An explicit weight $0$ Riemann-Roch theorem

Kathrin Bringmann, Ben Kane

Published 2015-05-02Version 1

In this paper, we prove an explicit version of the Riemann-Roch Theorem in weight 0, providing explicit meromorphic modular forms satisfying the principal parts conditions dictated by Riemann-Roch. Following work of Petersson, we construct the elements via Poincar\'e series. There are two main aspects of our investigation which differ from his approach. Firstly, the naive definition of the Poincar\'e series diverges and one must analytically continue via Hecke's trick. Hecke's trick is further complicated in our situation by the fact that the Fourier expansion does not converge everywhere due to singularities in the upper half-plane. To explain the second difference, we recall that Petersson constructed linear combinations from a family of meromorphic functions which happen to be modular if the principal parts condition is satisfied. In contrast to this, we construct linear combinations from a family of non-meromorphic modular forms, known as polar harmonic Maass forms, which are meromorphic whenever the principal parts condition is satisfied.