Étale cohomology and reduction of abelian varieties

A. Silverberg; Yu. G. Zarhin

Bulletin de la Société Mathématique de France (2001)

  • Volume: 129, Issue: 1, page 141-157
  • ISSN: 0037-9484

Abstract

top
In this paper we study the étale cohomology groups associated to abelian varieties. We obtain necessary and sufficient conditions for an abelian variety to have semistable reduction (or purely additive reduction which becomes semistable over a quadratic extension) in terms of the action of the absolute inertia group on the étale cohomology groups with finite coefficients.

How to cite

top

Silverberg, A., and Zarhin, Yu. G.. "Étale cohomology and reduction of abelian varieties." Bulletin de la Société Mathématique de France 129.1 (2001): 141-157. <http://eudml.org/doc/272465>.

@article{Silverberg2001,
abstract = {In this paper we study the étale cohomology groups associated to abelian varieties. We obtain necessary and sufficient conditions for an abelian variety to have semistable reduction (or purely additive reduction which becomes semistable over a quadratic extension) in terms of the action of the absolute inertia group on the étale cohomology groups with finite coefficients.},
author = {Silverberg, A., Zarhin, Yu. G.},
journal = {Bulletin de la Société Mathématique de France},
keywords = {abelian varieties; semistable reduction; étale cohomology; monodromy},
language = {eng},
number = {1},
pages = {141-157},
publisher = {Société mathématique de France},
title = {Étale cohomology and reduction of abelian varieties},
url = {http://eudml.org/doc/272465},
volume = {129},
year = {2001},
}

TY - JOUR
AU - Silverberg, A.
AU - Zarhin, Yu. G.
TI - Étale cohomology and reduction of abelian varieties
JO - Bulletin de la Société Mathématique de France
PY - 2001
PB - Société mathématique de France
VL - 129
IS - 1
SP - 141
EP - 157
AB - In this paper we study the étale cohomology groups associated to abelian varieties. We obtain necessary and sufficient conditions for an abelian variety to have semistable reduction (or purely additive reduction which becomes semistable over a quadratic extension) in terms of the action of the absolute inertia group on the étale cohomology groups with finite coefficients.
LA - eng
KW - abelian varieties; semistable reduction; étale cohomology; monodromy
UR - http://eudml.org/doc/272465
ER -

References

top
  1. [1] G. Almkvist & R. Fossum – « Decomposition of exterior and symmetric powers of indecomposable 𝐙 / p 𝐙 -modules in characteristic p and relations to invariants », Séminaire d’Algèbre Paul Dubreil, 30ème année (Paris, 1976–1977), Lecture Notes in Math., vol. 641, Springer, Berlin, 1978, p. 1–111. Zbl0381.16015MR499459
  2. [2] S. Bosch, W. Lütkebohmert & M. Raynaud – Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21, Springer-Verlag, Berlin, 1990. Zbl0705.14001
  3. [3] P. Deligne – « Introduction », Groupes de monodromie en géométrie algébrique. I, Lecture Notes in Mathematics, vol. 288, Springer Verlag, Berlin, 1972, p. v–vii. Zbl0258.00005MR354656
  4. [4] —, « Résumé des premiers exposés de a. grothendieck », Groupes de monodromie en géométrie algébrique. I, Lecture Notes in Mathematics, Vol. 288, Springer-Verlag, Berlin, 1972, p. 1–24. Zbl0237.00013
  5. [5] A. Grothendieck – Groupes de monodromie en géométrie algébrique. I, Lecture Notes in Mathematics, vol. 288, Springer-Verlag, Berlin, 1972, Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 I), Dirigé par A. Grothendieck. Avec la collaboration de M. Raynaud et D. S. Rim. Zbl0237.00013MR354656
  6. [6] —, « Modèles de néron et monodromie », Groupes de monodromie en géométrie algébrique. I, Lecture Notes in Mathematics, vol. 288, Springer-Verlag, Berlin, 1972, p. 313–523. Zbl0258.00005
  7. [7] H. W. Lenstra, Jr. & F. Oort – « Abelian varieties having purely additive reduction », J. Pure Appl. Algebra 36 (1985), no. 3, p. 281–298. Zbl0557.14022MR790619
  8. [8] J.-P. Serre – Lie algebras and Lie groups, second éd., Lecture Notes in Mathematics, vol. 1500, Springer-Verlag, Berlin, 1992, 1964 lectures given at Harvard University. Zbl0742.17008MR1176100
  9. [9] —, « Propriétés conjecturales des groupes de Galois motiviques et des représentations l -adiques », Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, p. 377–400. Zbl0812.14002
  10. [10] J.-P. Serre & J. Tate – « Good reduction of abelian varieties », Ann. of Math. (2) 88 (1968), p. 492–517. Zbl0172.46101MR236190
  11. [11] A. Silverberg & Y. G. Zarhin – « Variations on a theme of Minkowski and Serre », J. Pure Appl. Algebra 111 (1996), no. 1-3, p. 285–302. Zbl0885.14006MR1394358
  12. [12] —, « Semistable reduction of abelian varieties over extensions of small degree », J. Pure Appl. Algebra 132 (1998), no. 2, p. 179–193. Zbl0938.11029MR1640087
  13. [13] —, « Subgroups of inertia groups arising from abelian varieties », J. Algebra 209 (1998), no. 1, p. 94–107. Zbl0966.14029MR1652189
  14. [14] —, « Reduction of abelian varieties », The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), NATO Sci. Ser. C Math. Phys. Sci., vol. 548, Kluwer Acad. Publ., Dordrecht, 2000, p. 495–513. Zbl1029.14014MR1744959

NotesEmbed ?

top

You must be logged in to post comments.