The Eisenstein cycles and Manin Drinfeld properties

TL;DR

This paper develops an analytic criterion for torsion divisors on modular curves using Eisenstein cycles, with applications to Fermat curves and implications for curves over number fields.

Contribution

It introduces Eisenstein cycles and their explicit description via modular symbols, linking homology classes to torsion properties in Jacobians.

Findings

Eisenstein cycles characterized in terms of modular symbols

Criterion for torsion divisors on modular curves established

Application to Fermat curves demonstrated

Abstract

Consider a subgroup of finite index of modular group. We give an analytic criterion for a cuspidal divisor to be torsion in the Jacobian of the corresponding modular curve. By BelyI theorem, such a criterion would apply to any curve over a number field. 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 , where we can consider convenient generalized Jacobians instead of Jacobian. The Eisenstein classes are the real part of certain homology classes with complex coefficients. The imaginary part of those classes are related to the scattering constants attached to Eisenstein series. Finally, we illustrate our theory by considering Fermat curves.