Local-global principles for torsors over arithmetic curves

Curves over Every Global Field Violating the Local-global Principle

There is an algorithm that takes as input a global field k and produces a curve over k violating the local-global principle. Also, given a global field k and a nonnegative integer n, one can effectively construct a curve X over k such that #X(k) = n.

Counterexamples to the local-global divisibility over elliptic curves

Let p ≥ 5 be a prime number. We find all the possible subgroups G of GL2(Z/pZ) such that there exists a number field k and an elliptic curve E defined over k such that the Gal(k(E [p])/k)-module E [p] is isomorphic to the G-module (Z/pZ)2 and there exists n ∈ N such that the local-global divisibility by pn does not hold over E(k).

A Local{global Principle for Niteness Properties of S{arithmetic Groups over Number Elds

Torsors over the Punctured Affine Line

We classify G-torsors over the punctured affine line Spec (k[t]) where G is a reductive algebraic group defined over a field k of good characteristic. Our classification is in terms of the Galois cohomology of the complete field k((t)) with values in G.

Cycle groups of curves over global fields

The purpose of this investigation is to study certain low-dimensional cycle groups of smooth projective curves over global fields of positive characteristic, that is, fields which are finitely generated of transcendence degree one over a finite field. After a brief introduction, we overview some established results in this area and extend them appropriately. Throughout this paper, an algebraic ...