The \(p\)-adic Gross-Zagier formula for elliptic curves at supersingular primes
type: Item
instance of: scholarly article
title: The \(p\)-adic Gross-Zagier formula for elliptic curves at supersingular primes
zbMATH Open document ID: 1300.11053
published in: Inventiones Mathematicae
publication date: 2013-03-20T00:00:00Z
review text: 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 me…
reviewed by: Andrea Bandini
Mathematics Subject Classification ID: 11F85
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11G40
Mathematics Subject Classification ID: 11G50
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 14L05
zbMATH DE Number: 6147370
zbMATH Keywords: Gross-Zagier formula
zbMATH Keywords: \(p\)-adic \(L\)-functions
zbMATH Keywords: Heegner points
zbMATH Keywords: \(p\)-adic height pairings
author: Shinichi Kobayashi
MaRDI profile type: Publication
full work available at URL: s00222-012-0400-9
OpenAlex ID: W1984039258
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The \(p\)-adic Gross-Zagier formula for elliptic curves at supersingular primes (DOI)
DOI: 10.1007/S00222-012-0400-9