Effective bounds for Brauer groups of Kummer surfaces over number fields

Abstract

We study effective bounds for Brauer groups of Kummer surfaces associated to Jacobians of genus 2 curves defined over number fields.We obtain the following corollary as a consequence of results in [33,49]:Corollary 1.2. Given a smooth projective curve C of genus 2 defined over a number field k, there is an effective description of the setwhere X is the Kummer surface associated to the Jacobian Jac(C) of the curve C.Note that given a curve C of genus 2, the surface Y = Jac(C)/{±1} can be realized as a quartic surface in P 3 (see [20, Section 2]) and the Kummer surface X associated to Jac(C) is the minimal resolution of Y , so one can find defining equations for X explicitly.The quartic surface Y has sixteen nodes, and by considering the projection from one of these nodes, we may realize Y as a double cover of the plane. Thus X can be realized as a degree 2 K3 surface and our Theorem 1.1 follows from [23]. It is remarked in [23] that using the algebraic correspondence between X and Jac(C) it is possible to make [23] into an actual algorithm for Kummer surfaces. However we take a different approach from [23], and instead of using the Kuga-Satake construction we use a result of [52] reducing our problem to the case of abelian surfaces. In particular, our algorithm provides a large, but explicit bound for the Brauer group of X. (See the example we discuss in Section 6.)The method in this paper combines many results from the literature. The first key observation is that the Brauer group Br(X) admits the following stratification:Definition 1.3. Let X denote X × k Spec k where k is a given separable closure of k. Then we write Br 0 (X) = im(Br(k) → Br(X)) and Br 1 (X) = ker(Br(X) → Br(X)).Elements in Br 1 (X) are called algebraic elements; those in the complement Br(X) \ Br 1 (X) are called transcendental elements.Thus to obtain an effective bound for Br(X)/ Br 0 (X), it suffices to study Br 1 (X)/ Br 0 (X) and Br(X)/ Br 1 (X). The group Br 1 (X)/ Br 0 (X) is well-studied, and it admits the following isomorphism:Br 1 (X)/ Br 0 (X) ∼ = H 1 (k, Pic(X)).Note that for a K3 surface X, we have an isomorphism Pic(X) = NS(X). Thus as soon as we compute NS(X) as a Galois module, we are able to compute Br 1 (X)/ Br 0 (X). An algorithm to compute NS(X) is obtained in [49], but we consider another algorithm which is based on [10].To study Br(X)/ Br 1 (X), we use effective versions of Faltings' theorem and combine them with techniques in [23,51]. Namely, we have an injectionwhere Γ is the absolute Galois group of k. As a consequence of [52], we have an isomorphism of Galois modules Br(X) = Br(A), where A = Jac(C) is the Jacobian of C. Thus it suffice to bound the size of Br(A) Γ . To bound the cardinal of this group, we consider the following exact sequence as in [51]:Proposition 6.2 (Skorobogatov-Zarhin). For 3, we have that Br(A) Γ ( ) = 0.Proof. It suffices to show that the assumptions of [54, Proposition 4.2] are satisfied when image of Gal(Q/Q) in Aut(A[ ]) is GSp 4 (F ). This follows from PSp 4 (F ) being a simple nonabelian gr...

Kummer varieties and their Brauer groups

“…Thus no non-trivial linear combination of the classes [D i ] is zero in Pic(X). [2].) We generalise this result to g ≥ 2.…”

“…Let T be a k-torsor for the group k-scheme A [2]. We define the attached 2covering of A as the quotient Y = (A × k T )/A [2] by the diagonal action of A [2].…”

“…Let T be a k-torsor for the group k-scheme A [2]. We define the attached 2covering of A as the quotient Y = (A × k T )/A [2] by the diagonal action of A [2]. The first projection defines a morphism f : Y → A which is a torsor for A [2] such that T = f −1 (0).…”

“…The natural action of A on Y makes Y a k-torsor for A. In particular, there is an isomorphism of varieties Y ∼ = A. Alternatively, Y is the twisted form of A defined by a 1-cocycle with coefficients in A [2] representing the class of T in…”

“…Due to their intimate relation to abelian varieties, Kummer surfaces are a popular testing ground for conjectures on the geometry and arithmetic of K3 surfaces. Rational points and Brauer groups of Kummer surfaces were studied in [23,25,8,6,2,30].…”