Modular units and the \(q\)-difference equations of Selberg (Q2275694)

To describe the motivation and the result of the paper under review, let us first consider the \(q\)-recurrence equation \[ R(z)=R(zq)+zqR(zq^2). \] The famous Rogers-Ramanujan identities state that \[ R(1)^{-1}=\prod_{n=1}^\infty(1-q^{5n-1})(1-q^{5n-4}), \qquad R(q)^{-1}=\prod_{n=1}^\infty(1-q^{5n-2})(1-q^{5n-3}). \] Let \(r(\tau)=q^{1/5}R(q)/R(1)\), \(q=e^{2\pi i\tau}\) with \(\mathrm{Im}\,\tau>0\). It is well-known that the function \(r(\tau)\) generates the field of modular functions on the principal congruence subgroup \(\Gamma(5)\). In fact, \(r(\tau)\) is a modular unit since its divisor is supported on the cusps of the modular curve \(X(5)\). The goal of the paper under review is to generalize this result, that is, find \(q\)-recurrence equations such that the ratios of their solutions generate the fields of modular functions on principal congruence subgroups of prime levels. For an odd prime \(\ell\) with \(\ell=2k+1\), the author considers Selberg's \(q\)-recurrence equation \[ \sum_{m=0}^ks_{k,m-1}(z)S_k(zq^m)=0, \] where \(s_{k,m-1}\) are certain functions that first appeared in [\textit{A. Selberg}, ``Über einige arithmetische Identitäten'', Avh. Norske Vidensk. - Akad. Oslo I 1936, Nr. 8, 23 S (1936; JFM 62.1068.04)]. (The functions are too complicated to be displayed here.) Define \[ r_{\ell,1}=(-1)^{k-1}q^{-k(k-1)/2\ell}\frac{S_k(1)}{S_k(q)}, \qquad \ell\geq 5, \] and \[ r_{\ell,2}=(-1)^{k-2}q^{(k+1)(k-2)/2\ell}\frac{S_k(1)-q^{k-1}S_k(q^2)} {S_k(q)}, \qquad \ell\geq 7. \] The author shows that \(r_{\ell_1}\) and \(r_{\ell,2}\) are both modular units on the modular curve \(X(\ell)\) associated to the principal congruence subgroup of level \(\ell\). More generally, she defines \(r_{\ell,j}\) for \(j=1,\ldots,(\ell-3)/2\) in terms of the action of elements from the Galois group of the cover \(X(\ell)\to X(1)\) on \(S_k(q)\). (Again, the definition is too complicated to be stated here.) As the main result of the paper, she proves that these functions are all modular units on \(X(\ell)\) and they generate the group of modular units on \(X(\ell)\) that have divisors supported on cusps lying above the cusp \(\infty\) of \(X_0(\ell)\). We remark that in the case of \(X_1(N)\) for general positive integers \(N\), an explicit basis for the group of modular units on \(X_1(N)\) having divisors supported on the cusps lying above \(\infty\) of \(X_0(N)\) is given in the reviewer's paper [``Modular units and cuspidal divisor class groups of \(X_1(N)\)'', J. Algebra 322, No. 2, 514-553 (2009; Zbl 1208.11076)]. Since \(\Gamma(\ell)\) is conjugate to a certain subgroup of \(\Gamma_1(\ell^2)\), the reviewer's results can also be used to a basis for the group of modular units on \(X(\ell)\) having divisors supported on the cusps lying above \(\infty\) of \(X_0(\ell)\).