부산대학교 도서관

In this paper, the Riemann hypothesis on combinatorial zeta-functionsassociated to finite quotients of the affine building of $GL_n$ isstudied.\parFor a non-archimedean local field $F$ with residue field ${\cal O}/\pi{\cal O}$ of size $q$, let $G=GL_n(F)$. The Bruhat-Tits building of$G$, which is denoted by ${\cal B}={\cal B}(G)$, is an${(n-1)}$@-dimensional simply connected simplicial complex. Let a fixeddiscrete torsion-free cocompact modulo $Z$ subgroup $\Gamma$ of $G$intersect $Z$ trivially (here $Z$ is a center of $G$). The quotient${\Gamma \backslash {\cal B}={\cal B}_\Gamma}$ is a finite complex andis locally isomorphic to ${\cal B}$. A finite complex ${\cal B}_\Gamma$is called Ramanujan if it is equivalent to the condition that thematrix coefficients of irreducible subrepresentations $L^2(\Gamma\backslash G/Z)$ with non-zero vectors fixed by any maximal compactsubgroup are $(2+\epsilon)$-integrable.\parDefine the partial $k$th combinatorial zeta-function of weight $r$ bythe product$$Z_{k,r}({\cal B}_\Gamma,u)\coloneq\prod_{[\vartheta]}\left(1-u^{l_\Lambda(\vartheta)}\right)^{-1},$$where $[\vartheta]$ runs through all equivalence classes of primitiveclosed combinatorial $k$@-geodesics of weight $r$ in ${\cal B}_\Gamma$,and $l_\Lambda(\vartheta)$ is the length of $\vartheta$.\parIf the non-trivial poles of $Z_{k,r}({\cal B}_\Gamma,q^{-s})$ belong tothe finite set$$\Re(s)\in \left[ \frac{n-r-1}{2}, \frac{n-r}{2} \right]\cap\left( \frac{1}{2r}\Bbb Z \right)$$for all $k,r$, then ${\cal B}_\Gamma$ satisfies the Riemann hypothesis.\parIt is proved that if ${\cal B}_\Gamma$ is a strongly Ramanujan complex,then ${\cal B}_\Gamma$ satisfies the Riemann hypothesis. The converseis true if every irreducible subrepresentation in $L^2(\Gamma\backslash G/Z)$ is generic.