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Special automorphisms on Shimura curves and non-triviality of Heegner points. (English) Zbl 1416.11094
Summary: We define the notion of special automorphisms on Shimura curves. Using this notion, for a wild class of elliptic curves defined over $$\mathbb{Q}$$, we get rank one quadratic twists by discriminants having any prescribed number of prime factors. Finally, as an application, we obtain some new results on Birch and Swinnerton-Dyer (BSD) conjecture for the rank one quadratic twists of the elliptic curve $$X_0(49)$$.
##### MSC:
 11G18 Arithmetic aspects of modular and Shimura varieties 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
##### Keywords:
Shimura curves; Heegner points; BSD conjecture
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