A visible factor of the special L-value

Abstract

Let A be a quotient of J₀(N) associated to a newform ƒ 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 ƒ 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 ƒ 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 to the real period of A. Both of these results are as implied by the second part of the Birch and Swinnerton-Dyer conjecture, and thus provide theoretical evidence towards the conjecture.