Hasse principle violations for Atkin-Lehner twists of Shimura curves

Author:

Clark Pete,Stankewicz James

Abstract

Let D > 546 D > 546 be the discriminant of an indefinite rational quaternion algebra. We show that there are infinitely many imaginary quadratic fields l / Q l/\mathbb {Q} such that the twist of the Shimura curve X D X^D by the main Atkin-Lehner involution w D w_D and l / Q l/\mathbb {Q} violates the Hasse Principle over Q \mathbb {Q} . More precisely, the number of squarefree d d with | d | X |d| \leq X such that the quadratic twist of ( X D , w D ) (X^D,w_D) by Q ( d ) \mathbb {Q}(\sqrt {d}) violates the Hasse Principle is \gg X / log α D X X/\log ^{\alpha _D} X and X / log β D X \ll X/\log ^{\beta _D} X for explicitly given 0 > β D > α D > 1 0 > \beta _D > \alpha _D > 1 such that α D β D 0 \alpha _D - \beta _D \rightarrow 0 as D D \rightarrow \infty .

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference16 articles.

1. [BGW] M. Bhargava, B. Gross, and X. Wang, Pencils of quadrics and the arithmetic of hyperelliptic curves, to appear, J. Amer. Math. Soc.

2. [Cl] P. L. Clark, Curves over global fields violating the Hasse Principle, \url{alpha.math.uga.edu/ pete/HasseBjornODD.pdf}

3. An “anti-Hasse principle” for prime twists;Clark, Pete L.;Int. J. Number Theory,2008

4. On the Hasse principle for Shimura curves;Clark, Pete L.;Israel J. Math.,2009

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