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A visible factor of the special L-value Amod Agashe

Summary: A visible factor of the special L-value
Amod Agashe
October 1, 2008
Let A be a quotient of J0(N) associated to a newform f such that
the special L-value of A (at s = 1) is non-zero. We give a formula for
the ratio of the special L-value 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 non-trivial 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 Shafarevich-Tate 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 Mordell-Weil
rank. Then we show that p divides the factor mentioned above; in
particular, p divides the numerator of the ratio of the special L-value


Source: Agashe, Amod - Department of Mathematics, Florida State University


Collections: Mathematics