Arithmetic of alternating forms and quaternion hermitian forms

Abstract

Hecke's Dirichlet series obtained from modular forms can be regarded as zeta-functions attached to the general linear group $GL(2, Q)$ over the rational number field $Q$ .In general, we may expect to obtain zeta-functions of this kind for a fairly wide class of algebraic groups defined over $Q$ .In order to realize this, it is necessary to develop, in the first place, the theory of ele- mentary divisors for any algebraic group $G$ in question.This is actually done in the case where $G$ is the multiplicative group of a semi-simple algebra.Further, the case of the orthogonal group is investigated in detail by M. Eichler [3].In both cases, there are fundamental theorems, due to Eichler $[4, 5]$ and M. Kneser [6], which may be called the approximation theorem in the group $G$ , from which one can easily derive an important conclusion about the class- number for $G$ .This approximation theorem plays an essential role also in the theory of Hecke-rings attached to quaternion algebras $[8, 9]$ .In fact, by means of the theorem, we can prove the isomorphism between the Hecke-ring 1.2.Lattices in a vector space.Let $\mathfrak{g}$ be a Dedekind domain and $F$ the quotient field of $\mathfrak{g}$ .Let $W$ be a vector space over $F$ .By a $\mathfrak{g}$ -lattice in $W$ , we understand a finitely generated g-submodule $L$ of $W$ such that $FL=W$.Let $\mathfrak{p}$ be a prime ideal of $\mathfrak{g}$ .We denote by $F_{\mathfrak{p}}$ and $\mathfrak{g}_{\mathfrak{p}}$ the $\downarrow$ )-completions of $F$ and $\mathfrak{g}$ , respectively.Put $W_{\mathfrak{p}}=W\bigotimes_{F}F_{\mathfrak{p}}$ .For every q-lattice $L$ in $W$ , put $L_{\mathfrak{p}}=\mathfrak{g}_{\mathfrak{p}}L$ ; then $L_{\mathfrak{p}}$ is a $\mathfrak{g}_{\mathfrak{p}}$ -lattice in $W_{\mathfrak{p}}$ .The following lemma is well-known.LEMMA 1.1.Let $L$ be a g-lattice in W. Take, for each prime ideal 1) of $[J$ , $a\mathfrak{g}_{\mathfrak{p}}$ -lattice $j\psi^{\mathfrak{p}}$ in $W_{\mathfrak{p}}$ .Then there exists a g-lattice $M$ in $W$ such that $j\psi_{\mathfrak{p}}=1\psi^{\mathfrak{p}}$ for every } $j$ if and only if $M^{\mathfrak{p}}=L_{\mathfrak{p}}$ for all except a finite number of $\mathfrak{p}$ .If such a lattice $1\psi$ exists, we have $j\psi=\bigcap_{\mathfrak{p}}(j\psi_{\mathfrak{p}\cap}W)$ .LEMMA 1.2.Let $L_{\mathfrak{p}}$ be a $\mathfrak{g}_{\mathfrak{p}}$ -lattice in $W_{\mathfrak{p}}$ ; let $\sigma$ and $\tau$ be elements of $GL(W_{P})$ .Suppose that $ L_{\mathfrak{p}}(\sigma-\tau)\subset \mathfrak{p}L_{\mathfrak{p}}\sigma$ .Then we have $ L_{\mathfrak{p}}\sigma=L_{\mathfrak{p}}\tau$ .PROOF.Let $x_{1}$ , $\cdot$ .. , $x_{m}$ be generators of $L_{0}$ over $\mathfrak{g}_{\mathfrak{p}}$ .Put $M=L_{\mathfrak{p}}\sigma,$ $ K=L_{\mathfrak{p}}\tau$ .Then we have $ x_{i}\sigma-x_{i}\tau\in \mathfrak{p}j\psi\subset 1\psi$ for every $i$ .As ] $\psi$ and $K$ are respectively generated by the $ x_{i}\sigma$ and the $X_{?}T$ , we get $K\subset M$ .Further we have $M\subset K+\mathfrak{p}M$ .From this we obtain inductively $1M\subset K+t)^{e}M$ for every positive integer $e$ .This implies $l\psi\subset K$ , so that $M=K$.1.3.Canonical base of a lattice with respect to an alternating form.We first prove a generalization of a well-known theorem of Frobenius.PROPOSITION 1.3.Let $\mathfrak{g}$ be a Dedekind domain and $F$ the quotient field of $\mathfrak{g}$ .Let $W$ be a vector space of dimension $2n$ over $F$ and $g(x, y)$ a non-degenerate alternating form on W. Let $M$ be a g-lattice in W. Then there exist a base $\{y_{1}$ , $\cdot$ ., $y_{n},$ $z_{1}$ , $\cdot$ .. , $z_{n}\}$ of $W$ over $F$ and (fractional) g-ideals $0_{1}$ , , $0_{n}$ such that $g(y_{i},y_{j})=g(z_{i}, z_{j})=0$ , $g(y_{i}, z_{j})=\delta_{ij}$ ,.The ideals $t1_{i}$ are uniquely determined by $M$ and $g$.PROOF.We prove this by induction on n. $Foreveryx\in M,$ $put\mathfrak{a}_{x}=g(x, M)$ .Obviously, $\mathfrak{a}_{x}$ is a g-ideal.As $M$ is a g-lattice, there exists a maximal one among the $t\ddagger_{x}$ , say $\mathfrak{a}_{1}$ ; and take an element $y_{1}$ of Mso that $\mathfrak{a}_{1}=g(y_{1},1\psi)$ .As we have $\mathfrak{g}=g(y_{1}, \mathfrak{a}_{1}^{-1}M)$ , there exists an element $z_{1}$ of $\mathfrak{a}_{1}^{-1}1\psi$ such that $g(y_{1}, z_{1})$ $=1$ .Put $t$ ) $=g(M, z_{1})$ .As $t$ ) $\ni g(y_{1}, z_{1})=1$ , we have $r_{j}\supset \mathfrak{g}$ , so that $\mathfrak{a}_{1}\mathfrak{b}=\mathfrak{a}_{1}g(M, z_{1})$ $=\mathfrak{a}_{1}$ .Assume that $\mathfrak{b}\neq \mathfrak{g}$ .Then $\mathfrak{a}_{1}g(1\psi, z_{1})\neq \mathfrak{a}_{1}$ , and hence there exist an element $u$ of $1\psi$ and an element $\alpha$ of $\mathfrak{a}_{1}$ such that $g(u, \alpha z_{1})\not\in \mathfrak{a}_{1}$ .Put $\beta$ $=-g(u, \alpha z_{1}),$ $\gamma=g(y_{1}, u)$ .$Wehavetheng(y_{1}+\alpha z_{1}, u-\gamma z_{1})=\beta$ .Since $\gamma\in tI_{1}$ and ( $\iota_{1}z_{1}\subset 1\psi$ , the element $u-\gamma z_{1}$ is contained in $ j\psi$ .We note that $g(y_{1}+\alpha z_{1}, \mathfrak{a}_{1}z_{1})$ $=(1_{1}$ .Therefore, we have $g(y_{1}+\alpha z_{1},1M)\supset 1\ddagger_{1}+\mathfrak{g}\beta\supset \mathfrak{a}_{1}$ , $(\ddagger_{1}+\mathfrak{g}\beta\neq \mathfrak{a}_{1}$ .This contradicts the maximality of $\mathfrak{a}_{1}$ .Hence we must have $g(M, z_{1})=\mathfrak{g}_{-}$ Now define a submodule $M^{\prime}$ of $M$ by $M^{\prime}=\{v\in M|g(y_{1}, v)=g(z_{1}, v)=0\}$ .For every $w\in M$, put $\xi=g(y_{1}, w),$ $\eta=g(z_{1}, w),$ $w_{0}=w+\eta y_{1}-\xi z_{1}$ .Then $\xi\in(t_{1},$ $\eta\in \mathfrak{g}$ , and we have $g(y_{1}, w_{0})=g(z_{1}, w_{0})=0$ , so that $w_{0}\in M^{\prime}$ .This shows that $M$ $=\mathfrak{g}y_{1}+\mathfrak{a}_{1}z_{1}+M^{\prime}$ .Applying our induction to $M^{\prime}$ , we get an expression $M^{\prime}$ $=\mathfrak{g}y_{2}+\cdots+\mathfrak{g}y_{n}+\mathfrak{a}_{2}z_{2}+\cdots+\mathfrak{a}_{n}z_{n}$ with the properties $\mathfrak{a}_{2}\supset\cdots\supset \mathfrak{a}_{n},$ $g(y_{i},y_{j})=g(z_{i}, z_{j})|$ $=0,$ $g(y_{i}, z_{j})=\delta_{ij}$ for $2\leqq i\leqq n,$ $2\leqq j\leqq n$ .Therefore, the first assertion is proved if we show $\mathfrak{a}_{1}\supset \mathfrak{a}_{2}$ .Let $u$ and $v$ be elements of $M^{\prime}$ .We have $g(y_{1}+u, M)$ $\supset g(y_{1}+u, \mathfrak{a}_{1}z_{1}+\mathfrak{g}v)\supset \mathfrak{a}_{1}+\mathfrak{g}g(u, v)\supset \mathfrak{a}_{1}$ .By the maximality of $\mathfrak{a}_{1}$ , we must have $g(u, v)\in \mathfrak{a}_{1}$ , namely $g(M^{\prime}, M^{\prime})\subset \mathfrak{a}_{1}$ .This implies $\mathfrak{a}_{1}\supset \mathfrak{a}_{2}$ and completes the proof of the first assertion.The invariance of the ideals $\mathfrak{a}_{i}$ is easily shown by " localization ".Namely, for every prime ideal $p$ of $\mathfrak{g}$ , consider $W_{\mathfrak{p}}=W\bigotimes_{F}F_{\triangleright}$ and a $\mathfrak{g}_{\mathfrak{p}}$ -lattice $M_{\mathfrak{p}}=\mathfrak{g}_{\mathfrak{p}}M$ in $W_{\mathfrak{p}}$ .Then the invariance is an immediate con- sequence of the theory of elementary divisors over a principal ideal domain (cf.[2, \S 5.1, Theorem 1]).We can also prove the invariance more directly with no use of localization.We call the ideals $\mathfrak{a}_{i}$ of Proposition 1.3 the invariant factors of $M$ (with respect to $g$), and call $\{y_{1}, \cdots , y_{n}, z_{1}, \cdots , z_{n}\}$ a canonical base of $M$ (with respect to $g$).1.4.Maximal lattices.Let $F,$ $\mathfrak{g},$ $W,$ $g$ be the same as in Proposition 1.3.For every g-lattice $M$ in $W$ , we see that the first member $\mathfrak{a}_{1}$ of the invariant factors of $M$ is the g-ideal generated by $g(x, y)$ for $x,y\in M$ We put $N_{g}(M)$ $=\mathfrak{a}_{1}$ and call $N_{g}(M)$ the norm of $M$ with respect to $g$.For simplicity, we fix $g$ and write $N(M)=N_{g}(M)$.We say that $M$ is maximal (with respect to g) if $M$ is a maximal one among the g-lattices in $W$ with the same norm (with respect to $g$ ).It is clear that $N(M\sigma)=N(M)N(\sigma)$ for every $\sigma\in G(W, g)$ .If $M$ is maximal, $ M\sigma$ is maximal for every $\sigma\in G(W, g)$ .By Proposition 1.3, we see easily that $M$ is maximal if and only if the invariant factors of $M$ are all equal to $N(M)$.Furthermore, if $M$ is a g-lattice in $W$ and $\mathfrak{a}$ is a g-ideal such that $\mathfrak{a}\supset N(M)$ , we can find a maximal lattice $L$ in $W$ such that $L\supset M,$ .$N(L)=\mathfrak{a}$ .PROPOSITION 1.4.Let $M_{1}$ and $M_{2}$ be maximal lattices in W. Then we have $M_{1}\sigma=M_{2}$ for an element $\sigma$ of $G(W,g)$, if and only if $N(M_{1})^{-1}N(M_{2})$ is a princi- pal ideal.PROOF.If $M_{1}\sigma=M_{2}$ for an element $\sigma\in G(W, g)$ , we have $N(M_{2})=N(M_{1}\sigma)$ , $=N(M_{1})N(\sigma)$ ; this proves the ' only if ' part.Conversely, put $\mathfrak{a}_{i}=N(M_{t})$ and $\mathfrak{a}_{t^{1}}\mathfrak{a}_{2}=\mathfrak{g}\alpha$ with $\alpha\in F$ .Let $\{y_{1}, \cdots , y_{n}, z_{1}, \cdots , z_{n}\}$ and $\{u_{1}, \cdots , u_{n}, v_{1}, \cdots , v_{n}\}$ be respectively canonical bases of $M_{1}$ and $M_{2}$ .Define an element $\sigma$ of $E(W)$ by $y_{i}\sigma=u_{i},$ $z_{i}\sigma=\alpha v_{i}$ for $1\leqq i\leqq n$ .Then we see easily $\sigma\in G(W, g),$ $N(\sigma)=\alpha,$ $ M_{1}\sigma$ $=M_{2}$ .This proves the ' if ' part.2.3.Elementary theory of maximal lattices.Let $g$ be a Dedekind domain and $F$ the quotient field of $\mathfrak{g}$ .Let $A$ be a quaternion algebra over $F$ and $V$ an A-space of dimension $n$ .Take a non-degenerate Q-hermitian form $f$ on $V$ .Let $L$ be a g-lattice in $V$ .Put $0=\{a\in A|aL\subset L\}$ .Then $0$ is an order in $A$ .We call $0$ the order of $L$ and say that $L$ is normal if $0$ is a maximal order in $A$ .Assume that $L$ is normal.We denote by $N_{f}(L)$ the two-sided o-ideal gen- erated by the elements $f(x, y)$ for $x\in L,$ $y\in L$ , and call $N_{f}(L)$ the norm of $L$ with respect to $f$ .We denote $N_{f}(L)$ simply by $N(L)$ when we fix $f$ and there is no fear of confusion.Now, for every prime ideal $\mathfrak{p}$ of $\mathfrak{g}$ , consider the $\mathfrak{p}$ -completion $F_{\mathfrak{p}}$ and $g_{\mathfrak{p}}$ of $F$ and $\mathfrak{g}$ .Put $A_{\mathfrak{p}}=A\bigotimes_{F}F_{\mathfrak{p}},$ $V_{\mathfrak{p}}=V\bigotimes_{F}F_{\mathfrak{p}}$ .Then $V_{\mathfrak{p}}$ can be considered as an $A_{\mathfrak{p}^{-}}$ space of dimension $n$ in a natural manner.Further $f$ is uniquely extended to a non-degenerate Q-hermitian form on $V\mathfrak{p}$ , which we denote again by $f$ .The following proposition is an easy consequence of our definition.PROPOSITION 2.2.Let $L$ be a g-lattice in V.If $0$ is the order of $L$ , then Op $(=g_{\mathfrak{p}}o)$ is the order of $L_{\mathfrak{p}}(=\mathfrak{g}_{\mathfrak{p}}L)$ .$L$ is normal if and only if $L_{\mathfrak{p}}$ is normal for every prime ideal $\mathfrak{p}$ of $\mathfrak{g}$ .If $L$ is normal, we have $N(L)_{\mathfrak{p}}=N(L_{\mathfrak{p}})$ .Let $L$ be a normal lattice in $V$ and $0$ the order of $L$ .We call $L$ maximal (with respect to f ) if $L$ is a maximal one among the normal lattices with the same order $\mathfrak{o}$ and the same norm $N(L)$.PROPOSITION 2.3.Let $L$ be a g-lattice in $V$ and $\sigma$ an element of $G(V,f)$.Then $ L\sigma$ is a g-lattice in $V$ with the same order as L. If $L$ is normal, so is $ L\sigma$ ; and we have $N(L\sigma)=N(L)N(\sigma)$ .Moreover, if $L$ is maximal, so is $ L\sigma$ .This is also an easy consequence of definition.Further, by Lemma 1.1 and Proposition 2.2, we obtain PROPOSITION 2.4.A normal g-lattice in $V$ is maximal if and only if $L_{\mathfrak{p}}$ is maximal for every prime ideal $\mathfrak{p}$ of $\mathfrak{g}$ .Hereafter,

Locations

• Journal of the Mathematical Society of Japan - View - PDF