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$$p$$-adic periods, $$p$$-adic $$L$$-functions, and the $$p$$-adic uniformization of Shimura curves. (English) Zbl 1037.11045
From the introduction: Let $$E/\mathbb Q$$ be a modular elliptic curve of conductor $$N$$, and let $$p$$ be a prime of split multiplicative reduction for $$E$$. Write $$\mathbb C_p$$ for a fixed completion of an algebraic closure of $$\mathbb Q_p$$. Tate’s theory of $$p$$-adic uniformization of elliptic curves yields a rigid-analytic, $$\text{Gal} (\mathbb C_p, \mathbb Q_p)$$-equivariant uniformization of the $$\mathbb C_p$$-points of $E: 0\to q^{\mathbb Z}\to \mathbb C_p^\times @>\Phi_{\text{Tate}}>> E(\mathbb C_p)\to 0, \tag{1}$ where $$q\in p\mathbb Z_p$$ is the $$p$$-adic period of $$E$$.
B. Mazur, J. Tate and J. Teitelbaum conjectured in [[MTT] Invent. Math. 84, 1–48 (1986; Zbl 0699.14028)] that the cyclotomic $$p$$-adic $$L$$-function $$E/\mathbb Q$$ vanishes at the central point to order one greater than that of its classical counterpart. Furthermore, they proposed a formula for the leading coefficient of such a $$p$$-adic $$L$$-function. In the special case where the analytic rank of $$E(\mathbb Q)$$ is zero, they predicted that the ratio of the special value of the first derivative of the cyclotomic $$p$$-adic $$L$$-function and the algebraic part of the special value of the complex $$L$$-function of $$E/\mathbb Q$$ is equal to the quantity $$\frac {\log_p(q)} {\text{ord}_p(q)}$$ (where $$\log_p$$ is Iwasawa’s cyclotomic logarithm), which is defined purely in terms of the $$p$$-adic uniformization of $$E$$. Greenberg and Stevens gave a proof of this special case.
The article [M. Bertolini and H. Darmon, Invent. Math. 126, 413–456 (1996; Zbl 0882.11034); ibid. 453–491 (1998; Zbl 0899.11029)] formulates an analogue of the conjectures of [MTT] in which the cyclotomic $$\mathbb Z_p$$-extension of $$\mathbb Q$$ is replaced by the anticyclotomic $$\mathbb Z_p$$-extension of an imaginary quadratic field $$K$$. When $$p$$ is split in $$K$$ and the sign of the functional equation of $$L(E/K,s)$$ is $$+1$$, this conjecture relates the first derivative of the anticyclotomic $$p$$-adic $$L$$-function of $$E$$ to the anticyclotomic logarithm of the $$p$$-adic period of $$E$$. The present paper supplies a proof of this conjecture. Our proof is based on the theory of $$p$$-adic uniformization of Shimura curves.
The main result is the following.
Theorem. Suppose that $$c=1$$. The equality (up to sign) ${\mathcal L}_p' (E/K)= \frac {\text{rec}_p(q)} {\text{ord}_p(q)} \sqrt{ L(E/K,1) \Omega_f^{-1} \cdot d^{1/2} u^2 n_f}$ holds in $$I/I^2 \otimes \mathbb Q$$.

##### MSC:
 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11G18 Arithmetic aspects of modular and Shimura varieties 11F33 Congruences for modular and $$p$$-adic modular forms
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