منابع مشابه
Galois Theory and Torsion Points on Curves
We begin with a brief history of the problem of determining the set of points of a curve that map to torsion points of the curve’s Jacobian. Let K be a number field, and suppose that X/K is an algebraic curve of genus g ≥ 2. Assume, furthermore, that X is embedded in its Jacobian variety J via a K-rational Albanese map i; thus there is a K-rational divisor D of degree one on X such that i = iD ...
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In contrast to the geometric theory, where the different kinds of number pairs (x, y) that can occur as solutions are viewed as homogeneous, the arithmetic b g gggj b b bbstudy classifies more carefully the structure of solutions of specific type. That is, one tries to understand the solutions to the equation (*) where (x, y) are constrained to lie in some arithmetically defined set. One common...
متن کاملElliptic Curves with Maximal Galois Action on Their Torsion Points
Given an elliptic curve E over a number field k, the Galois action on the torsion points of E induces a Galois representation, ρE : Gal(k/k) → GL2(b Z). For a fixed number field k, we describe the image of ρE for a “random” elliptic curve E over k. In particular, if k 6= Q is linearly disjoint from the cyclotomic extension of Q, then ρE will be surjective for “most” elliptic curves over k.
متن کاملTorsion Points on Curves
EXAMPLES. (i) The multiplicative group A = Gm is the algebraic group whose points over a field are the nonzero elements of the field. Then for any field K, Tor Gm(K) are the roots of unity contained in K. (ii) A = Gm×Gm then Tor(A) = Tor(Gm)×Tor(Gm) = {(x, y) : x, y ∈ K are roots of unity}. (iii) Let A be an elliptic curve. Over the complex numbers we can uniformize A as A = C/L where L is a la...
متن کاملGalois theory, discriminants and torsion subgroup of elliptic curves
We find a tight relationship between the torsion subgroup and the image of the mod 2 Galois representation associated to an elliptic curve defined over the rationals. This is shown using some characterizations for the squareness of the discriminant of the elliptic curve. © 2009 Elsevier B.V. All rights reserved.
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ژورنال
عنوان ژورنال: Journal de Théorie des Nombres de Bordeaux
سال: 2003
ISSN: 1246-7405
DOI: 10.5802/jtnb.384