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 \(F\).


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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