JACOBIAN DETERMINANT OF MODULAR-CURVES AND HECKE OPERATORS

Let N be a strictly positive integer. We consider the modular curve X0(N)Q of level N, defined over Q and its jacobian J0(N)Q. There are at least two natural ways to extend this Q-abelian variety into a group-scheme over Z. We can use the Neron model of J0(N)Q or we can first extend the modular curve X0(N)Q into an integral model X0(N), thanks to Drinfeld, and then introduce its jacobian. We compare those two group-schemes. At a prime p such that p parallel-to N, the Neron model has semi-abelian reduction and we compute the group of connected components of the closed fiber. Further we recall how to extend the Hecke correspondances from Q tout Z(p). We end this lecture by an analogue of the Neron universal property for quasi-finite flat group-schemes, which is valid for semi-abelian group-schemes, when the base is not too ramified.