CMI Silver Jubilee Lecture
Loic Merel, University of Paris VII, France
The explicit version of the Manin-Drinfeld theorem
Friday, August 28, 2015
Consider a congruence subgroup Γ of the modular group SL2(Z). It acts on the upper half-plane by homographies. The quotient Riemann surface, suitably compactified by the set C of cusps, is a modular curve XΓ. Manin and Drinfeld proved that any divisor D of degree zero supported on C is torsion in the jacobian of XΓ. Depending on one's point of view, one can see this result as
We adopt the latter point of view. The theorem of Manin and Drinfeld amount to show that cD has rational coefficients. But they do not determine it. We ask the question of finding explicitly cD, which should be called an Eisenstein cycle, as it obeys similar properties as Eisenstein series.
We will explain our approach to this problem and recent development due to D. Banerjee (Pune) and myself.