A note on the Eisenstein elements of prime square level

Abstract

We explicitly write the Eisenstein elements inside the space of modular symbols corresponding to each Eisenstein series for the congruence subgroup Γ 0 (p 2 ), answering a question of Merel. As a consequence, we also write the winding element explicitly for the congruence subgroup Γ 0 (p 2 ).

The Eisenstein cycles as modular symbols

“…Part (2) of Theorem 2 follows easily from part (1). They are proved by Propositions 13 and 14, respectively.…”

“…Already in that work, the introduction of an auxiliary ‐structure played a key role. For the modular curves again of the form , the Eisenstein classes are computed explicitly using a similar method by Banerjee for and by Banerjee–Krishnamoorty for with and distinct odd prime numbers.…”

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

“…In this coordinate chart, the map π is given by z → z 2 . Hence, the function (f •π) 2 f •π ′ has no zero or pole on P − .…”

“…We calculate π E (γ ′ ) and π E (hγ ′ h −1 ). From 27, it is easy to see that hγ ′ h −1 = 1+z qv 2 −4p 2 q(1+q) 2 takes the cusp i∞ to 1 p . We deduce that π E (γ ′ ) = 2qa 0 (E[ 1 p ]) and γ ′ z0 z0 k * (ω E ) = 3a 0 (E[ 1 p ]).…”

“…In [2], a description is given of Eisenstein elements in terms of certain integrals for M = p 2 . In this article, we give an explicit description in terms of two matrices γ x,k 1 and γ x,k 2 .…”