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Points de Heegner et dérivées de fonctions L p-adiques. (Heegner points and derivatives of p-adic L-functions). (French) Zbl 0645.14010
A p-adic analogue of the Gross-Zagier theorem is proven. As is well known, the results of Gross and Zagier yield a relation between the derivative at 1 of a certain L-series attached to a modular form for \(\Gamma_ 0(N)\) with some character, and the height of so-called Heegner points [cf. B. H. Gross and D. B. Zagier, Invent. Math. 84, 225-320 (1986; Zbl 0608.14019)].
A similar relation turns out to be true for p-adic versions of these L- functions and height pairings. A p-adic height pairing (on an abelian variety over a number field) is defined as a sum of local contributions at each place of the field; the main difference with the “usual” case is that now the local Néron symbols are p-adic numbers. Such local symbols can be defined in this case except maybe at a place above p if the reduction of the abelian variety there is not ordinary. It turns out that the computation of these local symbols for places which don’t divide p, on Heegner points on the Jacobian of \(X_ 0(N)\), is already done in the Gross-Zagier paper. The bigger part of this paper is therefore devoted to the analysis of the p-adic L-functions.
An important application of these results appears in very recent work of Kolyvagin.
Reviewer: J.Top

MSC:
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11F33 Congruences for modular and \(p\)-adic modular forms
11S40 Zeta functions and \(L\)-functions
14K15 Arithmetic ground fields for abelian varieties
14G20 Local ground fields in algebraic geometry
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