We discuss a famous problem about right triangles with rational side lengths. This elementarysounding problem is still not completely solved; the last remaining step involves the Birch and Swinnerton-Dyer conjecture, which is one of the most important open problems in number theory (right up there with the Riemann hypothesis). 6.

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 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 p be a prime number and E denote the elliptic curve y 2 = x 3 + . It is known that for which congruent to 1 , 9 modulo 16, rank of over Q equal 0 Under condition Birch Swinnerton-Dyer conjecture true, we give necessary sufficient in terms constant term some polynomial defined by recurrence formula.

This paper provides evidence for the Birch and Swinnerton-Dyer conjecture for analytic rank 0 abelian varieties Af that are optimal quotients of J0(N) attached to newforms. We prove theorems about the ratio L(Af , 1)/ΩAf , develop tools for computing with Af , and gather data about certain arithmetic invariants of the nearly 20, 000 abelian varieties Af of level ≤ 2333. Over half of these Af ha...

We numerically verify the Conjecture of Birch and SwinnertonDyer concerning the analytic and geometric rank of an elliptic curve. An algorithm (based on the work of Cremona) is developed in the PARI/GP language for computing the order of vanishing of the L-function for any (non-singular) curve. The analytic rank outputs for several families of curves are compared with readily available data on ...

Let N ≡ 1(mod 4) be a positive integer and let be the single even primitive quadratic Dirichlet character on (Z/NZ)×. Let f ∈ S2(Γ0(N), ) be a newform with nebentypus . By the Shimura construction, f corresponds to an abelian variety Af defined over Q whose dimension is [Kf : Q] where Kf is the number field associated with f . When dimAf = 2, the Fricke involution wN acts on Af and is defined o...