数理解析研究所講究録 第 797 巻 1992 年 110-125 110 EQUIVALENCE CLASSES OF PRIMITIVE BINARY QUADRATIC FORMS AND ZETA FUNCTIONS PIA BAUER Department of Mathematics Faculty of Science Kyushu University 33 Higashi-ku, Fukuoka 812, Japan December 1991 0. INTRODUCTION For an arbitrary number field $K$ with ring of integers denoted by $\mathcal{O}$ we study $GL(2, \mathcal{O})$ $(=:G)$-and $SL(2, \mathcal{O})$ -equivalence classes of binary quadratic forms $\Phi(x, y)=ax^{2}+bxy+cy^{2}$ defined over $\mathcal{O}$ . After fixing $\Delta\in O$ we define the following zeta function for G-equivalence classes of the binary quadratic forms over $\mathcal{O}$ of discriminant $\Delta$ : $P_{\Delta}(s)$ $:=$ $\triangle\neq square$ , $\sum_{[\Phi],\Phi_{\triangle(\Phi)=\triangle}primitive}\sum_{(x,y)\in(\mathcal{O}x\mathcal{O})/E(\Phi)}|N_{K/Q}(\Phi(x, y))|^{-s},$ where $E(\Phi)$ $:=\{g\in G:g\Phi=\Phi\}$ . For the rational numbers and imaginary quadratic fields one can define this also for $SL(2, \mathcal{O})$ -equivalence, which for convenience we call l-equivalence. We will express it in closed form in terms of the zeta function of an order in the field $L=K(\sqrt{\Delta})$ associated to the primitive forms. For the latter one can also find a description by L-series of $L=K(\sqrt{\Delta})$ . The simplest method to give an identity involving L-series would be to take a representing form of each class, represent it as norm form of the associated module in the extension $L$ of $K$ by $\sqrt{\triangle}$ and employ the module zeta function in [Od]. Since each module has a difFerent conductor this does not lead to a closed form. But with this method one can omit the restriction to primitive forms, i.e. the ideal generated by the coefficients of the form is $\mathcal{O}$ . For primitive forms the corresponding modules all have the same endomorphism ring $R$ and their similarity classes form a subgroup of the ideal class group of R. This leads to the desired description of $P_{\triangle}(s)$ (Theorem 5.9). 111 In the first paragraph we give the basic definitions. In \S 2 we extend the concept of pairs introduced in [Kal] for $SL(2, \mathcal{O})$ -equivalence to G-equivalence and prove for given discrim- inant the 1-1-correspondence between G-equivalence classes of pairs and G-equivalence classes of binary quadratic forms (Lemma 2.2). In \S 3 we study the automorphism group of a form. Using the description of $+$ and $SL(2, \mathcal{O})$ -equivalence classes of pairs, the cor- respondence (Lemma 2.2) and the description of the automorphism group we show that the G-equivalence class of a form $\Phi$ splits into $[\mathcal{O}^{*} : N(O_{\Phi}^{*})]$ many $SL(2, \mathcal{O})$ -equivalence classes, $\mathcal{O}_{\Phi}$ denotes the endomorphism ring of the associated module (Lemma 3.2). In \S 4 we study the structure of the endomorphism ring $\mathcal{O}_{A}$ of a module $A$ coming from a binary quadratic form. We show that it is free over $\mathcal{O}$ if and only if the coefficients of the form generate a principal ideal in $\mathcal{O}$ . Fixing $\mathcal{O}_{A}$ we see that the finitely generated modules over $\mathcal{O}$ in $\mathcal{O}_{A}$ which have $\mathcal{O}_{A}$ as endomorphism ring (called ‘proper’) are exactly the $\mathcal{O}_{A}$ -regular ideals which are characterized by the existence of a similar ideal which is coprime to the $\mathcal{O}$ conductor of $\mathcal{O}_{A}$ in the integral closure of in L. With this characterisation one can construct a map $hom$ the ideal class group of $O_{A}$ to the ideal class group of $K$ which is $\mathcal{O}$ a homomorphism if and only if $\mathcal{O}_{A}$ is free over (Lemma 4.11). Its kernel is generated by the classes of free (over $\mathcal{O}$ ) proper modules of $O_{A}$ . In \S 5 we gather the results so far to show an 1-1-correspondence between G-equivalence classes of binary quadratic forms over $\mathcal{O}$ of discriminant $\triangle(\neq square)$ and similarity classes of fxee (over $\mathcal{O}$ ) proper modules of the unique oder of discriminant $\Delta$ that is free over $\mathcal{O}$ . We show $P_{\triangle}(S)=Q_{\triangle}(S)$ $:=$ $\sum$ $N_{K/Q}(N(A))^{-s}$ . $A\subset R$ , proper ideal free over $O$ In \S 6 we identify $Q_{\triangle}(s)$ as indicated in the beginning (Theorem 6.1) and by this give the analytic continuation of $P_{\triangle}(s)$ . 1. EQUIVALENCE OF BINARY QUADRATIC FORMS $\mathcal{O}$ Let $K$ be an algebraic number field, its ring of integers. For $a,$ $b,$ $c\in \mathcal{O}$ define the binary quadratic form $\Phi(x, y)=ax^{2}+bxy+cy^{2}$ . It has discriminant $\Delta(\Phi)=b^{2}-4ac$. We can assign the symmetric matrix $A=(\begin{array}{ll}a \frac{b}{2}\frac{b}{2} c\end{array})|$ to $\Phi$ and write $\Phi(x,y)=(x,y)A(x, y)^{t}$ . On such a form the $2\cross 2$-matrices with integral coefficients operate by $[g]\Phi(x,y):=\Phi((x, y)g)=(x,y)gAg^{t}(x,y)^{t}$, $g=(\begin{array}{ll}\alpha \gamma\beta \delta\end{array})\in M(2, \mathcal{O})$ , $=a’x^{2}+b'xy+c'y^{2}$ with $a’=a\alpha^{2}+b\alpha\gamma+c\gamma^{2}=\Phi(\alpha, \gamma)$ (1.1) $b’=2a\alpha\beta+b(\alpha\delta+\beta\gamma)+2c\gamma\delta$ $c’=a\beta^{2}+b\beta\delta+c\delta^{2}=\Phi(\beta, \delta)$ . 112 The discriminant behaves under this operation like $\Delta([g]\Phi)=\det(g)^{2}\Delta(\Phi)$ . We want to study zeta functions associated to equivalence classes of binary quadratic forms of a fixed discriminant. For this we introduce the following equivalence relation. We fix $\triangle\in \mathcal{O}$ and set $G:=GL(2, \mathcal{O})$ . $s\Phi$ $\Psi$ $\mathcal{O}$ $t\triangle$ 1.1. Definition. We call two binary $q$ uadratic form and over of discriminan G-equivalent if there exis $ts$ an element $g\in G$ such that $(g\Phi)(x, y)$ $:=\det(g)^{-1}\Phi((x, y)g)=\Psi(x, y)$ . If we only allow transformations of determinant 1, we call the forms l-equivalent. This is in fact an equivalence relation. Two equivalent forms represent up to a unit the same numbers. (1.2) $E(\Phi):=\{g\in G:g\Phi=\Phi\}$ and $E_{1}(\Phi):=\{g\in G:g\Phi=\Phi\det g=1\}$ are the groups of G-automorphisms and l-automorphisms of $\Phi$ . The numbers $hc$ and $h_{1}$ of G-and l-equivalence classes for a given discriminant $\Delta\neq 0$ are finite. This follows bom [Sp] (see [Ba, Chap. 5.1]). In paragraph 3 we will give a formula for the relation between $h_{G}$ and $h_{1}$ in the case that the discriminant is not a square. 2. BINARY QUADRATIC FORMS AND MODULES For the following discussion it is enough that $K$ is a field of characteristic $\neq 2$ and $\mathcal{O}$ an integral domain with quotient field K. To each binary quadratic form of non-zero discriminant $\triangle$ one naturally associates a free module of rank two over $\mathcal{O}$ by factorizing the form in the extension $L$ of $K$ by the square root of the discriminant. $L$ is separable over K. If the discriminant is a square then $L$ is isomorphic to $K\cross K$ , on which conjugation is given by exchanging the factors, if not, then $L$ is isomorphic to the algebraic number field * $K(\sqrt{\Delta})$ with conjugation given in the obvious way. For $x\in L$ let $N(x)$ $:=xx^{*}$ denote the relative norm of $x$ . In the sequel we restrict to the case that $\Delta$ is not a square. Then the coefficient $a$ of the form $\Phi(x, y)=ax^{2}+bxy+cy^{2}$ of discriminant $\Delta$ is always non-zero and the module $M_{\Phi}=[a, \frac{b-\sqrt{\triangle}}{2}]0$ in $L$ has the property that its elements fulfil (2.1) $\frac{1}{a}N(ax+\frac{b-\sqrt{\triangle}}{2}y)=\Phi(x, y)$ . This correspondence is not one to one since for example the conjugate module gives the same form. For l-equivalence I. Kaplansky [Kal] has introduced the notion of pairs and stated a one to one correspondence in these terms. We extended it to G-equivalence. We display it here by looking simultaneously at G-and l-equivalence. A pair $[A, a]$ consists 113 of a free module $A$ of rank 2 over $O$ in $L$ and a non-zero element $a\in K$ . The concepts of involution, norm, product, discriminant and different are extended to pairs: (2.2) $[A, a]^{*}=[A^{*}, a]$ , $N[A, a]= \frac{N(A)}{a}$ , $[A, a][B, b]=$ [$AB$ , ab]. If one fixes a basis $x,y$ of $A$ the discriminant of $A$ with respect to that basis is (2.3) $D(A)$ $:=(xy^{*}-x^{*}y)^{2}$ , the different $\delta(A)$ $:=(xy^{*}-x^{*}y)$ and one sets (2.4) $D([A, a]);=\frac{D(A)}{a^{2}}$ , $\delta([A, a]):=\frac{\delta(A)}{a}$ . Let $[A, a]$ be a pair of discriminant $\Delta$ . After fixing a square root $\delta$ of $\triangle$ a basis of $A$ is called admissible if (2.5) $\delta([A, a])=\delta$ . An admissible basis always exists [Kal, \S 5]. The essential point is that the different $\delta$ transforms in a good manner under base change by an element of G.