The \(p\)-adic Gross-Zagier formula for elliptic curves at supersingular primes
Let \(E\) be an elliptic curve of conductor \(N\) and defined over \(\mathbb{Q}\). Let \(p\) be a prime not dividing \(N\) and let \(K\) be an imaginary quadratic field unramified at \(p\) and verifying the Heegner condition (i.e., all primes dividing \(N\) split in \(K\)). The Gross-Zagier formula (proved in \textit{B. H. Gross} and \textit{D. B. Zagier} [Invent. Math. 84, 225--320 (1986; Zbl 0608.14019)]) provides a crucial relation between the first derivative of the Hasse-Weil \(L\)-function of \(E\) over \(K\) at \(s=1\) and the Néron-Tate height of an Heegner point of \(E\). A \(p\)-adic version of this result was given in \textit{B. Perrin-Riou} [Invent. Math. 89, No. 3, 455--510 (1987; Zbl 0645.14010)] for a prime \(p\) of good ordinary reduction for \(E\). The paper under review establishes an analogous formula for a prime \(p\) of supersingular reduction. As the author mentions in the introduction, the strategy of the proof is the same of the cited works of Gross and Zagier and of Perrin-Riou with some relevant new ingredients depending on the type of reduction. In particular to avoid non-admissible denominators in \(p\)-adic integration, the author introduces a 2-variable \(p\)-adic \(L\)-function \(\mathscr{L}_p(E,\varepsilon,\alpha,s,t)\) (as the product of a 1-variable \(p\)-adic \(L\)-function and of its quadratic twist by the character \(\varepsilon\) associated to \(K\)), which can be characterized by critical values and whose denominators are admissible outside the diagonal \(s=t\). This \(p\)-adic \(L\)-function also depends on the choice of a root \(\alpha\) of \(X^2 -a_pX+p\) (where \(a_p=p+1-\#E(\mathbb{F}_p)\,\)) and this choice determines different non-canonical definitions of \(p\)-adic height pairings: in Section 4 of the paper the author describes these constructions and proves that they are equivalent and nontrivial. The construction of the 2-variable \(L\)-function \(\mathcal{L}_p\) leads, via integration with respect to an Eisenstein measure, to the definition of a \(p\)-adic modular form related to the derivative of \(\mathcal{L}_p\) at \(s=1\) and whose Fourier coefficients are computed in Section 2. In Section 3 the author generalizes the theory of Coleman power series to formal groups of height 2 and then applies it to a system of Heegner points in the cyclotomic extension of \(K\), to define another \(p\)-adic modular form related to the \(p\)-adic height of an Heegner point in \(E(K)\). In the final section these two modular forms are shown to coincide and this leads to the main formula. An important consequence of this Gross-Zagier type formula is the equivalence between the \(p\)-adic and the full Birch and Swinnerton-Dyer conjecture when the analytic rank is 1. In particular, for CM elliptic curves of analytic rank 1 (for which the main conjecture of Iwasawa theory is known to hold by \textit{K. Rubin} [Invent. Math. 103, No. 1, 25--68 (1991; Zbl 0737.11030)] and \textit{R. Pollack} and \textit{K. Rubin} [Ann. Math. (2) 159, No. 1, 447--464 (2004; Zbl 1082.11035)]), the author shows that the full Birch and Swinnerton-Dyer conjecture is true up to an element of \(\mathbb{Z}\left[\frac{1}{2N}\right]^*\).
scientific article; zbMATH DE number 6147370
scientific article