Generalized arithmetic intersection numbers (Q2717084)

This is the full version of a note by the author [C. R. Acad. Sci., Paris, Sér. I, Math. 327, 283--288 (1998; Zbl 0926.14010)].NEWLINENEWLINEThe author develops an arithmetic intersection theory for hermitian line bundles on arithmetic surfaces in the more general setting in which the metrics are smooth away from a finite set of points, but logarithmic singularities are allowed at those points.NEWLINENEWLINEAs an application, the author considers modular curves and the line bundles of modular forms on them, metrized by the Petersson inner product. The self-intersection number of this line bundle is computed and shown to be \(\zeta(-1)+2\zeta'(-1)\) up to a positive trivial factor. Here \(\zeta\) denotes the Riemann zeta function.NEWLINENEWLINEFinally, the generalized arithmetic intersection product is shown to be compatible with the one proposed by \textit{J.-B. Bost} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 32, 241--312 (1999; Zbl 0931.14014)].