Growth of torsion groups of elliptic curves upon base change
نویسندگان
چکیده
منابع مشابه
Torsion Points of Elliptic Curves
Elliptic curves as an area of mathematical study are initially simple to understand, but reveal startling complexity when considered over different fields. This paper discusses the general properties and characteristics of projective space, elliptic curves, and the group structure that arises with certain binary operations on the curve. We discuss elliptic curves over Q, including the topic of ...
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2 Elliptic Curves and Maps Between Them 2 2.1 The Group Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Weierstrass Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 Maps Between Elliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Dual Isogenies . . . . . . . . . . . . . . ....
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Although it is not known which groups can appear as torsion groups of elliptic curves over cubic number fields, it is known which groups can appear for infinitely many non-isomorphic curves. We denote the set of these groups as S. In this paper we deal with three problems concerning the torsion of elliptic curves over cubic fields. First, we study the possible torsion groups of elliptic curves ...
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Let k be an algebraically closed field, char k 6= 2, 3, and let X ⊂ P be an elliptic curve with defining polynomial f . We show that any non-trivial torsion point of order r, determines up to equivalence, a unique minimal matrix Φr of size 3r×3r with linear polynomial entries such that det Φr = f . We also show that the identity, thought of as the trivial torsion point of order r, determines up...
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We exhibit several families of elliptic curves with torsion group isomorphic to Z/6Z and generic rank at least 3. Families of this kind have been constructed previously by several authors: Lecacheux, Kihara, Eroshkin and Woo. We mention the details of some of them and we add other examples developed more recently by Dujella and Peral, and MacLeod. Then we apply an algorithm of Gusi¢ and Tadi¢ a...
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ژورنال
عنوان ژورنال: Mathematics of Computation
سال: 2019
ISSN: 0025-5718,1088-6842
DOI: 10.1090/mcom/3478