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

TL;DR

This paper investigates hyperbolic cycles on modular curves, demonstrating that for any prime p, the p-ordinary part of the first homology group is generated by these cycles, revealing structural insights into the homology of congruence subgroups.

Contribution

It establishes that the p-ordinary part of the first homology group is generated by hyperbolic cycles for all primes p, providing new understanding of the homological structure of modular curves.

Findings

p-ordinary part of homology is generated by hyperbolic cycles

Results hold for all prime numbers p

Advances understanding of homological structure in modular curves

Abstract

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