The Atkin inner product for \(\Gamma_0(N)\) (Q5939019)

For a positive integer \(N\), let \({\mathcal M}^{(N)}\) be the set of modular functions (weight 0) on the congruence group \(\Gamma_0(N)\) which are holomorphic on the upper half plane \({\mathcal H}\) and at all cusps except for \(i\infty\). Let \(E_2^{(N)}\) be the Eisenstein series of weight 2 for \(\Gamma_0(N)\). It is holomorphic and transforms like a modular integral of weight 2 (not a modular form) on \(\Gamma_0(N)\). For \(f\), \(g\) in \({\mathcal M}^{(N)}\), the Atkin inner product \((f,g)_N\) is defined to be the constant term in the expansion of \(fgE_2^{(N)}\) as a Laurent series in \(q= e^{2\pi i\tau}\). The author proves two theorems. NEWLINENEWLINENEWLINE(1) The Hecke operators are self-adjoint with respect to this inner product. NEWLINENEWLINENEWLINE(2) The inner product is given by an integral NEWLINE\[NEWLINE\text{vol} (\Gamma_0(N)\setminus{\mathcal H})\cdot (f,g)_N= \lim_{\Omega\to{\mathcal F}} \int_\Omega f(\tau) g(\tau) \frac{dx dy}{y^2},NEWLINE\]NEWLINE where \({\mathcal F}\) is a fundamental domain of \(\Gamma_0(N)\) and \(\Omega\) runs through a sequence of ``truncations'' of \({\mathcal F}\) at the cusps. NEWLINENEWLINENEWLINETheorem (1) generalizes A. O. L. Atkin's result for \(N=1\), and (2) generalizes R. E. Borcherds' result for \(N=1\). The integral differs from that in the Petersson inner product just by a factor \(g(\tau)\) instead of \(\overline{g(\tau)}\). This establishes the convergence, and the limit can be evaluated by Stokes' theorem and the calculus of residues.