부산대학교 도서관

In this paper, the Riemann hypothesis on combinatorial zeta-functions associated to finite quotients of the affine building of $GL_n$ is studied. \par For 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 fixed discrete 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 and is locally isomorphic to ${\cal B}$. A finite complex ${\cal B}_\Gamma$ is called Ramanujan if it is equivalent to the condition that the matrix coefficients of irreducible subrepresentations $L^2(\Gamma \backslash G/Z)$ with non-zero vectors fixed by any maximal compact subgroup are $(2+\epsilon)$-integrable. \par Define the partial $k$th combinatorial zeta-function of weight $r$ by the 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 primitive closed combinatorial $k$@-geodesics of weight $r$ in ${\cal B}_\Gamma$, and $l_\Lambda(\vartheta)$ is the length of $\vartheta$. \par If the non-trivial poles of $Z_{k,r}({\cal B}_\Gamma,q^{-s})$ belong to the 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. \par It is proved that if ${\cal B}_\Gamma$ is a strongly Ramanujan complex, then ${\cal B}_\Gamma$ satisfies the Riemann hypothesis. The converse is true if every irreducible subrepresentation in $L^2(\Gamma \backslash G/Z)$ is generic.