Riemann-Roch isometries in the non-compact orbifold setting

Abstract

We generalize work of Deligne and Gillet-Soulé on a Riemann-Roch type isometry, to the case of the trivial sheaf on cusp compactifications of Riemann surfaces $Γ\backslash\mathbb{H}$, for $Γ\subset PSL_{2}(\mathbb{R})$ a fuchsian group of the first kind, equipped with the Poincaré metric. This metric is singular at cusps and elliptic fixed points, and the results of Deligne and Gillet-Soulé do not apply to this setting. Our theorem relates the determinant of cohomology of the trivial sheaf, with an explicit Quillen type metric in terms of the Selberg zeta function of $Γ$, to a metrized version of the $ψ$ line bundle of the theory of moduli spaces of pointed orbicurves, and the self-intersection bundle of a suitable twist of the canonical sheaf $ω_{X}$. We make use of surgery techniques through Mayer-Vietoris formulae for determinants of laplacians, in order to reduce to explicit evaluations of such for model hyperbolic cusps and cones. We derive an arithmetic Riemann-Roch formula, that applies in particular to integral models of modular curves with elliptic fixed points. As an application, we treat in detail the case of the modular curve $X(1)$. From this, we obtain the Selberg zeta special value $Z^{\prime}(1,PSL_{2}(\mathbb{Z}))$ in terms of logarithmic derivatives of Dirichlet $L$ functions. Our work finds its place in the program initiated by Burgos-Kramer-Kühn of extending arithmetic intersection theory to singular hermitian vector bundles.