Let $(𝒪,\Sigma ,F_{\infty })$ be an arithmetic ring of Krull dimension at most 1, $𝒮=\mathrm{Spec}𝒪$ and $(\pi :𝒳\to 𝒮;{\sigma }_1,...,{\sigma }_n)$ an $n$-pointed stable curve of genus $g$. Write $𝒰=𝒳\setminus {\cup }_j{\sigma }_j\left(𝒮\right)$. The invertible sheaf ${\omega }_{𝒳/𝒮}({\sigma }_1+\cdots +{\sigma }_n)$ inherits a hermitian structure ${\parallel ·\parallel }_{\mathrm{hyp}}$ from the dual of the hyperbolic metric on the Riemann surface ${𝒰}_{\infty }$. In this article we prove an arithmetic Riemann-Roch type theorem that computes the arithmetic self-intersection of ${\omega }_{𝒳/𝒮}{({\sigma }_1+...+{\sigma }_n)}_{\mathrm{hyp}}$. The theorem is applied to modular curves $X\left(\Gamma \right)$, $\Gamma ={\Gamma }_0\left(p\right)$ or ${\Gamma }_1\left(p\right)$, $p\ge 11$ prime, with sections given by the cusps. We show $Z^{\text{'}}(Y\left(\Gamma \right),1)\sim e^a{\pi }^b{\Gamma }_2{(1/2)}^{c}L(0,{ℳ}_{\Gamma })$, with $p\equiv 11mod12$ when $\Gamma ={\Gamma }_0\left(p\right)$. Here $Z\left(Y\right(\Gamma ),s)$ is the Selberg zeta...

Let $\Gamma \setminus {ℍ}^3$ be a finite-volume quotient of the upper-half space, where $\Gamma \subset \mathrm{SL}(2,ℂ)$ is a discrete subgroup. To a finite dimensional unitary representation $\chi$ of $\Gamma$ one associates the Selberg zeta function $Z(s;\Gamma ;\chi )$. In this paper we prove the Artin formalism for the Selberg zeta function. Namely, if $\tilde{\Gamma }$ is a finite index group extension of $\Gamma$ in $\mathrm{SL}(2,ℂ)$, and $\pi ={\mathrm{Ind}}_{\Gamma }^{\tilde{\Gamma }}\chi$ is the induced representation, then $Z(s;\Gamma ;\chi )=Z(s;\tilde{\Gamma };\pi )$. In the second part of the paper we prove by a direct method the analogous identity for the scattering function, namely $\phi (s;\Gamma ;\chi )=\phi (s;\tilde{\Gamma };\pi )$, for an appropriate...