Let
N be a positive integer, and
f a normalized newform of weight two on
0(
N). Attached to
f is an optimal quotient
A
f
of the Jacobian
J
0(
N). We prove two theorems concerning such optimal quotients. (A) Let
denote the algebra of endomorphisms of
J
0(
N) generated by the Hecke operators, and let
f
denote the ideal of fusion of
f in
. If Â
f
denotes the dual abelian variety to
A
f
, then the canonical polarization of
J
0(
N) induces a polarization
f
: Â
f
A
f
. We show that there is an embedding ker(
f
)
J
0
(
N) [
f
] whose cokernel is supported at maximal ideals
of
for which
J
0
(
N)[
] is
not two-dimensional. (B) If
N is prime, let
C denote the subgroup of
J
0(
N) generated by the divisor 0–
on
X
0(
N). Mazur has shown that
C is equal to the full torsion subgroup of
J
0(
N)(
), and that specialization modulo
N induces an isomorphism of
C with the group of connected components
J
0
(N) of the characteristic
N fibre of the Néron model of
J
0(
N). We prove that analogous results hold for every optimal quotient of prime conductor, thereby generalizing results of Mestre and Oesterlé (who treated the case of strong Weil curves) and confirming William Stein
s
refined Eisenstein conjecture. The key idea in the proof of these two theorems is encapsulated in corollary 2.5 below, which allows us to apply multiplicity one results in a novel way to the study of optimal quotients.
This work was partially supported by NSF grant DMS-0296095