Riemann-Roch isometries in the non-compact orbifold setting

Abstract

We generalize work of Deligne and Gillet-Soul\'e on a Riemann-Roch type isometry, to the case of the trivial sheaf on cusp compactifications of Riemann surfaces $\Gamma\backslash\mathbb{H}$, for $\Gamma\subset PSL_{2}(\mathbb{R})$ a fuchsian group of the first kind, equipped with the Poincar\'e metric. This metric is singular at cusps and elliptic fixed points, and the results of Deligne and Gillet-Soul\'e 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 $\Gamma$, to a metrized version of the $\psi$ line bundle of the theory of moduli spaces of pointed orbicurves, and the self-intersection bundle of a suitable twist of the canonical sheaf $\omega_{X}$. 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-K\"uhn of extending arithmetic intersection theory to singular hermitian vector bundles.