Proving the Birch and Swinnerton-Dyer conjecture for specific elliptic curves of analytic rank zero and one
نویسندگان
چکیده
منابع مشابه
Rank of Elliptic Curves and the Birch Swinnerton-dyer Conjecture
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 ...
متن کاملVisibility and the Birch and Swinnerton-Dyer conjecture for analytic rank zero
Let E be an optimal elliptic curve over Q of conductor N having analytic rank zero, i.e., such that the L-function LE(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 vis...
متن کاملVisibility and the Birch and Swinnerton-Dyer conjecture for analytic rank one
Let E be an optimal elliptic curve over Q of conductor N having analytic rank one, i.e., such that the L-function LE(s) 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 vanishes to order one at s = 1. Suppose there is another optimal elliptic curve over Q of the same conductor N whose Mord...
متن کاملThe Birch and Swinnerton-Dyer conjecture for Q-curves
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...
متن کاملThe Birch-swinnerton-dyer Conjecture
We give a brief description of the Birch-Swinnerton-Dyer conjecture which is one of the seven Clay problems.
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ژورنال
عنوان ژورنال: LMS Journal of Computation and Mathematics
سال: 2011
ISSN: 1461-1570
DOI: 10.1112/s1461157011000180