Failure of the Hasse principle for Châtelet surfaces in characteristic 2
نویسندگان
چکیده
Given any global field k of characteristic 2, we construct a Châtelet surface over k that fails to satisfy the Hasse principle. This failure is due to a Brauer-Manin obstruction. This construction extends a result of Poonen to characteristic 2, thereby showing that the étale-Brauer obstruction is insufficient to explain all failures of the Hasse principle over a global field of any characteristic.
منابع مشابه
0 Fe b 20 09 THE HASSE PRINCIPLE FOR CHÂTELET SURFACES IN CHARACTERISTIC 2
Given any global field k of characteristic 2, we construct a Châtelet surface over k which fails to satisfy the Hasse principle. This failure is due to a Brauer-Manin obstruction. This construction extends a result of Poonen to characteristic 2, thereby showing that the étale-Brauer obstruction is insufficient to explain all failures of the Hasse principle over a global field of any characteris...
متن کاملHigher Dimensional Analogues of Châtelet Surfaces
We discuss the geometry and arithmetic of higher-dimensional analogues of Châtelet surfaces; namely, we describe the structure of their Brauer and Picard groups and show that they can violate the Hasse principle. In addition, we use these varieties to give straightforward generalizations of two recent results of Poonen. Specifically, we prove that, assuming Schinzel’s hypothesis, the non-m powe...
متن کاملOn the Arithmetic of Del Pezzo Surfaces of Degree 2
Del Pezzo surfaces are smooth projective surfaces, isomorphic over the algebraic closure of the base ,eld to P P or the blow-up of P in up to eight points in general position. In the latter case the del Pezzo surface has degree equal to 9 minus the number of points in the blow-up. The arithmetic of del Pezzo surfaces over number ,elds is an active area of investigation. It is known that the Has...
متن کاملFailure of the Hasse Principle for Enriques Surfaces
We construct an Enriques surface over Q with empty étale-Brauer set (and hence no rational points) for which there is no algebraic Brauer-Manin obstruction to the Hasse principle. In addition, if there is a transcendental obstruction on our Enriques surface, then we obtain a K3 surface that has a transcendental obstruction to the Hasse principle.
متن کاملThe Brauer-manin Obstruction on Del Pezzo Surfaces of Degree 2
This paper explores the computation of the Brauer-Manin obstruction on Del Pezzo surfaces of degree 2, with examples coming from the class of “semi-diagonal” Del Pezzo surfaces of degree 2. It is conjectured that the failure of the Hasse principle for a broad class of varieties, including Del Pezzo surfaces, can always be explained by a nontrivial Brauer-Manin obstruction. We provide computatio...
متن کامل