Eisenstein cycles as modular symbols

TL;DR

This paper explicitly constructs Eisenstein cycles in the homology of modular curves as linear combinations of Manin symbols, providing an explicit version of the Manin-Drinfeld Theorem by characterizing these cycles as Hecke eigenvectors.

Contribution

It offers an explicit description of Eisenstein cycles for any odd level N, connecting them to Hecke eigenvectors and modular symbols, enhancing understanding of their structure.

Findings

Explicit formulas for Eisenstein cycles as Manin symbols

Characterization of Eisenstein cycles as Hecke eigenvectors

Connection to the Manin-Drinfeld Theorem

Abstract

For any odd integer N, we explicitly write down the Eisenstein cycles in the first homology group of modular curves of level N as linear combinations of Manin symbols. These cycles are, by definition, those over which every integral of holomorphic differential forms vanish. Our result can be seen as an explicit version of the Manin-Drinfeld Theorem. Our method is to characterize such Eisenstein cycles as eigenvectors for the Hecke operators. We make crucial use of expressions of Hecke actions on modular symbols and on auxiliary level 2 structures.