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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:
 [1] Bloch, S.: A note on height pairings. Tamagawa numbers and the Birch and Swinnerton-Dyer conjecture. Invent. math.58, 65-76 (1980) · Zbl 0444.14015 [2] Coates, J.: Infinite Descent on Elliptic Curves with Complex Multiplication. In: Arithmetic and Geometry, Papers, Dedicated to I.R. Shafarevich, vol. I. Progress in Math. vol. 35, pp. 107-137. Boston-Basel-Stuttgart: Birkhauser 1983 · Zbl 0541.14026 [3] Feerrero, B., Washington, L.: The Iwasawa invariant ? p vanishes for abelian number fieds. Ann. Math.109, 377-395 (1979) · Zbl 0443.12001 [4] Greenberg, R.: On a Certainl-Adic Representation. Invent. math.21, 117-124 (1973) · Zbl 0268.12004 [5] Imai, H.: A remark on the rational points of abelian arieties with values in cyclotomiz ? p -extensions. Proc. Japan Acad.51, 12-16 (1975) · Zbl 0323.14010 [6] Iwasawa, K.: On some properties of ?-finite modules. Ann. Math.70, 291-312 (1959) · Zbl 0202.33102 [7] Iwasawa, K.: On ? e -extensions of algebraic number fields. Ann. Math.98, 246-326 (1973) · Zbl 0285.12008 [8] Jannsen, U.: Über Galoisgruppen lokaler Körper. Invent. math.70, 53-69 (1982) · Zbl 0534.12009 [9] Konovalov, G.: The universal ?-norms of formal groups over a local field. Ukrain. Mat. Z.28, 396-398 (1976) · Zbl 0326.14014 [10] Lichtenbaum, S.: Values of zeta andL-functions at zero. Astérisque24-25, 133-138 (1975) · Zbl 0312.12016 [11] Lubin, J., Rosen, M.: The Norm Map for Ordinary Abelian Varieties. J. Algebra52, 236-240 (1978) · Zbl 0417.14035 [12] Mazur, B.: Local flat duality. Amer. J. Math.92, 343-361 (1970) · Zbl 0199.24501 [13] Mazur, B.: Rational Points of Abelian Varieties with Values in Towers of Number Fields. Invent. math.18, 183-266 (1972) · Zbl 0245.14015 [14] Mazur, B.: Notes on étale cohomology of number fields. Ann. sci. ENS6, 521-556 (1973) · Zbl 0282.14004 [15] Mazur, B., Messing, W.: Universal extensions and one-dimensional crystalline cohomology. Lecture Notes in Math., vol. 370. Berlin-Heidelberg-New York: Springer 1974 · Zbl 0301.14016 [16] Milne, J.S.: Etale cohomology. Princeton: Princeton Univ. Press 1980 · Zbl 0433.14012 [17] Perrin-Riou, B.: Arithmétique des courbes elliptiques et théorie d’Iwasawa Thése 1983 [18] Schneider, P.: Zur Vermutung von Birch and Swinnerton-Dyer über globalen Funktionenkörpern. Math. Ann.260, 495-510 (1982) · Zbl 0509.14022 [19] Schneider, P.:p-adic height pairings I. Invent. math.69, 401-409 (1982) · Zbl 0509.14048 [20] Schneider, P.: IwasawaL-functions of varieties over algebraic number fields. A first approach. Invent. math.71, 251-293 (1983) · Zbl 0511.14010 [21] Serre, J.-P.: Cohomologie Galoisienne. Lecture Notes in Math., vol 5. Berlin-Heidelberg-New York: Springer 1964 · Zbl 0143.05901 [22] Serre, J-P.: Sur les groupes de congruence des variétés abéliennes I, II. Izv. Akad. Nauk SSSR28, 3-20 (1964) and35, 731-737 (1971) [23] Serre, J-P.: Letter to Mazur, 1974 [24] Mazur, B., Tate, J.: Canonical Height Pairings via Biextensions. In Arithmetic and Geometry, Papers Dedicated to I.R. Shafarevich, vol. I. Progress in Math. vol. 35, pp. 195-237, Boston-Basel-Stuttgart: Birkhauser 1983 · Zbl 0574.14036 [25] Hazewinkel, M.: Norm maps for formal groups I: J. Algebra32, 89-108 (1974), II: J. reine angew. Math.268/69, 222-250 (1974). III: Duke Math. J.44, 305-314 (1977). IV: Michigan Math. J.25, 245-255 (1978) · Zbl 0288.12011 [26] SGA Grothendieck, A., Artin, M., Deligne, P., Demazure, M., Verdier, J.L.: Seminaire de Géométrie Algébrique du Bois Marie. 3I: Lecture Notes in Math. vol. 151. Berlin-Heidelberg-New York: Springer 1970, 4: Ibidem269, 270, 305 (1972-73). 41/2: Ibidem569 (1977). 7I: Ibidem288 (1972)
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