Abstract
It has recently become clear that the construction of a p-adic height on an Abelian variety A eventually reduces to a splitting of the Hodge filtration of its de Rham cohomology. The present paper provides a natural description of this connection, based on the study of the universal vectorial extension of A, and of rigidified extensions of algebraic groups. Following a request of the editor, a detailed introduction to these topics has been included, in order to make the text as self-contained as possible.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Bibliography
N. Bourbaki.— Groupes et algébres de Lie, Chapitre 3, Hermann, Paris, 1972
R. Coleman and B. Gross — p-adic heights on curves, Preprint, MSRI, Berkeley, August, 1987.
S. Lang — Fundamentals of diophantine geometry, Springer-Verlag, 1983.
Yu.I. Manin.— The refined structure of the Néron-Tate height, Math. Sbornik 83 (1970), 332-248 (Math. USSR Sbornik 12 (1971), 325 - 342 ).
B. Mazur and J. Tate.— Canonical height pairings via biextensions, Arithmetic and Geometry (vol. 1), Progress in Mathematics (Birkhäuser) 35 (1983), 195 - 238.
W. Messing — The universal extension of an abelian variety by a vector group. Symposia Mathematica 11 (1973), 359 - 372.
A. Néron .— Hauteurs et fonctions théta, Rend. Sci. Mat. Milano 46 (1976), 111–135.
Perrin-Riou.— Hauteurs p-adiques, Séminaire de théorie des nombres, Paris 1982-83, Progress in Mathematics (Birkhäuser) 51 (1984), 233 - 257.
P. Schneider .— p-adic height pairings, I, II, Invent. Math. 69 (1982), 401–409; 79 (1985), 329 - 374.
J.—P. Serre.— Groupes algébriques et corps de classes, Hermann, Paris, 1958.
Yu.G. Zarhin.— Néron pairing and quasicharacters, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 497-509 (Math. USSR Izvestija, 6 (1972), 491 - 503 ).
N. Katz (with an appendix by L. Illusie).— Internal reconstruction of the unit root F-crystal via expansion coefficients, Annales Sci. ENS (4), 18 (1985), 245-268 (269-285).
H. Imai.— On the p-adic heights of some abelian varieties, Proc. Amer. Math. Soc. 100 (1987), 1 - 7.
M. Rosenlicht — Extensions of vector groups by Abelian varieties, Amer. J. of Math. 80 (1958), 685–714.
J. Oesterlé.— Constructions de hauteurs archimédiennes et p-adiques suivant la méthode de Bloch, Séminaire de Théorie des Nombres, Paris 1980-81, Birkh0user Prog. Math., 22, 1982, 175 - 192.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Birkhäuser Boston
About this chapter
Cite this chapter
Zarhin, Y.G. (1990). p-Adic Heights On Abelian Varieties. In: Goldstein, C. (eds) Séminaire de Théorie des Nombres, Paris 1987–88. Progress in Mathematics, vol 81. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-3460-9_16
Download citation
DOI: https://doi.org/10.1007/978-1-4612-3460-9_16
Publisher Name: Birkhäuser Boston
Print ISBN: 978-1-4612-8032-3
Online ISBN: 978-1-4612-3460-9
eBook Packages: Springer Book Archive