Regularized Petersson inner products for meromorphic modular forms

Abstract

We investigate the history of inner products within the theory of modular forms. We first give the history of the applications of Petersson's original definition for the inner product of $S_{2k}$ and then recall Zagier's extension to a non-degenerate (but not necessarily positive-definite) inner product on all holomorphic modular forms. We then recall the history of the so-called ``regularization'' of the inner product to extend it to weakly holomorphic modular forms originally by Petersson and then later independently rediscovered by Harvey--Moore and Borcherds, as well as its applications to theta lifts by Borcherds, Bruinier--Funke, and many more recent authors. This has been recently extended to a well-defined inner product on all weakly holomorphic modular forms by Bringmann, Diamantis, and Ehlen. Finally, we consider inner products on meromorphic modular forms which have poles in the upper half-plane. Petersson also defined a regularization in this case by cutting out small neighborhoods around each pole occurring in the fundamental domain; Bringmann, von Pippich, and the author have recently constructed an extension of this regularization, which, when combined with the regularization of Bringmann, Diamantis, and Ehlen, yields an inner product that is well-defined and finite on all meromorphic modular forms.