On the Hasse principle for Shimura curves |
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Authors: | Pete L Clark |
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Institution: | (1) 1126 Burnside Hall, Department of Mathematics and Statistics, McGill University, 805 Sherbrooke West, Montreal, QC, Canada, H3A 2K6 |
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Abstract: | 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 has points everywhere locally but not globally. We show that our conjecture holds for all but finitely many Shimura curves
of the form X
0
D
(N)/ℚ or X
1
D
(N)/ℚ, where D > 1 and N are coprime squarefree positive integers. The proof uses a variation on a theorem of Frey, a gonality bound of Abramovich,
and an analysis of local points of small degree. |
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Keywords: | |
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