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On a Shimura Curve That is a Counterexample to the Hasse Principle
Author(s) -
Siksek Samir,
Skorobogatov Alexei
Publication year - 2003
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609302001893
Subject(s) - mathematics , counterexample , hasse principle , divisor (algebraic geometry) , pure mathematics , corollary , quaternion , number theory , degree (music) , algebra over a field , combinatorics , algebraic number field , geometry , physics , acoustics
Let X be the Shimura curve corresponding to the quaternion algebra over Q ramified only at 3 and 13. B. Jordan showed thatX Q (− 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 Q (− 13) ‐rational divisor classes of odd degree. A corollary of this is the fact that this counterexample is explained by the Manin obstruction. 2000 Mathematics Subject Classification 11G18, 11G05, 11G30.