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My primary objective is to verify the BSD conjecture for
specific modular abelian varieties by using the
rich theory of their arithmetic.
The BSD conjecture asserts that if *A* is a modular abelian
variety with
,
then

Here
is the group of rational torsion points on *A*;
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 certain component groups associated to *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*.

*William A. Stein*

*1999-12-01*