 
Summary: A visible factor of the special Lvalue
Amod Agashe
October 1, 2008
Abstract
Let A be a quotient of J0(N) associated to a newform f such that
the special Lvalue of A (at s = 1) is nonzero. We give a formula for
the ratio of the special Lvalue to the real period of A that expresses
this ratio as a rational number. We extract an integer factor from the
numerator of this formula; this factor is nontrivial in general and is
related to certain congruences of f with eigenforms of positive analytic
rank. We use the techniques of visibility to show that, under certain
hypotheses (which includes the first part of the Birch and Swinnerton
Dyer conjecture on rank), if an odd prime q divides this factor, then
q divides either the order of the ShafarevichTate group or the order
of a component group of A. Suppose p is an odd prime such that p2
does not divide N, p does not divide the order of the rational torsion
subgroup of A, and f is congruent modulo a prime ideal over p to an
eigenform whose associated abelian variety has positive MordellWeil
rank. Then we show that p divides the factor mentioned above; in
particular, p divides the numerator of the ratio of the special Lvalue
