We present a proof, which is conditional on the Birch and Swinnerton-Dyer Conjecture for a specific abelian variety, that there do not exist rational numbers x and c such that x has exact period N = 6 under the iteration x 7→ x + c. This extends earlier results by Morton for N = 4 and by Flynn, Poonen and Schaefer for N = 5.

We explain how recent work on 3-descent and 4-descent for elliptic curves over Q can be combined to search for generators of the Mordell-Weil group of large height. As an application we show that every elliptic curve of prime conductor in the SteinWatkins database has rank at least as large as predicted by the conjecture of Birch and Swinnerton-Dyer.

Let E/Q be an optimal elliptic curve of analytic rank zero. It follows from the Birch and Swinnerton-Dyer conjecture for curves zero that order torsion subgroup divides product Shafarevich–Tate group E/Q, (global) Tamagawa number at infinity. This consequence was noticed by Agashe Stein in 2005. In this paper, we prove divisibility statement unconditionally many cases, including case where is s...

We improve Kolyvagin’s upper bound on the order of the p-primary part of the Shafarevich-Tate group of an elliptic curve of rank one over a quadratic imaginary field. In many cases, our bound is precisely the one predicted by the Birch and Swinnerton-Dyer conjectural formula.

Let E be an optimal elliptic curve of conductor N , such that the L-function of E vanishes to order one at s = 1. Let K be a quadratic imaginary field in which all the primes dividing N split and such that the L-function of E over K also vanishes to order one at s = 1. In view of the Gross-Zagier theorem, the second part of the Birch and Swinnerton-Dyer conjecture says that the index in E(K) of...

Let E E be an elliptic curve over alttext="double-struck upper Q"> <mml:mi mathvariant="double-struc...

Chromatic Selmer groups are modified with local information for supersingular primes p. We sketch their role in establishing the p-primary part of Birch–Swinnerton-Dyer formula Sections 2–5, and then study growth Mordell–Weil rank along ℤ p 2 -extension a quadratic imaginary number field which splits Section 6.