We define the notion of special automorphisms on Shimura curves. Using this notion, for a wild class of elliptic curves defined over Q, we get rank one quadratic twists by discriminants having any prescribed number of...We define the notion of special automorphisms on Shimura curves. Using this notion, for a wild class of elliptic curves defined over 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).展开更多
We prove an anticyclotomic Iwasawa main conjecture proposed by Perrin-Riou for Heegner points for semi-stable elliptic curves E over a quadratic imaginary field K satisfying a certain generalized Heegner hypothesis,at...We prove an anticyclotomic Iwasawa main conjecture proposed by Perrin-Riou for Heegner points for semi-stable elliptic curves E over a quadratic imaginary field K satisfying a certain generalized Heegner hypothesis,at an ordinary prime p.It states that the square of the index of the anticyclotomic family of Heegner points in E equals the characteristic ideal of the torsion part of its Bloch–Kato Selmer group(see Theorem 1.3 for precise statement).As a byproduct we also prove the equality in the Greenberg–Iwasawa main conjecture for certain Rankin–Selberg product(Theorem 1.7)under some local conditions,and an improvement of Skinner’s result on a converse of Gross–Zagier and Kolyvagin theorem(Corollary 1.11).展开更多
文摘We define the notion of special automorphisms on Shimura curves. Using this notion, for a wild class of elliptic curves defined over 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).
基金the Chinese Academy of Science(Grant No.Y729025EE1)NSFC(Grant Nos.11688101,11621061)an NSFC grant associated to the recruitment Program of Global Experts。
文摘We prove an anticyclotomic Iwasawa main conjecture proposed by Perrin-Riou for Heegner points for semi-stable elliptic curves E over a quadratic imaginary field K satisfying a certain generalized Heegner hypothesis,at an ordinary prime p.It states that the square of the index of the anticyclotomic family of Heegner points in E equals the characteristic ideal of the torsion part of its Bloch–Kato Selmer group(see Theorem 1.3 for precise statement).As a byproduct we also prove the equality in the Greenberg–Iwasawa main conjecture for certain Rankin–Selberg product(Theorem 1.7)under some local conditions,and an improvement of Skinner’s result on a converse of Gross–Zagier and Kolyvagin theorem(Corollary 1.11).
