2004/06/30 by Víctor Rotger, V. Rotger, Alexei N. Skorobogatov +6
Mathematics · #11G18 4G35 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT #msc:11G18 #msc:4G35
paper · pdf · doi:10.48550/arxiv.math/0406625
To appear in Moscow Math. Journal. Completely revised and improved version
openalex publication_date 2004/06/30 · arxiv created 2005/02/22 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show how to construct counter-examples to the Hasse principle over the field of rational numbers on Atkin-Lehner quotients of Shimura curves and on twisted forms of Shimura curves by Atkin-Lehner involutions. A particular example is the quotient of the Shimura curve X attached to the indefinite rational quaternion algebra of discriminant 23*107 by the Atkin-Lehner involution w107. The quadratic twist of X by Q(sqrt-23) with respect to this involution is also a counter-example to the Hasse principle over Q.