The Eisenstein and winding elements of modular symbols for odd square-free level

TL;DR

This paper explicitly computes Eisenstein and winding elements of modular symbols for odd square-free level congruence subgroups, providing concrete answers to Merel's question and generalizing previous results.

Contribution

It provides explicit formulas for Eisenstein and winding elements of modular symbols for odd square-free levels, extending prior work to new cases.

Findings

Explicit Eisenstein elements for $ ext{Gamma}_0(N)$ with odd square-free N

Explicit winding elements for these congruence subgroups

Generalization of Manin-Drinfeld Theorem to these cases

Abstract

We explicitly write down the Eisenstein elements inside the space of modular symbols for Eisenstein series with integer coefficients for the congruence subgroups $\Gamma_0(N)$ with $N$ odd square-free. We also compute the winding elements explicitly for these congruence subgroups. This gives an answer to a question of Merel in these cases. Our results are explicit versions of the Manin-Drinfeld Theorem [Thm. 6]. These results are the generalization of the paper [1] results to odd square-free level.