Eichler–Selberg type identities for mixed mock modular forms

Abstract

Using holomorphic projection, we work out a parametrization for all relations of products (resp. Rankin-Cohen brackets) of weight 3 2 mock modular forms with holomorphic shadow and weight 1 2 modular forms in the spirit of the Kronecker-Hurwitz class number relations. In particular we obtain new proofs for several class number relations among which some are classical, others are relatively new. We also obtain similar results for the mock theta functions.

Proof of the umbral moonshine conjecture

“…The following theorem is a special case of a more general theorem due to Mertens (cf. Theorem 6.3 of [42]). See also [41].…”

“…To prove Theorem 1.1 it suffices to prove that these multiplicities are non-negative integers. To prove Theorem 1.2 we apply recent work [42] of Mertens on the holomorphic projection of weight 1 2 mock modular forms, which generalizes earlier work [41] of Imamoglu, Raum and Richter.…”

Special Values of Shifted Convolution Dirichlet Series

“…Recently, Imamoglu, Raum, and Richter [15] have obtained general theorems in this direction with applications to Ramanujan's mock theta functions in mind. This work, as well as the preprint [20] by the first author, are based on unpublished notes of Zagier. Here we continue this theme, and we show that the L (ν) (f 1 , f 2 ; τ ), again for ν = k 1 −k 2 2 , also arise in this way.…”

Asymptotic bounds for special values of shifted convolution Dirichlet series

“…Recall that if f is a smooth weight k ≥ 2 modular form for Γ 0 (N ) with moderate growth at cusps, then its holormorphic projection π hol f lies in M k (Γ 0 (N )). For more on the classical holomorphic projection operator, see [25], [16], [21] and [12].…”