Heegner points on modular curves

Cai, Li; Chen, Yihua; Liu, Yu

doi:10.1090/tran/7053

Heegner points on modular curves
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by Li Cai, Yihua Chen and Yu Liu PDF

Trans. Amer. Math. Soc. 370 (2018), 3721-3743 Request permission

Abstract:

In this paper, we study the Heegner points on more general modular curves other than $X_0(N)$, which generalizes Gross’ work “Heegner points on $X_0(N)$”. The explicit Gross-Zagier formula and the Euler system property are stated in this case. Using such a kind of Heegner points, we construct certain families of quadratic twists of $X_0(36)$, with the ranks of Mordell-Weil groups being zero and one respectively, and show that the $2$-part of their BSD conjectures hold.

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