An arithmetic Riemann-Roch theorem on modular curves via heat kernel regularization

The arithmetic Riemann{Roch theorem has been established by H. Gillet/C. Soule as well as by G. Faltings for projective and generically smooth morphisms f : X->Y of arithmetic varieties and hermitian vector bundles E = (E ; k k) equipped with smooth hermitian metrics. In our talk, we will present a variant of an arithmetic Riemann{Roch theorem in a singular setting, namely, the case of the line bundle of modular forms of weight k (an even integer) on (regular, projective models of) modular curves equipped with the Petersson metric, which becomes logarithmically singular at the cusps. The proof starts from the known arithmetic Riemann-Roch formula for smooth hermitian metrics approximating the hyperbolic metric under consideration and then proceeds with an investigation of the degeneration behavior of the starting formula using heat kernel regularization techniques while the smoothened metric approaches the singular metric.