Integrality properties of the CM-values of certain weak Maass forms

Abstract

In a recent paper, Bruinier and Ono prove that the coefficients of certain weight −1/2 harmonic Maass forms are traces of singular moduli for weak Maass forms. In particular, for the partition function p(n), they prove thatwhere P is a weak Maass form and α Q ranges over a finite set of discriminant −24n + 1 CM points. Moreover, they show that 6 · (24n − 1) · P (α Q ) is always an algebraic integer, and they conjecture that (24n − 1) · P (α Q ) is always an algebraic integer. Here we prove a general theorem which implies this conjecture as a corollary.

Singular moduli for a distinguished non-holomorphic modular function

“…By the expression for β as a product of differences of j-values (2.5), it suffices to show that if p is a split prime and p is a prime above p in Q( √ D) that j(α Q ) ≡ j(α Q ) (mod p) for α Q not SL 2 (Z)-equivalent to α Q . This is exactly the situation of Lemma 3.2 of [6], which is also stated in Theorem 13.21 of [4], and is essentially a result of Deuring lifting theory.…”

“…This function was important in [6] and [3], and its singular moduli were first studied in Masser ([8, App. 1]).…”

Class invariants for certain non-holomorphic modular functions

“…They, together with Sutherland, conjectured that the associated polynomials always generate ring class fields and are irreducible ( [6], p. 20), which was shown by Mertens and Rolen in [17]. Many others have studied properties of these non-holomorphic singular moduli (see [1,7,12,13,14]). Here, we consider the general problem of constructing class invariants (i.e., modular forms whose CM-values generate Hilbert class fields) from non-holomorphic modular forms of a special type.…”

Class polynomials for nonholomorphic modular functions

“…As shown by Larson and Rolen, the function P (z) may be decomposed as where the product ranges over right coset representatives α for Γ 0 (6) in SL 2 (Z). The polynomials Ψ g may be expressed as polynomials in X whose coefficients are integer polynomials in j(z), and we regard them as elements of Z[X, J]; see Appendix A of [27] for the exact values of Ψ A and Ψ B . Here each occurrence of j(z) is replaced by the indeterminate J.…”