The Eisenstein elements of modular symbols for level product of two distinct odd primes

TL;DR

This paper explicitly describes Eisenstein elements and computes winding elements in the space of modular symbols for level pq, where p and q are distinct odd primes, providing concrete versions of the Manin-Drinfeld Theorem.

Contribution

It provides explicit formulas for Eisenstein and winding elements in modular symbols for level product of two odd primes, answering Merel's question.

Findings

Explicit Eisenstein elements for level pq

Computed winding elements explicitly

Provides concrete versions of the Manin-Drinfeld Theorem

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 (pq) with p and q distinct odd primes, giving an answer to a question of Merel in these cases. We also compute the winding elements explicitly for these congruence subgroups. Our results are explicit versions of the Manin-Drinfeld Theorem.