Geodesic
Twists of Shimura Curves
Canadian journal of mathematics, Tome 66 (2014) no. 4, pp. 924-960

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DOI

Consider a Shimura curve $X_{0}^{D}\left( N \right)$ over the rational numbers. We determine criteria for the twist by an Atkin–Lenher involution to have points over a local field. As a corollary we give a new proof of the theorem of Jordan and Livné on ${{\mathbf{Q}}_{p}}$ points when $p|D$ and for the first time give criteria for ${{\mathbf{Q}}_{p}}$ points when $p|N$ . We also give congruence conditions for roots modulo $p$ of Hilbert class polynomials.
DOI : 10.4153/CJM-2013-023-8
Mots-clés : 11G18, 14G35, 11G15, 11G10, Shimura curves, complex multiplication, modular curves, elliptic curves 
Stankewicz, James. Twists of Shimura Curves. Canadian journal of mathematics, Tome 66 (2014) no. 4, pp. 924-960. doi: 10.4153/CJM-2013-023-8
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     title = {Twists of {Shimura} {Curves}},
     journal = {Canadian journal of mathematics},
     pages = {924--960},
     year = {2014},
     volume = {66},
     number = {4},
     doi = {10.4153/CJM-2013-023-8},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2013-023-8/}
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