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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