Modular invariance, modular identities and supersingular \(j\)-invariants (Q862051)

Let \(V\) be a modular invariant vector space with basis \(\{ f_i(\tau) \}_{i=1}^k\). Let \(W_V\) (resp. \(W_V'\)) denote the Wronskian determinant of \(f_1,f_2,\dots,f_k\) (resp. \(f_1',f_2',\dots,f_k'\)), with respect to the derivative \(q\frac{d}{dq}\), where \(q:=e^{2\pi i \tau}\). The function \(\frac{W_V'(\tau)}{W_V(\tau)}\) is a (meromorphic) modular form on \(\text{SL}_2(\mathbb{Z}\)) of weight \(2k\). Number-theoretic properties of these quotients of Wronskians were first treated by the author, \textit{E. Mortenson} and \textit{K. Ono} [Int. J. Number Theory 4, No. 2, 323--337 (2008; Zbl 1214.11050)], who took \(V=V_k\) to be the set of irreducible characters of \(\mathcal{M}(2,2k+1)\) Virasoro minimal models. Here the author studies these same quotients, with particular attention to the case where \(V\) is the \(m\)-th symmetric power of a modular invariant space \(U\) of dimension \(2\). He re-derives some modular identities (such as the Ramanujan-Watson identities) and relates some of the Wronskian quotients to modular forms studied by \textit{M. Kaneko} and \textit{D. Zagier} [AMS/IP Stud. Adv. Math. 7, 97--126 (1998; Zbl 0955.11018)].