Selmer Groups of Quadratic Twists of Elliptic Curves
نویسندگان
چکیده
(1.1) E : y + a1xy + a3y = x + a2x + a4x+ a6 where a1, a2, a3, a4, a6 ∈ Z. Let N(E) denote the conductor of E, j(E) the j-invariant of E, and L(E, s) = ∑∞ n=1 a(n)n −s the Hasse-Weil L-function of E. If E is modular, then let FE(z) = ∑∞ n=1 aE(n)q n ∈ S2(N(E), χ1) be the associated weight 2 cusp form. Here χ1 denotes the trivial Dirichlet character. Throughout, D will denote a square-free integer, and χD shall denote the Kronecker character for the field Q( √ D). Let h(D) denote the order of Cl(D), the ideal class group of Q( √ D). If ` is prime, then define h(D)` by h(D)` := |Cl(D)/` · Cl(D)| = `,
منابع مشابه
2-selmer Groups of Quadratic Twists of Elliptic Curves
In this paper we investigate families of quadratic twists of elliptic curves. Addressing a speculation of Ono, we identify a large class of elliptic curves for which the parities of the “algebraic parts” of the central values L(E/Q, 1), as d varies, have essentially the same multiplicative structure as the coefficients ad of L(E/Q, s). We achieve this by controlling the 2-Selmer rank (à la Mazu...
متن کاملOn the Quadratic Twists of a Family of Elliptic Curves
In this paper, we consider the average size of the 2-Selmer groups of a class of quadratic twists of each elliptic curve over Q with Q-torsion group Z2 × Z2. We prove the existence of a positive proportion of quadratic twists of such a curve, each of which has rank 0 Mordell-Weil group.
متن کاملSelmer Companion Curves
We say that two elliptic curves E1, E2 over a number field K are n-Selmer companions for a positive integer n if for every quadratic character χ of K, there is an isomorphism Seln(E χ 1 /K) ∼= Seln(E 2 /K) between the nSelmer groups of the quadratic twists E 1 , E χ 2 . We give sufficient conditions for two elliptic curves to be n-Selmer companions, and give a number of examples of non-isogenou...
متن کاملRANKS OF QUADRATIC TWISTS OF ELLIPTIC CURVES by
— We report on a large-scale project to investigate the ranks of elliptic curves in a quadratic twist family, focussing on the congruent number curve. Our methods to exclude candidate curves include 2-Selmer, 4-Selmer, and 8-Selmer tests, the use of the Guinand-Weil explicit formula, and even 3-descent in a couple of cases. We find that rank 6 quadratic twists are reasonably common (though stil...
متن کاملDisparity in Selmer ranks of quadratic twists of elliptic curves
We study the parity of 2-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. We prove that the fraction of twists (of a given elliptic curve over a fixed number field) having even 2-Selmer rank exists as a stable limit over the family of twists, and we compute this fraction as an explicit product of local factors. We give an example ...
متن کاملRanks of Twists of Elliptic Curves and Hilbert’s Tenth Problem
In this paper we investigate the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2-Selmer rank, and we give lower bounds for the number of twists (with bounded conductor) that have a given 2-Selmer rank. As a consequence, under appropriate hypotheses we can find m...
متن کامل