}(v;z)$$\delta_{\gamma}(\alpha)=\mathrm{e}(\alpha_{1}’\alpha_{2}’-\alpha_{1}\alpha_{2}+\alpha_{1}(\alpha_{1}+\frac{1}{2})-\alpha_{1}’(\alpha_{1}’+\frac{1}{2}))$

where
and
the Jacobi theta functions:$\alpha’=\alpha\gamma$$\beta’=\alpha’+\beta_{11}$. Specially, if we take$\beta\in((1/2)\mathrm{Z})/\mathrm{Z})^{2}$

,

(18)

,

, we obtain the automorphy of

if $\beta=(1/21/2)=\beta_{11}$ ,$((a+\mathrm{c})(6+d-a-c+1)-1)/4$ , if $\beta=(00)$ ,$a(b-a+1)/4$ ,if $\beta=(1/20)$ ,$c(d-c+1)/4$ ,if $\beta=(01/2)$ ,1,$\delta_{\gamma}(\alpha)=\{$

for$\gamma=(\begin{array}{ll}a bc d\end{array})$$\in \mathrm{S}\mathrm{L}_{2}(\mathrm{Z})$.

, the automorphy is not within one theta function, and some authors describe it under theFor.
restriction to suitable subgroups of$\alpha\neq 0$$\mathrm{S}\mathrm{L}_{2}(\mathrm{Z})$109
Adirect proof of the automorphy (18) is given by Ibukiyama [Ib]. Our proof is the same as Weber [Wei],
Weil [We2], Rademacher [Ral] and others, but we avoid the calculation of .
$\epsilon_{\gamma}$

The sigma function has afactorization into automorphic funcitons. Put $f(x)=(1-x)e^{x+x^{2}/2}$ . We
then define the -th part of the sigma function:$n$$\sigma^{(n)}(u;\varpi):=\prod_{\mathrm{g}\mathrm{c}\mathrm{d}(m_{1},m_{2})=n}f(\frac{u}{m\varpi})=\sigma^{(1)}(\frac{u}{n};\varpi)$

We define$(\mathrm{n})\mathrm{g}(\mathrm{w}):=g$ $(\begin{array}{l}n\varpi_{2}n\varpi_{1}\end{array})$and put$T:= \sum_{n=1}^{\infty}\langle n\rangle$.

. Then the original sigma funciton is written by$\sigma(u;\varpi)=u(\sigma^{(1)}(u;\varpi))^{T}=u(\sigma^{(1)}(u;\varpi))^{\zeta(-1,\lambda)}$where we put Ap:$=\langle p\rangle$

and write$\mathrm{C}(\mathrm{s}, \lambda):=\prod_{p}(1-\lambda_{p}^{-s})$. This argument is the same for Eisenstein series.

We need aslightly different automorphy from above. We consider it in the next section.
In the rest of this section, we shall prove the quasi-periodicity of the sigma funciton. It is well-h own
and simply obtained in the system of Weierstrass’s elliptic functions. But we would like to get it directlyfrom our starting definiton (13). Shortly saying, the translation of
direction has no effect in the
Eisenstein product (because
and is an invariant of translation of integers), and the effectappears in the exponential factor. The translation of
direction appears in the Eisenstein productas afinite quotient. Above two are not independent. Finally, the Legendre relation (12) decides their
twist to the quasi-periodicity as awell-k own form.
, we have the quasi-periodicity
For$\varpi_{1}’ \mathrm{s}$$\varpi_{1}arrow 1$$q$$\varpi_{2}’ \mathrm{s}$$n’=(n_{1}’n_{2}’)\in \mathrm{z}^{2}$$\sigma(u+n’\varpi;\varpi)=(-1)n_{1}’+n_{2}’+n_{1}’n_{2}’e\int n’\eta(2u+n’\varpi)\sigma(u;\varpi)$

where$\eta=$$(\begin{array}{l}\eta_{2}\eta_{1}\end{array})$is the eta constants with respect to the basis $.

In fact, we see$\frac{\sigma(u+n’\varpi,\varpi)}{\sigma(u,\varpi)}.\cdot$.

$=$$\frac{u+n’\varpi}{u}\prod_{n}\frac{1-\frac{u+n’\varpi}{n\varpi}}{1-\frac{u}{n\varpi}}e^{\frac{\prime}{1}j_{\varpi}^{\mathrm{g}}+\frac{1}{2(n\mathrm{w})^{2}}\{2un’\varpi+(n’\varpi)^{2}\}}’$$=$$\prod_{\mathrm{e}}(1-\frac{-n’\varpi}{u+n\varpi})e^{\mathrm{e}}\neq\{2un’\varpi+(n’\varpi)^{2}\}$$=$$e^{(u+:n’\varpi)n’\varpi e_{2}} \prod_{n_{1}}\mathrm{e}\prod_{n_{2}}\mathrm{e}(1-\frac{-n’\varpi}{u+n_{1}\varpi_{2}+n_{2}\varpi_{1}})$$=$$e^{(u+:n’\varpi)(n’\eta+_{\varpi_{1}}^{\mathrm{Z}\mathrm{E}1}n_{1}’)} \prod_{n_{1}}\mathrm{e}\frac{\sin\pi(v+n_{1}z-n_{1}’z-n_{2}’)}{\sin\pi(v+n_{1}z)}$$=$$e^{(u+:n’\varpi)n’\eta+2\pi:vn_{1}+\pi j(n_{1}’z+n_{2}’)n_{1}’}(-1)^{n_{2\frac{\sin\pi(v-n_{1}’z)}{\sin\pi v}}’}$$\mathrm{x}\lim_{Narrow\infty}\prod_{n_{1}=1}^{N}\frac{\sin\pi(v+(n_{1}’+n_{1})z)\sin\pi(v+(n_{1}’-n_{1})z)}{\sin\pi(v+n_{1}z)\sin\pi(v-n_{1}z)}$$=$$(-1)n_{2}’e(u+_{2}^{1}n’ \varpi)n’\eta x^{n_{1}’}q(\dotplus^{r2}\iota-1)n_{1}’n_{2}’q’.\frac{1-xq^{-n_{1}’}}{1-x}\neq^{l}$

,

(19)

110
$\mathrm{x}\lim_{Narrow\infty}\prod_{n_{1}=1}^{N}\frac{(1-xq^{n_{1}-n_{1}’})(1-x^{-1}q^{n_{1}-n_{1}’})}{(1-xq^{n_{1}})(1-x^{-1}q^{n_{1}})}$$(-1)^{n_{2}’+n_{1}’n_{2}’}e^{(u+\}n’\varpi)n’\eta}x^{n_{1}’}q^{\}n_{1}’(n_{1}’+1)_{\frac{1-xq^{-n_{1}’}}{1-x}}}$$=$$\mathrm{x}\lim_{Narrow\infty}\prod_{l=1}^{N}\frac{(1-xq^{N+l})(1-x^{-1}q^{1-l})}{(1-xq^{l})(1-x^{-1}q^{N-l})}$$=$$(-1)n_{2}’+n_{1}’n_{2}’e(u+:n’ \varpi)n’\eta xq\frac{1-xq^{-n_{1}’}}{1-x}n_{1}’\}n_{1}’(n_{1}’+1)\frac{1-x^{-1}}{1-x^{-1}q^{-n_{1}’}}\prod_{l=1}^{n_{1}’}(-x^{-1}q^{-\downarrow)}$$=$$(-1)^{n_{1}’+n_{2}’+n_{1}’n_{2}’}e^{(u+\}n’\varpi)n’\eta}$.

As mentioned above, we used the Legendre relation (12).
Remark.
The first infinite product (13) of the Weierstrass sigma function has the most higher symmetric shape.
When we fix aspecified generator of the lattice, this symmetry breaks. But the Legendre relation
restores it in many scenes. Symmetry means “indetermination”, therefore we named the process of “fixbasis” and “apply the Eisensten summation (product)” “observation”. We consider that arelation of
two observations has aright to be called canonical commutative relations as quantum mechanics. But
we do not know the exact correspondence between these terminology.
Indetermination has two faces for the sigma function: the first is indetermination between and(where 7is in), and the second is that between and. The first means the automorphy,
and the second means the periodicity.
Observation breaks the symmetrical -expression to aspecial -expression. Therefore, trivial things
on may be complex on .
Needless to say, there are many remaining problems. In particular, we did not consider the automorphyof the original Dedekind eta function. Does the eta function
have ahomogeneous -expression?
Our only knowledge is thatis aproduct of 7Klein forms.
We see the sigma function as asolution of the multiplicative Cousin problem for lattices on C. Therefore, we can imagine that if we find an explicit solusion of the multiplacative Cousin problem for lattices
onthen it is an automorphic function of degree .
$\varpi$$\mathrm{G}\mathrm{L}_{2}(\mathrm{Z})$$u$$u+\mathrm{n}\mathrm{m}$$\varpi$$\varpi$$\gamma\varpi$

$q$

$q$

$\eta(z)$$\varpi$$\eta^{2}$$\mathrm{C}^{g}$$g$

6Klein forms
In this section we introduce the Klein forms and show their automorphy. Fix the notations: $A(M)=$$[0,1)^{2}$resp.
or(resp.
foror). Forand , we define the Klein form$\mathrm{Q}^{2}\cap[0,$$($$A(\mathrm{R})$$1)^{2}$$((1/N)\mathrm{Z}/\mathrm{Z})^{2}\cap[0, 1)^{2})$$M=\mathrm{R}$$\mathrm{Q}$$(1/N)\mathrm{Z}$$\alpha=(\alpha_{1}\alpha_{2})\in$$\varpi$.

$\mathrm{k}_{\alpha}(\varpi)=e^{-\}\alpha\eta\alpha\varpi}\sigma(\alpha\varpi_{j}\varpi)$

We see these functions are automorphic functions for
Theorem 6. For$\mathrm{S}\mathrm{L}_{2}(\mathrm{Z})$$\gamma\in \mathrm{S}\mathrm{L}_{2}(\mathrm{Z})$$\mathrm{k}_{\alpha}(\gamma\varpi)=(-1)^{n_{1}+n_{2}+n_{1}n_{2}}\mathrm{e}(\frac{1}{2}\alpha\gamma$

where$\alpha’\in \mathrm{A}(\mathrm{R})$in the sense as follows:

eryith $\alpha\gamma-\alpha’=n=(n_{1}n_{2})\in \mathrm{Z}^{2}$ .

$(\begin{array}{ll}0 1-1 0\end{array})t\alpha’)\mathrm{k}_{\alpha^{l}}(\varpi)$

,

(20)

111
Proof. In fact, by the quasi-periodicity (19), we have$=$$\mathrm{k}_{\alpha}(\gamma\varpi)$$e^{-_{2}^{1_{\alpha\gamma\eta\alpha\gamma\varpi}}}\sigma(\alpha\gamma\varpi;\gamma\varpi)$$=$$(-1)n_{1}+n_{2}+n_{1}n_{2}e\#(\alpha\gamma\eta\alpha’\varpi-\alpha’\eta\alpha\gamma\varpi)\mathrm{k}_{\alpha’}(\varpi)$$=$$(-1)^{n_{1}+n_{2}+n_{1}n_{2}} \mathrm{e}(\frac{1}{2}\alpha\gamma$ $(\begin{array}{ll}0 1-1 0\end{array})t\alpha’)\mathrm{k}_{\alpha’}(\varpi)$,

where we used the Legendre relation (12) in the last equality.

$\square$

The automorphic factor
has complex absolute value 1. Furthermore, by
the Legendre relation (12) again, the Klein forms are written by$\epsilon(\gamma, \alpha):=\mathrm{k}_{\alpha}(\gamma\varpi)/\mathrm{k}_{\alpha’}(\varpi)$

.

(21)

$\mathrm{k}_{\alpha}(\varpi)=-\frac{\varpi_{1}}{2\pi i\eta(z)^{2}}q^{1_{B_{2}(\alpha_{1})}}2\mathrm{e}(\frac{1}{2}\alpha_{2}(\alpha_{1}-1))\prod_{n=1}^{\infty}(1-x^{-1}q^{n})(1-xq^{n-1})$. Therefore,
If we choose at in -4(Q) [resp. $A((1/N)\mathrm{Z})$ ], then is in the same set for every
if we consider suitable products or quotients of Klein forms, then they become true modular functions.
offor some congruence subgroups of$\alpha’$$\gamma\in \mathrm{S}\mathrm{L}_{2}(\mathrm{Z})$$\mathrm{S}\mathrm{L}_{2}(\mathrm{Z})$$\mathrm{S}\mathrm{L}_{2}(\mathrm{Z})$7Integrality of $L(-1, \chi_{D})/2$
Theorem 7. For every fundamental discriminant $D>8$ , $L(-1, \chi_{D})/2$ is an integer.

Proof. The proof is devided into two parts. The first part shows the theorem for prime discriminants.
The second part reduces the general case to the case of prime discriminant (which is possibly negative).
Part one. Let $D=p>5$ be an odd prime number. We see$\frac{1}{2}L(-1, \chi_{p})$where $p’=(p-1)/2$ and$r$$\frac{p}{2}\acute{\sum_{n=1}^{p}}\chi_{p}(n)B_{2}(n/p)$$\equiv$-$=$$- \frac{1}{2p}\sum\chi_{p}(n)n^{2}+\frac{1}{2}\sum\chi_{p}(n)n$$\equiv$$- \frac{1}{4p}\mathrm{I}(n^{4}-rn^{4})+\frac{1}{2}\sum n$$(\mathrm{m}\mathrm{o}\mathrm{d} \mathrm{Z})$$(\in \mathrm{Z}_{2}, \mathrm{Z}_{3}, \mathrm{Z}_{5})$$(\mathrm{m}\mathrm{o}\mathrm{d} \mathrm{Z})$$=$$\frac{r-1}{120p}p’(p’+1)p(3p^{\prime 2}+3p’-1)+\frac{p^{2}-1}{16}$$=$$( \frac{r-1}{2}\cdot\frac{3p^{2}-7}{60}+1)\frac{p^{2}-1}{16}$$(\in \mathrm{Z}_{p})$,is the square of an odd primitive root modulo . Hence we obtain the result.
$p$andsuch that
Part two. Write $D=D_{1}D_{2}$ with (possibly negative) fundamental discriminants$|D_{i}|>1$ $(i=1, 2)$ and they are prime to each other. By the Chinese remainder theorem, we can show
$D_{1}$$\frac{1}{2}L(-1, \chi_{D})\equiv D_{1}Cm_{1}^{2}L(-1, \chi_{D2})$$(\mathrm{m}\mathrm{o}\mathrm{d} \mathrm{Z})$$D_{2}$,

with some integers $C$ and. If $D_{2}<0$ , then $L(-1, \chi_{D_{2}})$ vanishes by the gamma factor. Only
exceptional discriminant is 40 and we calculate directly in this case. In the other case, we can put$D_{2}=p>5$ and reduce to the part one.
$m_{1}$$\square$112
8Proof of the automorphy
We are now ready to complete the proof of the automorphy ofof the Klein forms (Theorem 6). Put $\alpha=(\alpha_{1}/D0)$ and$\eta_{XD}$(Theorems 1and 3) by using those$\varpi=(\begin{array}{l}Dz1\end{array})$, then by (21)$\mathrm{k}_{\alpha}(\varpi)=-\frac{q^{\}B_{2}(\alpha_{1})}}{2\pi i\eta(Dz)^{2}}(1-q^{\alpha_{1}})\prod_{n}(1-q^{Dn-\alpha_{1}})(1-q^{Dn+\alpha_{1}})$

Therefore, we can express our$\eta_{\mathrm{X}D}$.

as aproduct of Klein forms:$\eta_{XD}(z)=\prod_{n\in(\mathrm{Z}/D\mathrm{Z})^{\mathrm{X}}/\pm 1}(\mathrm{k}_{(n/D0)}$$(\begin{array}{l}Dz1\end{array})$

Then we deduce from Theorem 6that the automorphic factor of$\chi_{D}(a)$$\frac{\eta_{XD}(\gamma z)}{\eta_{XD}(z)^{\chi D(a)}}=\mathrm{e}(-\frac{ab}{2}L(-1, \chi_{D}))\mathrm{x}\{$

1,

, if

$\eta_{XD}$$)^{XD(n)}$for$\gamma$$\in\Gamma_{0}(D)$isaprime number,otherwise,$D\mathrm{i}\epsilon$(22)

when $D\neq 8$ . By Theorem 7in the previous section, the exponential factor is 1except for the cases$D=5$ or 8. In these cases, we calculate directly. This gives acomplete proof of Theorems 1and 3.
$\square$

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[Ral] H. Rademacher, Topics in Analytic Number Theory, Grundlehren 169, Springer-Verlag, 1973

[Ta] T. Takenouchi, Daen kans\^uron(in Japanese, Elliptic Functions, [TAKETAN]), Iwanami Japan,

1936[T-M] J. Tannery and J. Molk, Elements de la thSorie des Fonctions Elliptiques, Paris, 1893
[Wel] H. Weber, Lehrbuch der Algebra, Vol. In, 2ed edition, Vieweg, Braunschwieg, 1908. (reprinted
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und ihrer Grenzgebiete 88, Springer-Verlag Berlin Heidelberg New York, 1976
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B. C. Berndt, Character transformation formulae similar to those for the Dedekind eta function, in
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[Be2] B. C. Berndt, On Eisenstein series with characters and the values of Dirichlet -functions, Acta Arith.

28 (1975), 299320[Be3] B. C. Berndt, Ramanujan’s Notebooks, Part III, Springer-Verlag, New York, 1991
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R. Borcherds, Automorphic forms on[CN]

J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. London Math. Soc. 11 (1979), 308339$O_{\iota+2.2}(\mathrm{R})$and infinite products, Invent. Math. 120 (1995), 161-210

113[G-R] L. J. Goldstein and M. Razar, The theory of Hecke integrals, Nagoya Math. J. 63 (1976), 93-121[H-K]

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Hamburg 70 (2000), 315339[It]

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Pacific J. Math. 194 (2000), 137-164[Na]

C. Nagasaka, On generalized Dedekind sums attached to Dirichlet characters, J. Number Theory 19(1984), 374-383[Ra2] K. G. Ramanathan, Ramanujan’s continued fraction, Indian J. Pure Appl. Math. 16 (7) (1985), 695724[Ra3] S. Ramanujan, Notebooks VolJI, Tata Institute of Fundamental Research, Bombay, 1957
[S-S]

-P. Serre and H. M. Stark, Modular forms of weight 1/2, Lecture Note in Math. 627 Modular forms of
one variable VI (1977), 29-67$\mathrm{J}$Authors’ address:
Taro Horie
Suzuka National College of Technology
Shiroko, Susuka-City, Mie
-0294
Japan
e-mail:horie@genl.suzuka-ct.ac.j$(ffi\grave{\gamma}\mathrm{I}\mathrm{X}\mathrm{E}\#)$$\mathit{5}\mathit{1}\mathit{0}$$(\mathrm{f}_{11}^{\Delta}\ovalbox{\tt\small REJECT} \mathrm{I}\ovalbox{\tt\small REJECT} E\Leftrightarrow\Leftrightarrow \mathrm{f}\ovalbox{\tt\small REJECT} l+^{\mathrm{t}}\mathrm{R}\mathrm{u})$$(510- 0294\underline{=}\ovalbox{\tt\small REJECT} \mathrm{k}\mathrm{B}8_{\mathrm{F}}^{\Delta}\ovalbox{\tt\small REJECT} \mathrm{f}\mathrm{f}\mathrm{i}\mathrm{B}\mp \mathrm{N}\mathrm{I})$Naruo Kanou(7)O 納或男)
Graduate School of Mathematics, Nagoya University (名古屋大学大学院多元数理科学研究科)
Chikusa-ku, Nagoya 464-8602(464-8602 名古屋市千種区)
Japan-mail..kanou@math nagoya-u. ac.j$e$