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

Debargha Banerjee and Srilakshmi Krishnamoorthy

Vol. 281 (2016), No. 2, 257–285

DOI: 10.2140/pjm.2016.281.257

Abstract

We explicitly write down the Eisenstein elements inside the space of modular symbols for Eisenstein series with integer coefficients for the congruence subgroups ${\Gamma }_0\left(pq\right)$ 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.

Keywords

Eisenstein series, modular symbols, special values of $L$-functions

Mathematical Subject Classification 2010

Primary: 11F67

Secondary: 11F11, 11F20, 11F30

Milestones

Received: 14 April 2015

Revised: 25 June 2015

Accepted: 3 July 2015

Published: 16 February 2016