Shimura integrals of cusp forms

V. V. Shokurov

Izvestiya. Mathematics , Tome 16 (1981) no. 3, pp. 603-646

Résumé

This paper studies integrals of the form $\int_\alpha^{i\infty}\Phi z^k\,dz$ on the upper half-plane, where $\alpha$ is a rational number, $0\leqslant k\leqslant w$ is integral, and $\Phi$ is a cusp form of weight $w+2$ with respect to some modular group $\Gamma\subset\mathrm{SL}(2,\mathbf Z)$. The main result is that if $\Gamma$ is a congruence subgroup and $\Phi$ is an eigenvector of all the Hecke operators, then all these integrals are representable as linear combinations of two complex numbers with coefficients in some field of algebraic numbers. Bibliography: 13 titles.