Regularized inner products and modular invariants for real quadratic fields

In this paper, we study the regularized inner products of Borcherds-Zagier bases of the space of weakly holomorphic modular forms of weight 1 / 2 1/2 . We prove a general result, which, in particular, will imply that the regularized inner products of these forms can be expressed in terms of traces of integrals of Klein’s j j -function over real quadratic geometric invariants (certain modular surfaces) and Hurwitz-Kronecker class numbers. This complements a result of Duke, Imamoğlu and Tóth, describing the regularized inner products of (dual) weight 3 / 2 3/2 weakly holomorphic modular forms in terms of real quadratic analog of traces of singular moduli. Further, we establish an identity relating the traces of harmonic Maass forms J ~ J̃ and sesquiharmonic Maass forms ∂ ∂ z J ^ m ∂ /∂ z Ĵₘ , both of weight 2 2 , which will play an important role in proving the above results. The form J ^ 1 Ĵ₁ is essentially the pre-image of the j j -function under the hyperbolic Laplacian while J ~ J̃ is related to the j j -function as a dual object via the ξ ξ -operator. As an application, we also interpret the regularized inner products of Borcherds-Zagier bases in ter