Colloquium abstracts

Debargha Banerjee*
July 23, 2020

The Eisenstein cycles and Manin-Drinfeld properties:  Consider any subgroup of finite index inside the full modular group. We write down the Eisenstein cycles in the first homology group of modular curves associated to any cuspidal divisor for the subgroup. Our expression of these cycles are in terms of values at cusps of certain real valued function. This function is the harmonic conjugate of real analytic Eisenstein series of weight zero associated to the divisor. As an application, we give a criteria for a divisor to be torsion or not in the cuspidal group in terms of these Eisenstein cycles. As an application, we also compute these Eisenstein cycles for non-congruence subgroups associated to Fermat's curves and quotients of Fermat's curves and Heisenberg curves. We answered a question raised by KumarMurty-Ramakrishnan regarding the cuspidal subgroups associated to noncongruence subgroups corresponding to the Heisenberg curves. This is a joint work with Loic Merel.