منابع مشابه
On the Hasse Principle for Shimura Curves
Let C be an algebraic curve defined over a number field K, of positive genus and without K-rational points. We conjecture that there exists some extension field L over which C violates the Hasse principle, i.e., has points everywhere locally but not globally. We show that our conjecture holds for all but finitely many Shimura curves of the form XD 0 (N)/Q or X D 1 (N)/Q, where D > 1 and N are c...
متن کاملThe Hasse Principle and the Brauer-manin Obstruction for Curves
We discuss a range of ways, extending existing methods, to demonstrate violations of the Hasse principle on curves. Of particular interest are curves which contain a rational divisor class of degree 1, even though they contain no rational point. For such curves we construct new types of examples of violations of the Hasse principle which are due to the Brauer-Manin obstruction, subject to the c...
متن کاملOn a Shimura Curve That Is a Counterexample to the Hasse Principle
Let X be the Shimura curve corresponding to the quaternion algebra over ramified only at 3 and 13. B. Jordan showed that X ( √ −13) is a counterexample to the Hasse principle. Using an equation of X found by A. Kurihara, it is shown here, by elementary means, that X has no ( √ −13)-rational divisor classes of odd degree. A corollary of this is the fact that this counterexample is explained by t...
متن کاملCORRESPONDENCES ON SHIMURA CURVES AND MAZUR’S PRINCIPLE AT p
Under an additional hypothesis, the same result is true when [F(μ ) : F] = 2. In [12], we proved this result with an additional hypothesis if [F : Q] is even; this was removed in [10]. When [F : Q] is odd, the results of this paper do not depend on unpublished work, but we need Fujiwara’s results when [F : Q] is even. Subsequently, Rajaei removed the restriction that NF/Q(p) ≡ 1 (mod ), thus ge...
متن کاملCurves over Global Fields Violating the Hasse Principle
We exhibit for each global field k an algebraic curve over k which violates the Hasse Principle. We can find such examples among Atkin-Lehner twists of certain elliptic modular curves and Drinfeld modular curves. Our main tool is a refinement of the “Twist Anti-Hasse Principle” (TAHP). We then use TAHP to construct further Hasse Principle violations, e.g. among curves over any number field of a...
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ژورنال
عنوان ژورنال: Israel Journal of Mathematics
سال: 2009
ISSN: 0021-2172,1565-8511
DOI: 10.1007/s11856-009-0053-6