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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.

##### MSC:
 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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