On the Hasse principle for Shimura curves

Details On the Hasse principle for Shimura curves On the Hasse principle for Shimura curves

2005-10-12

arxiv.org/abs/math/0510239v2

Mathematics

Number Theory

10 pages

Scientific paper

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 X^D_0(N)_{/Q} or X^D_1(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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