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\(p\)-adic Rankin \(L\)-series and rational points on CM elliptic curves. (English) Zbl 1326.11026

The purpose of this paper is to give a new proof of K. Rubin’s theorem [Invent. Math. 107, No. 2, 323–350 (1992; Zbl 0770.11033)] establishing that if \(\nu_A\) is a Hecke character of type \((1,0)\) attached to an elliptic curve \(A/{\mathbb Q}\) with complex multiplication and if \(L(A,s)\) vanishes to order one at the central point \(s=1\), then there exists a global point \(P\in A({\mathbb Q})\) of infinite order such that \[ {\mathcal L}_p(\nu_A^{\ast})=\Omega_p(A)^{-1}\log_{\omega_A}(P)^2 \bmod K^{\ast}, \] where \(\Omega_p(A)\) is the \(p\)-adic period attached to \(A\), \(\omega_A\in \Omega^1(A/{\mathbb Q})\) is a regular differential on \(A\) over \({\mathbb Q}\), and \(\log_{\omega_A}: A({\mathbb Q}_p) \to {\mathbb Q}_p\) denotes the \(p\)-adic formal group logarithm with respect to \(\omega_A\).
The original proof relies on a comparison between Heegner points and Euler systems of elliptic units. The new proof is based on the \(p\)-adic Gross-Zagier type formula given by the authors in [Duke Math. J. 162, No. 6, 1033–1148 (2013; Zbl 1302.11043)], only makes use of Heegner points, and requires neither elliptic units nor Perrin-Riou’s \(p\)-adic height computations. The result is given in Theorem 2. This new formula gives \(p\)-adic logarithms of Heegner points in terms of the special values of a \(p\)-adic Rankin \(L\)-function attached to a cusp form \(f\) and an imaginary quadratic field \(K\), applies to arbitrary, not necessary CM, elliptic curves over \({\mathbb Q}\), and removes the vanishing condition of order one.
The paper also includes a generalization to the case of modular abelian varieties with complex multiplication.

MSC:

11G05 Elliptic curves over global fields
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
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