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Heights of Heegner points on Shimura curves. (English) Zbl 1036.11029
Let $$F$$ be a totally real number field, and let $$N$$ be a nonzero ideal of the ring $$\mathcal O_F$$ of integers in $$F$$. Let $$f$$ be a new form on $$\text{GL}_2 (\mathbb A_F)$$ of weight two and level $$K_0 (N)$$ with trivial central character, where $$K_0 (N) = \{ \left( \begin{smallmatrix} a&b\\ c&d \end{smallmatrix} \right) \in\text{GL} (\mathcal O_F \times \prod_p \mathbb Z_p) \mid c \in N \times \prod_p \mathbb Z_p \}$$. Let $$\mathcal O_f$$ be the subalgebra of $$\mathbb C$$ over $$\mathbb Z$$ generated by the eigenvalues $$a(m,f)$$ of $$f$$ under the Hecke operators $$T(m)$$ with $$(m,N) =1$$. Each embedding $$\sigma: \mathcal O_f \to \mathbb C$$ determines a new form $$f^\sigma$$ such that $$a(f^\sigma,m) = a (f,m)^\sigma$$. Assume that either $$[F:\mathbb Q]$$ is odd of $$\text{ord}_v (N) =1$$ for at least one finite place $$v$$ of $$F$$. Then there exists an abelian variety $$A$$ over $$F$$ of dimension $$[\mathcal O_f:\mathbb Z]$$ such that its $$L$$-function $$L(s,A)$$ coincides with $$\prod_{\sigma :\mathcal O_f \to \mathbb C} L(s,f^\sigma)$$ modulo the factors at the places dividing $$N$$. In this paper, under the assumption that the $$L$$-function $$L(s,f)$$ has order at most one at $$s=1$$, the author proves that the rank of the Mordell-Weil group $$A(F)$$ is equal to $$[\mathcal O_f:\mathbb Z] \text{ord}_{s=1} L(s,f)$$ and that the Shafarevich-Tate group of $$A$$ is finite. The proof is carried out by studying Heegner points over an imaginary quadratic extension of $$F$$.

##### MSC:
 11G18 Arithmetic aspects of modular and Shimura varieties 14G35 Modular and Shimura varieties 11F11 Holomorphic modular forms of integral weight 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11G50 Heights
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