\(p\)-adic periods, \(p\)-adic \(L\)-functions, and the \(p\)-adic uniformization of Shimura curves.

*(English)*Zbl 1037.11045From 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 \(I/I^2 \otimes \mathbb Q\).

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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\textit{M. Bertolini} and \textit{H. Darmon}, Duke Math. J. 98, No. 2, 305--334 (1999; Zbl 1037.11045)

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