The Atkin orthogonal polynomials for congruence subgroups of low levels (Q926487)

Let \({\mathcal M}_N\) denote the ring of modular functions for the group \(\Gamma_0(N)\) which are holomorphic on the upper half plane and at all cusps except for \(i\infty\). The Atkin inner product \(A^{(N)}(f,g)\) is defined for \(f,g\in{\mathcal M}_N\). It has several properties in commen with the Petersson inner product for cusp forms. For \(N= 1\), A. O. L. Atkin found the inner product and its relation to supersingular elliptic curves over finite fields, but he did not publish his results. Atkin's results and further properties for \(N= 1\) were elaborated by \textit{M. Kaneko} and \textit{D. Zagier} [AMS/IP Stud. Adv. Math. 7, 97--126 (1998; Zbl 0955.11018)]. The author [J. Math. Kyoto Univ. 40, No. 4, 751--773 (2000; Zbl 0993.11021)] defined the Atkin inner product for arbitrary \(N\) and showed that the Hecke operators are self-adjoint with respect to this product. In the paper under review he generalizes Kaneko's and Zagier's results to those groups \(\Gamma_0(N)\) which are noncompact triangle groups, which embraces just the cases \(N\leq 4\). Then a function \(j_N\) is explicitly defined such that \({\mathcal M}_N\) coincides with the polynomial ring \(\mathbb C[j_N]\). (Of course, \(j_1\) is the elliptic invariant.) In Theorem 1, the Atkin inner product is defined by a certain Laurent coefficient, it is identified with a line integral in the upper half plane, and it is shown to be positive definite on \(\mathbb{R}[j_N]\). In Theorem 2, a sequence of orthogonal polynomials \(A^{(N)}_n\) for \(A^{N)}\) is introduced by a recursion and described by explicit formulas. In Theorem 3 it is shown that the degree \(n\) polynomial \(A^{(N)}_n\) has \(n\) distinct roots in a certain open interval. Theorem 4 gives the relation between \(A^{(N)}_{n_p}(X)\) and supersingular elliptic curves over a field \(K\) with prime characteristic \(p\). These curves have no torsion over the algebraic closure of \(K\). Their number \(n_p\) is finite. Theorem 4 states that \(A^{(N)}_{n_p}(X)\) is congruent modulo \(p\) to a product of factors \(X- j_N\) taken over certain supersingular curves. The group \(\Gamma_0(4)\) is conjugate to the principal congruence subgroup \(\Gamma(2)\). In this case the supersingular polynomial dates back to \textit{M. Deuring} [Abh. Math. Semin. Hansische Univ. 14, 197--272 (1941; Zbl 0025.02003 and JFM 67.0107.01)]. The restriction to the cases \(N\leq 4\) seems to be reasonable since otherwise one cannot expect positive definiteness even when the genus of \(\Gamma_0(N)\) is \(0\). The author shows that the Atkin inner product is in fact degenerate for \(N=5\).