$p$-Ordinary Part of Hyperbolic Cycles on Modular Curves

In this paper, we study hyperbolic cycles in the first homology group with local coefficients of congruence subgroups of ${\mathrm{S}\mathrm{L}}_2(\mathbb{Z})$. We prove that, for any prime number $p$, the $p$-ordinary part of the first homology group is generated by hyperbolic cycles.