Chennai Mathematical Institute

Silver Jubilee Events

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

  • either a part of the theory of special functions (one can exhibit a rational function on XΓ whose divisor is a multiple of D)
  • or to the theory of automorphic forms. Find a 1-cycle cD on XΓ with boundary D such that the integral of any holomorphic differential form along cD vanishes. Such cD can be characterized by its annihilation by certain Hecke operators.

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.