Galois Lines for Normal Elliptic Space Curves
نویسندگان
چکیده
منابع مشابه
Galois Representations and Elliptic Curves
An elliptic curve over a field K is a projective nonsingular genus 1 curve E over K along with a chosen K-rational point O of E, which automatically becomes an algebraic group with identity O. If K has characteristic 0, the n-torsion of E, denoted E[n], is isomorphic to (Z/nZ) over K. The absolute Galois group GK acts on these points as a group automorphism, hence it acts on the inverse limit l...
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Galois representations ρ : GQ → GL2(Z/n) with cyclotomic determinant all arise from the n-torsion of elliptic curves for n = 2, 3, 5. For n = 4, we show the existence of more than a million such representations which are surjective and do not arise from any elliptic curve.
متن کاملGalois Theory, Elliptic Curves, and Root Numbers
The inverse problem of Galois theory asks whether an arbitrary finite group G can be realized as Gal(K/Q) for some Galois extension K of Q. When such a realization has been given for a particular G then a natural sequel is to find arithmetical realizations of the irreducible representations of G. One possibility is to ask for realizations in the Mordell-Weil groups of elliptic curves over Q: Gi...
متن کاملElliptic Curves with Surjective Adelic Galois Representations
Let K be a number field. The Gal(K/K)-action on the the torsion of an elliptic curve E/K gives rise to an adelic representation ρE : Gal(K/K) → GL2(Ẑ). From an analysis of maximal closed subgroups of GL2(Ẑ) we derive useful necessary and sufficient conditions for ρE to be surjective. Using these conditions, we compute an example of a number field K and an elliptic curve E/K that admits a surjec...
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ژورنال
عنوان ژورنال: Algebra Colloquium
سال: 2005
ISSN: 1005-3867,0219-1733
DOI: 10.1142/s1005386705000192