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...

Let E/Q be an elliptic curve defined by the equation y 2 = x 3 + ax + b. For a prime p, p ∤ ∆ = −16(4a 3 + 27b 2) = 0, define N p = p + 1 − a p = |E(F p)|. As a precursor to their celebrated conjecture, Birch and Swinnerton-Dyer originally conjectured that for some constant c, p≤x,p∤∆ N p p ∼ c(log x) r , x → ∞. Let α p and β p be the eigenvalues of the Frobenius at p. Define ˜ c n =    α k ...

Abstract In the present paper, we generalize celebrated classical lemma of Birch and Heegner on quadratic twists elliptic curves over ℚ {{\mathbb{Q}}} . We prove existence explicit infinite families with analytic ranks 0 1 for a large class curves, use points to explicitly construct rational order rank 1. addition, sho...

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...