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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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