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Component groups of optimal quotients of Jacobians

Component groups of optimal quotients of Jacobians

W.A. Stein

Let A be an abelian variety over a finite extension K of the p-adic numbers Qp. Let O be the ring of integers of K, m its maximal ideal and k=O/m the residue class field. The N'{e}ron model of A is a smooth commutative group scheme A over O such that A is its generic fiber and satisfying the property: the restriction map
           Hom_O(S,A) -----> Hom_K(S/K,A)
is bijective for all schemes S over O. The special fiber A_k is a group scheme over k, which need not be connected. Denote by A_k^0 the connected component containing the identity. There is an exact sequence
           0 ----->A_k^0 ----->A_k ----->Phi_A ----->0
with Phi_A a finite '{e}tale group scheme over k, i.e., a finite abelian group equipped with an action of Gal(kbar/k).

    In this paper we study the group Phi_A with particular emphasis on quotients A of Jacobians of modular curves X_0(N). When A has semistable reduction Grothendieck described the component group in terms of a monodromy pairing on certain free abelian groups. When A=J is the Jacobian of X_0(N), this pairing can be explicitly computed, hence so can Phi_J, as has been done in many cases in cite{mazur:eisenstein} and cite{edixhoven:eisen}. Suppose now that A is a simple quotient of J and that the kernel of the map J ---> A is connected. There is a natural map Phi_J ---> Phi_A. In this paper we give a formula which can be used to compute the image and the order of the cokernel.

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