Two torsion in the Brauer group of a hyperelliptic curve

Abstract

Abstract. We construct unramified central simple algebras representing 2-torsion classes in the Brauer group of a hyperelliptic curve, and show that every 2-torsion class can be constructed this way when the curve has a rational Weierstrass point or when the base field is C 1 . In general, we show that a large (but in general proper) subgroup of the 2-torsion classes are given by the construction. Examples demonstrating applications to the arithmetic of hyperelliptic curves defined over number fields are given.

Relative Brauer groups of torsors of period two

“…For any ∈ L × , let A be the L(x)-algebra obtained by adjoining anticommuting square roots of x − θ and to L(x). The corestriction A := Cor L(x)/K(x) (A ) is a tensor product of quaternion algebras over K(x) which can be explicitly computed using Rosset-Tate reciprocity (see [11,Proposition 3.1]). In [11] it is shown that, under suitable conditions on the norm of , the algebra A pulls back to an unramified central simple k(X)-algebra A := A ⊗ K(x) k(X).…”

“…In general there exist Brauer-like torsors of period 2 under J which are not of the form V . For example, see [11,Proposition 5.1 and Remark 5.4]. We also note that the hypothesis Br 0 (X/K) = 0 is satisfied quite frequently, as suggested by the following lemma.…”

“…This is a central simple algebra over k(X) which can be written explicitly as a tensor product of quaternion algebras (see [11,Proposition 2.4]). In this section we show that for suitable torsors V the class A V ∈ Br(X) can be represented by an algebra of the form A .…”

Insufficiency of the Brauer–Manin Obstruction for Rational Points on Enriques Surfaces

“…in finding an explicit representative for [the nontrivial] Brauer class of [X a ]." Recent work of Creutz and the last author [CV14a,CV14b], and Ingalls, Obus, Ozman, and the last author [IOOV] makes this problem more tractable. Building on techniques from [CV14a, CV14b, IOOV], we prove:…”

On Brauer groups of double covers of ruled surfaces

“…v := ψ(1 : −6 : 1 + i : −6 − i : 2 √ 4255 − 4160i). The point Q v lies over t = 36 and B t consists of 2 F v -points R 1 and R 2 and one quadratic point R; they have s values 19Since these points are unramified in B → P 1 and are away from the support of ℓ and α, the cocycle description of γ ′ (ℓ) in[CV, Lem. 3.5]shows thatA(Q v ) = (ℓ(−2), 25) 2 + (ℓ(−10), 35) 2 + γ ′ (ℓ)(Q v ) = (ℓ(10), 35) 2 + Cor Q 2 (i, √ 109)/Q 2 (i) ((x(Q v ) − α(R), ℓ(R)) 2 ) + 2 j=1 (x(Q v ) − α(R j ), ℓ(R j )) 2 .…”