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Let p ∈ N be a prime with p ≠ 2 , and let K be a p -adic field. Let F be the function field of a curve over K. Let Ω F be the set of all divisorial discrete valuations of F. In this paper, we ask whether the Hasse principle holds for semisimple adjoint linear algebraic groups over F. We give a positive answer to this question for a class of adjoint classical groups. [ABSTRACT FROM AUTHOR]

For any number field, we prove that there exists an elliptic curve defined over this field such that its Shafarevich-Tate group has a nontrivial 2-torsion subgroup. [ABSTRACT FROM AUTHOR]

For any number field not containing Q (i) , we give an explicit construction to prove that there exists an elliptic curve defined over this field such that its Shafarevich-Tate group is nontrivial. [ABSTRACT FROM AUTHOR]

We prove a local-global principle for torsors under the prosolvable geometric fundamental group of an affine curve over a number field. [ABSTRACT FROM AUTHOR]

In a recent paper, Colliot-Thélène, Parimala and Suresh conjectured that a local–global principle holds for projective homogeneous spaces under connected linear algebraic groups over function fields of p-adic curves. In this paper, we show that the conjecture is true for any linear algebraic group whose almost simple factors of its semisimple part are isogenous to unitary groups or special unitary groups of hermitian or skew-hermitian spaces over central simple algebras with involutions. The proof implements patching techniques of Harbater, Hartmann and Krashen. As an application, we obtain a Springer-type theorem for isotropy of hermitian spaces over odd degree extensions of function fields of p-adic curves. [ABSTRACT FROM AUTHOR]

Résumé Soit K le corps des fonctions d'une courbe projective lisse X sur un corps local supérieur k . On exhibe des exemples et des contre-exemples au principe local–global pour les algèbres simples centrales sur K , qui est une hypothèse importante pour disposer de suites exactes de Poitou–Tate pour les tores sur K . Let K be the function field of a smooth projective curve X over a higher-dimensional local field k . We give examples and counter-examples to the local–global principle for central simple algebras over K , which is an important hypothesis for having Poitou–Tate exact sequences for tori over K . [ABSTRACT FROM AUTHOR]

Abstract: Let be a field and a finite group. The question of ‘admissibility’ of over was originally posed by Schacher, who gave partial results in the case . In this paper, we give necessary conditions for admissibility of a finite group over function fields of curves over complete discretely valued fields. Using this criterion, we give an example of a finite group which is not admissible over . We also prove a certain Hasse principle for division algebras over such fields. [Copyright &y& Elsevier]

Abstract: For two semi-simple algebras A and B over an arbitrary ground field F, we give a numerical criterion when , the set of F-algebra homomorphisms between them, is non-empty. We also determine when the orbit set is finite and give an explicit formula for its cardinality. A few applications of main results are given. [Copyright &y& Elsevier]

Abstract: We discuss some results on the triviality and finiteness for Galois cohomology of connected unipotent groups over non-perfect (and especially local and global function) fields, and their relation with the closedness of orbits, which extend some well known results of Serre, Raynaud and Oesterlé. As one of the applications, we show that a separable additive polynomial over a global field k of characteristic in two variables is universal over k if and only if it is so over all completions of k. [Copyright &y& Elsevier]

Let K be a function field with an A-algebra structure. The ring A arises in the definition of the Drinfeld module φ over K. By E(K) we denote K together with the A-module structure induced on it by φ. For any principal prime ideal (a)⊂A, we study the question whether an element x∈E(K) which is an a-fold in E(Kν) for every place ν of K, is an a-fold in E(K). In particular, we study the groupS(a,K)≔kerE(K)/aE(K)→∏lower limit ν E(Kν)/aE(Kν)for Drinfeld modules of rank 2. We show that this finite group is trivial in many cases, but can become arbitrarily large. [Copyright &y& Elsevier]