Alladi Ramakrishnan Hall

Eisenstein cycles and Manin-Drinfeld properties

Loic Merel

Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université de Paris

By a theorem of Belyi, every algebraic curve over a number field can be realized as the compactification $X_{\Gamma}$; of the quotient of the upper half-plane by a finite index subgroup $\Gamma$ in $SL2(Z)$. Such a situatiion is encoded by a "dessin d'enfant", a graph with simple properties (it is connected, finite, the vertices are bicolored, with distinct colors for the two vertices of a given edge, the set of edges attached to a given edge is endowed with a transitive action of $Z$). The cusps $P_{\Gamma}$ of $X_{\Gamma}$ correspond to the vertices of the graph. How to determine when a divisor D of degree 0 is torsion in the jacobian variety $J_{\Gamma}$ of $X_{\Gamma}$? The Manin-Drinfeld theorem asserts it is always the case when $\Gamma$ is a congruence subgroup. This question has been considered by Scholl and, separately, by K. Murty and Ramakrishnan. It is intimately connected to the determination of the Eisenstein class associated to D : the class E_D in $H_1(X_{\Gamma}, P_{\Gamma}; R)$ of boundary D such that the integration over E_D of $\omega$ vanishes for every holomorphic differential form
$\omega$ on $X_{\Gamma}$.

We will see how to reformulate this problem in a pleasant way when
$\Gamma$ is a subgroup of the principal congruence subgroup $\Gamma(2)$, and when one makes use of certain generalized Jacobian instead of $J_{\Gamma}$ The answer is of an analytic nature and involves what (some call) the Kloosterman zeta function.

We will see as an application what happens for the Fermat curve already considered by Rohrlich, Vélu, Posingies, and for the Heisenberg covering of the Fermat curve, introduced by K. Murty and Ramakrishnan.
This is a report on a joint work with Debargha Banerjee.