On Quaternion Algebras Split by a Given Extension, Clifford Algebras and Hyperelliptic Curves

Darrell Haile, Louis Rowen, Jean Pierre Tignol

Research output: Contribution to journalArticlepeer-review

Abstract

Given a monic separable polynomial π of degree 2n over an arbitrary field and a scalar α, we define generic algebras Hπ and Aαπ for the decomposition of π into a product of two polynomials of degree n and for the factorization απ = 2 respectively. We investigate representations of degree 1 or 2 of these generic algebras. Every representation of degree 1 of Hπ factors through an étale algebra of degree (2nn), whereas Aαπ has no representation of degree 1. We show that every representation of degree 2 of Hπ or Aαπ factors through the Clifford algebra of some quadratic form, pointed or not, and thus obtain a description of the quaternion algebras that are split by the étale algebra Fπ defined by π of by the function field of the hyperelliptic curve Xαπ with equation y2 = απ(x). We prove that every quaternion algebra split by the function field of Xαπ is also split by Fπ, and provide an example to show that a quaternion algebra split by Fπ may not be split by the function field of any curve Xαπ.

Original languageEnglish
Pages (from-to)1807-1826
Number of pages20
JournalAlgebras and Representation Theory
Volume23
Issue number4
DOIs
StatePublished - 1 Aug 2020

Bibliographical note

Publisher Copyright:
© 2019, Springer Nature B.V.

Keywords

  • Clifford algebra
  • Hyperelliptic curve
  • Pointed quadratic form
  • Quaternion algebra
  • étale algebra

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