zbMATH — the first resource for mathematics

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 \(X_0(49)\).
11G18 Arithmetic aspects of modular and Shimura varieties
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
Full Text: DOI
[1] Birch, B, Elliptic curves and modular functions, Symp Math, 4, 27-32, (1970) · Zbl 0225.14016
[2] Birch, B; Swinnerton-Dyer, P, Notes on elliptic curves (II), J Reine Angew Math, 218, 79-108, (1965) · Zbl 0147.02506
[3] Cai L, Chen Y, Liu Y. Euler system of Gross points and quadratic twist of elliptic curves. ArXiv:1601.04415, 2016
[4] Cai, L; Shu, J; Tian, Y, Explicit Gross-Zagier and waldspurger formulae, Algebra Number Theory, 8, 2523-2572, (2014) · Zbl 1311.11054
[5] Coates, J; Li, Y; Tian, Y; etal., Quadratic twists of elliptic curves, Proc Lond Math Soc (3), 110, 357-394, (2015) · Zbl 1311.11046
[6] Gonzalez-Aviles, C, On the conjecture of Birch and Swinnerton-Dyer, Trans Amer Math Soc, 349, 4181-4200, (1997) · Zbl 0918.11037
[7] Gross, B, Local orders, root numbers, and modular curves, Amer J Math, 110, 1153-1182, (1988) · Zbl 0675.12011
[8] Heegner, K, Diophantische analysis und modulfunktionen, Math Z, 56, 227-253, (1952) · Zbl 0049.16202
[9] Kobayshi, S, The p-adic Gross-Zagier formula for elliptic curves at supersingular primes, Invent Math, 191, 527-629, (2013) · Zbl 1300.11053
[10] Kolyvagain, V, Finiteness of \(E\)(Q) andx(E,Q) for a subclass of Weil curves (in Russian), Izv Akad Nauk SSSR Ser Mat, 52, 522-540, (1988)
[11] Kolyvagain, V, Euler system, 435-483, (1990), Boston
[12] Nekovr, J, The Euler system method for CM points on Shimura curves, 471-547, (2007), Cambridge · Zbl 1152.11023
[13] Perrin-Riou, B, Points de Heegner et dérivées de fonctions lp-adiques, Invent Math, 89, 455-510, (1987) · Zbl 0645.14010
[14] Prasad, D, Some applications of seesaw duality to braching laws, Math Ann, 304, 1-20, (1996) · Zbl 0838.22005
[15] Tian, Y, Congruent numbers with many prime factors, Proc Nat Acad Sci India Sect A, 109, 21256-21258, (2012) · Zbl 1298.11053
[16] Tian, Y, Congruent numbers and Heegner points, Cambridge J Math, 2, 117-161, (2014) · Zbl 1303.11067
[17] Tian Y, Yuan X, Zhang S. Genus periods, genus points, and congruent number problem. ArXiv:1411.4728, 2014 · Zbl 1441.11172
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.