Note on linear relations in {\'e}tale $K$-theory of curves

Communications in Contemporary Mathematics, 2021

In this paper we investigate a local to global principle for Galois cohomology of number fields with coefficients in the Tate module of an abelian variety. In [BK13] G. Banaszak and the author obtained the sufficient condition for the validity of the local to global principle forétale K-theory of a curve. This condition in fact has been established by means of an analysis of the corresponding problem in the Galois cohomology. We show that in some cases this result is the best possible i.e if this condition does not hold we obtain counterexamples. We also give some examples of curves and their Jacobians. Finally, we prove the dynamical version of the local to global principle forétale K-theory of a curve. The dynamical local to global principle for the groups of Mordell-Weil type has recently been considered by S. Barańczuk in [B17]. We show that all our results remain valid for Quillen K-theory of X if the Bass and Quillen-Lichtenbaum conjectures hold true for X .

Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology, 2013

Let X be a smooth, proper and geometrically irreducible curve X defined over a number field F and let χ be a regular and proper model of X over OF,Sl. In this paper we address the problem of detecting the linear dependence over ℤl of elements in the étale K-theory of χ. To be more specific, let P ∊ Ket2n(χ) and let ⋀̂ ⊂ Ket2n(χ) be a ℤl-submodule. Let rυ: Ket2n(χ) → Ket2n(χυ) be the reduction map for υ ∉ Sl. We prove, under some conditions on X, that if rυ($\(P)_circumflex\$) ∈ rυ (⋀̂) for almost all υ of $\mathematical script capital(O) F,Sl\$ then $\(P)_circumflex\$ ∈ ⋀̂ + Ket2n(χ)tor.

Journal of K-theory and its Applications to Algebra Geometry and Topology 12 (01) pp.183-201, 2013

Let X be a smooth, proper and geometrically irreducible curve X defined over a number field F and let X be a regular and proper model of X over O F;S l : In this paper we address the problem of detecting the linear dependence over Z l of elements in the étale K-theory of X : To be more specific, let P 2 K et 2n .X / and let O ƒ K et 2n .X / be a Z l-submodule. Let r v W K et 2n .X / ! K et 2n .X v / be the reduction map for v … S l. We prove, under some conditions on X; that if r v. O P / 2 r v. O ƒ/ for almost all v of O F;S l then O P 2 O ƒ C K et 2n .X / tor :

In the present work, we investigate the reduction map on the etale K- theory of a curve dened over a global eld. We prove that on the even-dimensional K-groups the map has nite kernel and reduce the odd-dimensional case to a con- jecture of Jannsen.

Mathematical Proceedings of the Cambridge Philosophical Society, 1999

Let K ⊂ C be a field finitely generated over Q, K(a) ⊂ C the algebraic closure of K, G(K) = Gal(K(a)/K) its Galois group. For each positive integer m we write K(µ m ) for the subfield of K(a) obtained by adjoining to K all mth roots of unity. For each prime ℓ we write K(ℓ) for the subfield of K(a) obtained by adjoining to K all ℓ-power roots of unity. We write K(c) for the subfield of K(a) obtained by adjoining to K all roots of unity in K(a). Let K(ab) ⊂ K(a) be the maximal abelian extension of K. The field K(ab ℓ for the cyclotomic character defining the Galois action on all ℓ-power roots of unity. We write χℓ = χ ℓ mod ℓ : G(K) → Z * ℓ → (Z/ℓZ) * for the cyclotomic character defining the Galois action on the ℓ-th roots of unity. The character χ ℓ identifies Gal(K(ℓ)/K) with a subgroup of Let g be a positive integer, X a g-dimensional abelian variety over K. We write End K (X) for the ring of all endomorphisms of X defined over K and End 0 (X) for the finite-dimensional semisimple Q-algebra End K (X) ⊗ Q. Its center F = F X is a field if and only if X is K-isogenous to a power of a K-simple abelian variety. If so, F is either a totally real number field or a CM-field. We write Lie(X) for the tangent space to X at the origin. It is the g-dimensional K-vector space. By functoriality, End 0 (X) acts faithfully on Lie(X). We write for the corresponding trace map. The embedding End 0 (X) ֒→ End K (Lie(X)) gives rise to a natural structure of (not necessarily faithful) End 0 (X) ⊗ Q K-module on Lie(X). The well-known Mordell-Weil-Néron-Lang theorem asserts that X(K) is a finitely generated commutative group. In particular, its torsion subgroup TORS(X(K)) is finite. Hereafter we will write TORS(A) for the torsion subgroup of a commutative group A. This implies that TORS(X(L)) is finite for any finite algebraic extension L of K. Mazur [7] has raised the question of whether the groups X(K(ℓ)) are Supported by the NSF.