Traces of singular moduli and moonshine for the Thompson group (Q305095)

\textit{D. Zagier} [in: Motives, polylogarithms and Hodge theory. Part I: Motives and polylogarithms. Papers from the International Press conference, Irvine, USA, 1998. Somerville, MA: International Press. 211--244 (2002; Zbl 1048.11035)] constructed functions \(f_d(\tau)\) for \(d \equiv 0,3 \pmod{4}\) with Fourier expansions of the form \(q^{-d} + \sum_{n > 0} A(n,d) q^n\) which satisfy modular transformations like the Jacobi theta function \(\theta(\tau) = \sum_{n \in {\mathbb Z}} q^{n^2}\), \(q = e^{2\pi i \tau}\). In fact, \(f_0(\tau) = \theta(\tau)\). In the paper under review, the authors study a function \(F_3(\tau) = 2f_3(\tau) + 248 \theta(\tau)\) with Fourier expansion \(\sum_{m = -3}^\infty c(m) q^m\), where \(c(m)\) is an integer with \(c(m) = 0\) unless \(m \equiv 0,1 \pmod{4}\). The coefficients \(c(m)\) can be expressed as traces of singular moduli. The authors verify that each \(c(m)\) for \(m \leq 12\) is, up to sign, a sum of dimensions of at most two irreducible representations of the Thompson group. Moreover, the authors conjecture the existence of a \({\mathbb Z}\)-graded module \(W = \oplus_{m = -3}^\infty W_m\) for the Thompson group with \(\dim W_m = |c(m)|\) and calculate the McKay-Thompson series for the \(48\) conjugacy classes of the Thompson group associated with \(W\). It is shown that those McKay-Thompson series are weakly holomorphic modular forms of weight one-half. A discriminant property related to \(F_3(\tau)\) is also discussed.