$L$-functions in Scattering on $p$-adic Multiloop Surfaces

Abstract

We study scattering processes on $p$-adic multiloop surfaces represented as multiloop infinite graphs with total valence in each vertex equal $p+1$. They all are spaces of the constant negative curvature since they are quotients of the $p$-adic hyperbolic plane over free acting discrete subgroup of $PGL(2, {\bf Q}_p)$. Releasing the closed part of this graph containing all loops which is called reduced graph $T_{red}$ we can obtain $L$-function corresponding to this closed graph. For the total graph we introduce the notion of the spherical functions being eigenfunctions of the Laplace operator acting on the graph and consider $s$--wave scattering processes therefore defining scattering matrices $c_i$. The number of possibilities coincides with $|\T_{red}|$ --- the number of vertices of the reduced graph. Taking the product over all $c_i$ we define the total scattering matrix which appears to be essentially presented as a ratio of $L$--functions: $C\sim L(α_+)/L(α_-)$, where the function $L$ itself depends only on the shape of $\Tr$ and not on the initial infinite graph, and the only dependence of initial $p$ is contained in arguments $α_\pm$ defined by $p$ and eigenvalue $t$ of the Laplacian. We also present a proof by H.Bass of the theorem expressing $L$--functions on arbitrary finite graphs via determinants of some local operators on these graphs.