The Eisenstein cycles as modular symbols

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.

Higher Eisenstein elements, higher Eichler formulas and rank of Hecke algebras

“…The obvious generalization does not hold. More precisely, there seems to be no link between the vanishing of N −1 2 k=1 k • log(k) 3 and the fact that g p > 3. For instance, if p = 5 and N = 3671, we have g p = 5 but N −1 2 k=1 k • log(k) 3 ≡ 0 (modulo p), and if p = 7 and N = 4229, we have g p = 3 and…”

“…k=1 k • log(k) 3 ≡ 0 (modulo p). Frank Calegari and Matthew Emerton have identified T with a universal deformation ring for the residual representation ρ = χ p 0 0 1 , where χ p is the reduction modulo p of the pth cyclotomic character.…”

“…Thus, there should be a way to compute m + 1 using Eisenstein elements in H 1 (X 1 (N ), cusps, C) + . Formulas for all the Eisenstein elements of level Γ(N ) have recently been given in [3], where Γ(N ) ⊂ SL 2 (Z) is the subgroup of matrices congruent to the identity modulo N .…”

The Eisenstein cycles and Manin Drinfeld properties

“…We computed the functions F + D and F − D when Γ is congruence subgroup [2,3,4]. Here is a classical instance of a non-congruence subgroup where the computation of these formulas is possible.…”

“…Scholl [23], Murty-Ramakrishnan [19] and recently Burrin [5] have given criteria for D being torsion in J Γ , without appealing to E D . Our approach to this question is different from those authors and is a continuation of [4] and [17] (where we limited ourselves to congruence subgroups).…”