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We give an explicit example of a diagonal quartic surface defined over the rationals for which a necessarily transcendental 3 torsion element of its Brauer group gives rise to an obstruction to weak approximation. We also show that no non constant 2 primary element exists in its Brauer group demonstrating the need to consider all of the Brauer group for obstructions on diagonal quartics over the rationals and not just the 2 torsion part. We give a concrete number theoretic application of our construction. more...

List of contributors Preface Part I. Lecture Notes: 1. Three lectures on Cox rings Jurgen Hausen 2. A very brief introduction to etale homotopy Tomer M. Schlank and Alexei N. Skorobogatov 3. Torsors and representation theory of reductive groups Vera Serganova Part II. Contributed Papers: 4. Torsors over luna strata Ivan V. Arzhantsev 5. Abelianisation des espaces homogenes et applications arithmetiques Cyril Demarche 6. Gaussian rational points on a singular cubic surface Ulrich Derenthal and Felix Janda 7. Actions algebriques de groupes arithmetiques Philippe Gille and Laurent Moret-Bailly 8. Descent theory for open varieties David Harari and Alexei N. Skorobogatov 9. Factorially graded rings of complexity one Jurgen Hausen and Elaine Herppich 10. Nef and semiample divisors on rational surfaces Antonio Laface and Damiano Testa 11. Example of a transcendental 3-torsion Brauer-Manin obstruction on a diagonal quartic surface Thomas Preu 12. Homotopy obstructions to rational points Yonatan Harpaz and Tomer M. Schlank. more...