Displaying similar documents to “On the classification of 3-dimensional non-associative division algebras over p -adic fields”

On elementary equivalence, isomorphism and isogeny

Pete L. Clark (2006)

Journal de Théorie des Nombres de Bordeaux

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Motivated by recent work of Florian Pop, we study the connections between three notions of equivalence of function fields: isomorphism, elementary equivalence, and the condition that each of a pair of fields can be embedded in the other, which we call isogeny. Some of our results are purely geometric: we give an isogeny classification of Severi-Brauer varieties and quadric surfaces. These results are applied to deduce new instances of “elementary equivalence implies isomorphism”: for...

Twists of Hessian Elliptic Curves and Cubic Fields

Katsuya Miyake (2009)

Annales mathématiques Blaise Pascal

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In this paper we investigate Hesse’s elliptic curves H μ : U 3 + V 3 + W 3 = 3 μ U V W , μ ∈ Q - { 1 } , and construct their twists, H μ , t over quadratic fields, and H ˜ ( μ , t ) , μ , t ∈ Q over the Galois closures of cubic fields. We also show that H μ is a twist of H ˜ ( μ , t ) over the related cubic field when the quadratic field is contained in the Galois closure of the cubic field. We utilize a cubic polynomial, R ( t ; X ) : = X 3 + t X + t , t ∈ Q - { 0 , - 27 / 4 } , to parametrize all of quadratic fields and cubic ones. It should be noted that H ˜ ( μ , t ) is a twist of H μ as algebraic curves because it may not always have any...

Generalized Kummer theory and its applications

Toru Komatsu (2009)

Annales mathématiques Blaise Pascal

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In this report we study the arithmetic of Rikuna’s generic polynomial for the cyclic group of order n and obtain a generalized Kummer theory. It is useful under the condition that ζ ∉ k and ω ∈ k where ζ is a primitive n -th root of unity and ω = ζ + ζ - 1 . In particular, this result with ζ ∈ k implies the classical Kummer theory. We also present a method for calculating not only the conductor but also the Artin symbols of the cyclic extension which is defined by the Rikuna polynomial.

Failure of the Hasse principle for Châtelet surfaces in characteristic 2

Bianca Viray (2012)

Journal de Théorie des Nombres de Bordeaux

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Given any global field k of characteristic 2 , we construct a Châtelet surface over k that fails to satisfy the Hasse principle. This failure is due to a Brauer-Manin obstruction. This construction extends a result of Poonen to characteristic 2 , thereby showing that the étale-Brauer obstruction is insufficient to explain all failures of the Hasse principle over a global field of any characteristic.