Vanishing and non-vanishing Dirichlet twists of <i>L</i>
-functions of elliptic curves

Fearnley, Jack; Kisilevsky, Hershy; Kuwata, Masato

doi:10.1112/jlms/jds018

Cited by 15 publications

(11 citation statements)

References 19 publications

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“…This can be a difficult problem in general depending on what X, d and H are. For example for X = P 1 this problem is equivalent to the inverse Galois problem for H, and for X = E an elliptic curve and H = Z/dZ this is closely related to the rank growth of E over cyclic extensions -see for example [8,Section 4].…”

Section: Strategy For Determining the Galois Groups Of Degree D Point...mentioning

confidence: 99%

Torsion of elliptic curves over cyclic cubic fields

Derickx

Najman

2018

Math. Comp.

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We determine all the possible torsion groups of elliptic curves over cyclic cubic fields, over non-cyclic totally real cubic fields and over complex cubic fields. IntroductionAn important problem in the theory of elliptic curves is to determine the possible torsion groups of elliptic curves over number fields of degree d. Mazur solved the problem for d = 1 [14] and Kamienny [12], building on the work of Kenku and Momose [13], solved the problem for d = 2. Recently, Derickx, Etropolski, van Hoeij, Morrow and Zuerick-Brown announced the solution of the problem for d = 3 [5], building on the work of Parent [17, 18]. For d > 3 the problem remains unsolved at the moment.A question that naturally arises is which of the groups that arise as torsion groups of elliptic curves over number fields of degree d arise over some natural subset of the set of number fields of degree d. These subsets of the set of all number fields of degree d can be chosen to be, perhaps most naturally, the subset of real (or totally real) number fields, the subset of complex number fields, and the subset of number fields whose normal closure over Q has Galois group isomorphic to some prescribed group G. Throughout the paper, by abuse of language we say that a number fields K has Galois group G over Q if the Galois group over Q of the normal closure of K over Q is G.These problems are of course meaningless for the case d = 1, and for d = 2 one can only consider the subdivision of quadratic fields into real and imaginary quadratic fields. The possibilities in each of these cases (for d = 2) follow directly from [3] and are summed up in Theorem 3.1. Thus the problem has been solved completely for d = 2, so it is natural to consider the case d = 3, which is the last case where all the possible torsion groups are known. The main result of the paper is the determination of all possible torsion groups of elliptic curves over a) All cubic fields with Galois group Z/3Z. b) All complex cubic fields. c) All totally real cubic fields with Galois group S 3 .Parts of some proofs of our results are based on computations in Magma [1]. Some of these computations depend on a magma package writen by Solomon Vishkautsan and the first author [7]. All computations done in Magma for this paper can be found at https://github.com/wishcow79/chabauty/tree/master/pWe would like to thank Solomon Vishkautsan for making his Chabauty code public and his help in explaining on how to use it.

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Section: Strategy For Determining the Galois Groups Of Degree D Point...mentioning

confidence: 99%

Torsion of elliptic curves over cyclic cubic fields

Derickx

Najman

2018

Math. Comp.

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“…This is not the only way of constructing points on the double cover, but it is the one that I focus on in this paper. For a different approach, refer to the paper of Fearnley, Kisilevsky, and Kuwata [FKK12].…”

Section: Dihedral Quotientsmentioning

confidence: 99%

“…The case G = Z/3Z has been studied by Fearnley, Kisilevsky, and Kuwata in [FKK12]. The authors prove that if K contains its cube roots of unity then there are infinitely many cyclic cubic extensions of K over which E gains rank.…”

Section: Introductionmentioning

confidence: 99%

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Ranks of elliptic curves over cyclic cubic, quartic, and sextic extensions

Kashyap¹

2014

Preprint

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For a given group G and an elliptic curve E defined over a number field K, I discuss the problem of finding G-extensions of K over which E gains rank. I prove the following theorem, extending a result of Fearnley, Kisilevsky, and Kuwata: Theorem 1. Let n = 3, 4, or 6. If K contains its n th -roots of unity then, for any elliptic curve E over K, there are infinitely many Z/nZ-extensions of K over which E gains rank.

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“…The surface X C is the quotient mentioned in Theorem 1.1. It is also used in [7], where the number of rational points on X C is related to random matrix theory. The fixed points of ρ are exactly the nine points (P, P ) where P runs through the flexes of C. Let P be such a flex and let r and s be two copies of a uniformizer at P .…”

Section: A K3 Surface Associated To a Plane Cubic Curvementioning

confidence: 99%