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Heegner points, \(p\)-adic \(L\)-functions, and the Cherednik-Drinfeld uniformization. (English) Zbl 0899.11029
Let \(E/\mathbb Q\) be a modular elliptic curve of conductor \(N\), and let \(K\) be an imaginary quadratic field. The analytic continuation and functional equation for the Hasse-Weil zeta function \(L(E/K,s)\) can be determined by Rankin’s method. When the sign of this functional equation is \(-1\), a Heegner point \(\alpha_K\) is defined on \(E(K)\). In the case where all the primes dividing \(N\) are split in \(K\), the Heegner point comes from a modular curve parametrization of \(E\), and the Gross-Zagier formula relates its Néron-Tate canonical height to the first derivative of \(L(E/K,s)\) at \(s=1\).
B. Perrin-Riou [Invent. Math. 89, 455-510 (1987; Zbl 0645.14010)] obtained a \(p\)-adic analogue of the Gross-Zagier formula, expressing the \(p\)-adic height of \(\alpha_K\) in terms of a derivative of the 2-variable \(p\)-adic \(L\)-function attached to \(E/K\). At about the same time, B. Mazur, J. Tate and J. Teitelbaum [Invent. Math. 84, 1-48 (1986; Zbl 0699.14028)] formulated a \(p\)-adic Birch-Swinnerton-Dyer conjecture for the \(p\)-adic \(L\)-function of \(E\) associated to the cyclotomic \(\mathbb Z_p\)-extension of \(\mathbb Q\). In an earlier paper, M. Bertolini and H. Darmon [Invent. Math. 126, 413-456 (1996; Zbl 0882.11034)] proposed analogues of the Mazur-Tate-Teitelbaum conjectures for the \(p\)-adic \(L\)-function of \(E\) associated to the anticyclotomic \(\mathbb Z_p\)-extension of \(K\). In a special case, they predicted a \(p\)-adic analytic construction of the Heegner point \(\alpha_K\) from the first derivative of the anticyclotomic \(p\)-adic \(L\)-function. This paper provides a proof of this conjecture.

11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11G05 Elliptic curves over global fields
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
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