Heegner points on Hijikata-Pizer-Shemanske curves and the Birch and Swinnerton-Dyer conjecture
نویسندگان
چکیده
منابع مشابه
The Birch–swinnerton-dyer Conjecture and Heegner Points: a Survey
For a (connected) smooth projective curve C over the rational numbers Q, it is known that the rational points CpQq depends on the genus g “ gpCq of C: (1) If g “ 0, then the local-global principle holds for C, i.e.: CpQq ‰ H if and only if CpQpq ‰ H for all primes p ď 8 (we understand Qp “ R when p “ 8). In other words, C is globally solvable if and only if it is locally solvable everywhere. An...
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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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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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A polynomial relation f(x, y) = 0 in two variables defines a curve C. If the coefficients of the polynomial are rational numbers then one can ask for solutions of the equation f(x, y) = 0 with x, y ∈ Q, in other words for rational points on the curve. If we consider a non-singular projective model C of the curve then over C it is classified by its genus. Mordell conjectured, and in 1983 Falting...
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ژورنال
عنوان ژورنال: Publicacions Matemàtiques
سال: 2018
ISSN: 0214-1493
DOI: 10.5565/publmat6221803