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By [6] we now know
that every elliptic curve over
**Q** is
a quotient of the curve *X*_{0}(*N*) whose complex points
are the isomorphism classes of pairs consisting of a
(generalized) elliptic curve and a cyclic subgroup of order *N*.
Let *J*_{0}(*N*) denote the Jacobian of *X*_{0}(*N*); this is an abelian
variety of dimension equal to the genus of *X*_{0}(*N*) whose points
correspond to the degree 0 divisor classes on *X*_{0}(*N*).
An *optimal quotient* of *J*_{0}(*N*) is a quotient by an abelian subvariety.
Consider an optimal quotient *A* such that
.
By [13],
*A*(**Q**) and
are both finite.
The BSD conjecture asserts that

Here the Shafarevich-Tate group
is a measure of the failure
of the local-to-global principle; the Tamagawa numbers *c*_{p} are the
orders of the component groups of *A*; the real number
is
the volume of
*A*(**R**) with respect to a basis of differentials having
everywhere nonzero good reduction; and
is the dual of *A*.
My goal is to verify the full
conjecture for many specific abelian varieties on a case-by-case basis.
This is the first step in a program to verify the above
conjecture for an infinite family of quotients of *J*_{0}(*N*).

** Next:** The ratio
** Up:** Invariants of modular abelian
** Previous:** Invariants of modular abelian
*William A. Stein*

*1999-12-01*