Displaying similar documents to “On octahedral extensions of ℚ and quadratic ℚ -curves”

On elliptic Galois representations and genus-zero modular units

Julio Fernández, Joan-C. Lario (2007)

Journal de Théorie des Nombres de Bordeaux

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Given an odd prime   p   and a representation ϱ   of the absolute Galois group of a number field k onto PGL 2 ( 𝔽 p ) with cyclotomic determinant, the moduli space of elliptic curves defined over k with p -torsion giving rise to ϱ consists of two twists of the modular curve X ( p ) . We make here explicit the only genus-zero cases p = 3 and p = 5 , which are also the only cases: PGL 2 ( 𝔽 p ) ≃ 𝒮 n for n = 4 or n = 5 , respectively. This is done by studying the corresponding twisted Galois actions on the function field of the curve, for which...

An analogue of Pfister's local-global principle in the burnside ring

Martin Epkenhans (1999)

Journal de théorie des nombres de Bordeaux

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Let N / K be a Galois extension with Galois group 𝒢 . We study the set 𝒯 ( 𝒢 ) of ℤ -linear combinations of characters in the Burnside ring ℬ ( 𝒢 ) which give rise to ℤ -linear combinations of trace forms of subextensions of N / K which are trivial in the Witt ring W ( K ) of K . In particular, we prove that the torsion subgroup of ℬ ( 𝒢 ) / 𝒯 ( 𝒢 ) coincides with the kernel of the total signature homomorphism.

Fields of definition of ℚ -curves

Jordi Quer (2001)

Journal de théorie des nombres de Bordeaux

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Let C be a ℚ -curve with no complex multiplication. In this note we characterize the number fields K such that there is a curve C ' isogenous to C having all the isogenies between its Galois conjugates defined over K , and also the curves C ' isogenous to C defined over a number field K such that the abelian variety Res K / ℚ ( C ' / K ) obtained by restriction of scalars is a product of abelian varieties of GL 2 -type.

Polynomials over Q solving an embedding problem

Nuria Vila (1985)

Annales de l'institut Fourier

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The fields defined by the polynomials constructed in E. Nart and the author in J. Number Theory 16, (1983), 6–13, Th. 2.1, with absolute Galois group the alternating group A n , can be embedded in any central extension of A n if and only if n ≡ 0 ( m o d 8 ) , or n ≡ 2 ( m o d 8 ) and n is a sum of two squares. Consequently, for theses values of n , every central extension of A n occurs as a Galois group over Q .

Relative Galois module structure of integers of abelian fields

Nigel P. Byott, Günter Lettl (1996)

Journal de théorie des nombres de Bordeaux

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Let L / K be an extension of algebraic number fields, where L is abelian over ℚ . In this paper we give an explicit description of the associated order 𝒜 L / K of this extension when K is a cyclotomic field, and prove that o L , the ring of integers of L , is then isomorphic to 𝒜 L / K . This generalizes previous results of Leopoldt, Chan Lim and Bley. Furthermore we show that 𝒜 L / K is the maximal order if L / K is a cyclic and totally wildly ramified extension which is linearly disjoint to ℚ ( m ' ) / K , where m ' is the conductor...

The Brauer–Manin obstruction for curves having split Jacobians

Samir Siksek (2004)

Journal de Théorie des Nombres de Bordeaux

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Let X → 𝒜 be a non-constant morphism from a curve X to an abelian variety 𝒜 , all defined over a number field k . Suppose that X is a counterexample to the Hasse principle. We give sufficient conditions for the failure of the Hasse principle on X to be accounted for by the Brauer–Manin obstruction. These sufficiency conditions are slightly stronger than assuming that 𝒜 ( k ) and Ш ( 𝒜 / k ) are finite.