Chennai Mathematical Institute


3.30 pm, Seminar Hall
CMI Silver Jubilee Lecture
The explicit version of the Manin-Drinfeld theorem

Loic Merel
University of Paris VII, France.


Consider a congruence subgroup $\Gamma$ of the modular group ${\rm SL}_2({\bf Z})$. 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 $X_{\Gamma}$. Manin and Drinfeld proved that any divisor $D$ of degree zero supported on $C$ is torsion in the jacobian of $X_{\Gamma}$. 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 $X_{\Gamma}$ whose divisor is a multiple of $D$)

- or to the theory of automorphic forms. Find a 1-cycle $c_D$ on $X_{\Gamma}$ with boundary $D$ such that the integral of any holomorphic differential form along $c_D$ vanishes. Such $c_D$ 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 $c_D$ has rational coefficients. But they do not determine it. We ask the question of finding explicitly $c_D$, which should be called an {\it 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.