 
Summary: Modp reducibility, the torsion subgroup, and the
ShafarevichTate group
Amod Agashe
May 26, 2009
Abstract
Let E be an optimal elliptic curve over Q of prime conductor N. We
show that if for an odd prime p, the mod p representation associated
to E is reducible (in particular, if p divides the order of the torsion
subgroup of E(Q)), then the pprimary component of the Shafarevich
Tate group of E is trivial. We also state a related result for more
general abelian subvarieties of J0(N) and mention what to expect if N
is not prime.
1 Introduction and results
Let N be a positive integer. Let X0(N) be the modular curve over Q
associated to 0(N), and let J = J0(N) denote the Jacobian of X0(N),
which is an abelian variety over Q. Let T denote the Hecke algebra, which
is the subring of endomorphisms of J0(N) generated by the Hecke operators
(usually denoted T for N and Up for pN). If f is a newform of weight 2
on 0(N), then let If = AnnTf and let Af denote the associated newform
quotient J/If J, which is an abelian variety over Q. If the newform f has
