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Journal of the Ramanujan Mathematical Society

Volume 36, Issue 2, June 2021  pp. 169–178.

Visibility and the Birch and Swinnerton-Dyer conjecture for analytic rank zero

Authors:  Amod Agashe
Author institution:Department of Mathematics, Florida State University, 208 Love Building, 1017 Academic Way Tallahassee, Florida 32306-4510, U.S.A.

Summary:  Let E be an optimal elliptic curve over Q of conductor N having analytic rank zero, i.e., such that the L-function L{E} (s) of E does not vanish at s = 1. Suppose there is another optimal elliptic curve over Q of the same conductor N whose Mordell-Weil rank is greater than zero and whose associated newform is congruent to the newform associated to E modulo a power r of a prime p. The theory of visibility then shows that under certain additional hypotheses involving p, r divides the product of the order of the Shafarevich-Tate group of E and the orders of the arithmetic component groups of E. We extract an explicit integer factor from the Birch and Swinnerton-Dyer conjectural formula for the product mentioned above, and under some hypotheses similar to the ones made in the situation above, we show that r divides this integer factor. This provides theoretical evidence for the second part of the Birch and Swinnerton-Dyer conjecture in the analytic rank zero case.

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