The Eisenstein cycles and Manin–Drinfeld properties

Abstract

Abstract Let Γ be a subgroup of finite index of SL 2 ⁡ ( 𝐙 ) {\operatorname{SL}_{2}(\mathbf{Z})} . We give an analytic criterion for a cuspidal divisor to be torsion in the Jacobian J Γ {J_{\Gamma}} of the corresponding modular curve X Γ {X_{\Gamma}} . Our main tool is the explicit description, in terms of modular symbols, of what we call Eisenstein cycles. The latter are representations of relative homology classes over which integration of any holomorphic differential forms vanishes. Our approach relies in an essential way on the specific case Γ ⊂ Γ ⁢ ( 2 ) {\Gamma\subset\Gamma(2)} , where we can consider convenient generalized Jacobians instead of J Γ {J_{\Gamma}} . We relate the Eisenstein classes to the scattering constants attached to Eisenstein series. Finally, we illustrate our approach by considering Fermat curves.