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\(p\)-adic height pairings. II. (English) Zbl 0571.14021
The author continues his study of \(p\)-adic \(L\)-series via flat cohomology, begun in [Invent. Math. 69, 401–409 (1982; Zbl 0509.14048) and ibid. 71, 251–293 (1983; Zbl 0511.14010)]. Let \(k\) denote an algebraic number-field, \({\mathfrak O}\) its rings of integers, \(A\) an abelian variety over \(k\), with good reduction over a prime \(p\), \(A^{\vee}\) its dual, \({\mathcal A}\) and \({\mathcal A}^{\vee}\) the Néron-models over \({\mathfrak O}\). For a \(\mathbb Z_ p\)-extension \(k_{\infty}\) of \(k\) and a character \(\kappa: \Gamma=\mathrm{Gal}(k^{\infty}/k)\to \mathbb Z_ p^*\) we have various pairings:
\[ \langle\langle\cdot,\cdot\rangle\rangle_{\kappa}: H^ 1({\mathfrak O},T_ p({\mathcal A}))\times H^ 1({\mathfrak O},T_ p({\mathcal A}^{\vee}))\to \mathbb Q_ p,\qquad \langle\cdot,\cdot\rangle_{\kappa}: \tilde A(k)\times A(k)\to\mathbb Q_ p, \] and if \(A\) is ordinary over \(p\): \[ (\cdot, \cdot)_{\kappa}: \tilde A(k)\times A(k)\to \mathbb Q_ p. \]
\(\langle\langle\cdot,\cdot\rangle\rangle_{\kappa}\) is defined via cohomology, \(\langle\cdot,\cdot\rangle_{\kappa}\) (the algebraic height pairing) is obtained from \(\langle\langle\cdot,\cdot\rangle\rangle_{\kappa}\) via the injection \(A(k)\otimes\mathbb Z_ p\to H^ 1({\mathfrak O},L_ p(A))\), and finally \((\cdot, \cdot)_{\kappa}\) is the analytic height. The main results of the paper are:
(i) If \(\langle\langle\cdot,\cdot\rangle\rangle_{\kappa}\) is non-degenerate and \(A\) ordinary, one obtains a Birch–Swinnerton-Dyer formula for the Iwasawa \(L\)- function.
(ii) A formula for the \(\mathbb Z_ p\)-rank of \(H^ 1(\Gamma,A(p^{\infty}))\) (it is zero if \(\langle\langle\cdot,\cdot\rangle\rangle_{\kappa}\) is non- degenerate).
(iii) The formula \((\cdot, \cdot)_{\kappa}=-\langle\cdot,\cdot\rangle_{\kappa}\).
Reading the paper requires some familiarity with the cohomological mechanism. The main question which remains open is of course whether any of the pairings is non-degenerate (or even non-zero).
Reviewer: Gerd Faltings

MSC:
14K15 Arithmetic ground fields for abelian varieties
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11G10 Abelian varieties of dimension \(> 1\)
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
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References:
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