Let denote the kernel of the quotient map . Consider the exact sequence , and the corresponding complex of Néron models. Because has semiabelian reduction (since ), Theorem A.1 of the appendix of [AU96, pg. 279-280], due to Raynaud, implies that there is an integer and an exact sequence
Here is the tangent space at the 0 section; it is a finite free -module of rank equal to the dimension. In particular, we have . Note that is -dual to the cotangent space, and the cotangent space is isomorphic to the space of global differential -forms. The theorem of Raynaud mentioned above is the generalization to of [Maz78, Cor. 1.1], which we used above in the proof of Theorem 3.5.
Let be the cokernel of . We have a diagram
and
Since is torsion free, by Lemma 4.1, the induced map
is injective. Since is a newform quotient, if then acts as a scalar on and on . Using Lemma 4.2, with , we see that the image of in under the composite of the maps in (1) is saturated. The Manin constant for at is the index of the image via -expansion of in in its saturation. Since the image of in is saturated, the image of is the saturation of the image of , so the Manin constant at is the index of in , which is by (4), hence is at most .
William Stein 2006-06-25