2-Selmer groups and the Birch–Swinnerton-Dyer Conjecture for the congruent number curves
نویسندگان
چکیده
منابع مشابه
2-selmer Groups and the Birch-swinnerton-dyer Conjecture for the Congruent Number Curves
We take an approach toward counting the number of n for which the curve En : y = x3−n2x has 2-Selmer groups of a given size. This question was also discussed in a pair of papers by Roger Heath-Brown [6, 7]. We discuss the connection between computing the size of these Selmer groups and verifying cases of the Birch and Swinnerton-Dyer Conjecture. The key ingredient for the asymptotic formulae is...
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Sn = {d ∈ M | Cd(Qp) 6= ∅ ∀p|2n, Cd(Q∞) 6= ∅}, S ′ n = {d ∈ M | C ′ d(Qp) 6= ∅ ∀p|2n, C ′ d(Q∞) 6= ∅}, where the equations Cd and C ′ d, in variables (w, t, z) are given by Cd : dw 2 = t + (2n/d)z, C ′ d : dw 2 = t − (n/d)z. We should note that (0, 0, 0) is always a solution to Cd(C ′ d). So, when we write Cd(Qp) 6= ∅ (C ′ d(Qp) 6= ∅), we mean there exists nontrivial solutions. There has been m...
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Using elliptic modular functions, Kronecker proved a number of recurrence relations for suitable class numbers of positive binary quadratic forms. For instance if F (N) denotes the number of uneven classes of positive binary quadratic forms with determinant −N, then
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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...
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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 ...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2009
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2009.01.015