Pairs of mod~3 and mod~5 representations arising from elliptic 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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We prove that quotients of complex vector spaces by lattices of maximal rank invariant with respect to irreducible (complex) reflection groups are isogenous to self-products of elliptic curves, and that for reflections groups which are not the complexifications of the Weyl groups of irreducible root systems, they are isomorphic to self-products of mutually isogenous elliptic curves with complex...
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Generically, one can attach to a Q-curve C octahedral representations ρ : Gal(Q/Q) −→ GL 2 (F 3) coming from the Galois action on the 3-torsion of those abelian varieties of GL 2-type whose building block is C. When C is defined over a quadratic field and has an isogeny of degree 2 to its Galois conjugate, there exist such representations ρ having image into GL 2 (F 9). Going the other way, we ...
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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.
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Explicit equations are given for the elliptic curves (in characteristic = 2, 3) with mod 2 representation isomorphic to that of a given one.
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ژورنال
عنوان ژورنال: Mathematical Research Letters
سال: 1999
ISSN: 1073-2780,1945-001X
DOI: 10.4310/mrl.1999.v6.n6.a12