The Fourier coefficients of modular forms and Niebur modular integrals having small positive weight. I. (English) Zbl 0944.11014

This article treats the difficult problem of calculating the Fourier coefficients of modular forms (and the closely related Niebur modular integrals) on the full modular group \(\Gamma(1)\), for small positive weights \(k\). (In this context, “small” means \(0< k< 2\).) Fix \(k\in \mathbb R\) and let \(v\) be a multiplier system (MS) corresponding to the weight \(k\) for the group \(\Gamma(1)\). Then with D. Niebur’s seminal work [Trans. Am. Math. Soc. 191, 373–385 (1974; Zbl 0306.30023)] in mind, the author calls a function \(F\) a “Niebur modular integral” of weight \(k\) and MS \(v\), provided (i) \(F\) is holomorphic in the upper half-plane \(H\), and (ii) there exists a cusp form \(G\) on \(\Gamma(1)\) of weight \(2-k\) and MS \(\overline{v}\) (the “complementary” weight and MS) such that \[ [F(\tau)- \overline{v}(V) (\gamma\tau+ \delta)^{-k} ]^-= \int_{V^{-1}(\infty)}^{i\infty} G(z) (z-\overline{\tau})^{-k} dz, \] for all \(V=\left(\begin{smallmatrix} * & * \\ \gamma & \delta\end{smallmatrix}\right)\in \Gamma(1)\). Here, \([.]^-\) denotes complex conjugation and the path of integration is a vertical line in \(H\). Since there are no nontrivial cusp forms of nonpositive weight on \(\Gamma(1)\), when \(k\geq 2\) Niebur modular integrals are simply modular forms. Note as well that when \(k\in \mathbb Z\), \(k\leq 0\), they reduce to “Eichler integrals” [see M. Eichler, Math. Z. 67, 267–298 (1957; Zbl 0080.06003)]. Thus, \(0< k< 2\) is the range of interest for the study of Niebur modular integrals.
The author achieves an explicit calculation of the Fourier coefficients for cusp forms of weight \(k\) on \(\Gamma(1)\) for \(0< k<2\), \(k\neq 1\). The expressions for the Fourier coefficients are distinct in the two ranges, \(0< k< 1\) and \(1< k< 2\), but both expressions involve Selberg’s Kloosterman zeta-function. In addition, the author obtains an expression for the Fourier coefficients of Niebur modular integrals on \(\Gamma(1)\) when \(1< k< 2\). (See §10 for explicit statements.) He applies these results to obtain new (!!) expressions for \(\eta^{2k}(\tau)\) when \(0< k< 2\), \(k\neq 1\), where \(\eta(\tau)\) is Dedekind’s much-studied cusp form of weight \(\frac 12\) on \(\Gamma(1)\).
Clearly, this important work should be extended to finitely generated Fuchsian groups of the first kind (i.e., with finite hyperbolic area) acting on it. The paper contains an interesting, useful and well-written historical introduction.


11F30 Fourier coefficients of automorphic forms
11F11 Holomorphic modular forms of integral weight
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
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