Hostname: page-component-857557d7f7-9f75d Total loading time: 0 Render date: 2025-11-20T17:36:10.703Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Selmer groups of abelian varieties over function fields of characteristic p > 0

Published online by Cambridge University Press:  01 January 2009

TADASHI OCHIAI  and
FABIEN TRIHAN
TADASHI OCHIAI
Affiliation:
Department of Mathematics, Osaka University, 1-1, Machikaneyama Toyonaka, Osaka 560-0043Japan e-mail: ochiai@math.sci.osaka-u.ac.jp
FABIEN TRIHAN
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, United Kingdom. e-mail: fabien.trihan@nottingham.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We study a (p-adic) geometric analogue for abelian varieties over a function field of characteristic p of the cyclotomic Iwasawa theory and the non-commutative Iwasawa theory for abelian varieties over a number field initiated by Mazur and Coates respectively. We will prove some analogue of the principal results obtained in the case over a number field and we study new phenomena which did not happen in the case of number field case. We also propose a conjecture (Conjecture 1.6) which might be considered as a counterpart of the principal conjecture in the case over a number field.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 146 , Issue 1 , January 2009 , pp. 23 - 43
Copyright
Copyright © Cambridge Philosophical Society 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

[BBM]Berthelot, P. P., Breen, L. and Messing, W.Théorie de Dieudonné cristalline. II, Lecture Notes in Mathematics, 930. (Springer-Verlag, 1982).CrossRefGoogle Scholar
[B–H]Balister, P. and Howson, S.Note on Nakayama's lemma for compact-modules. Asian J. Math. Vol. 1, no. 2 (1997).CrossRefGoogle Scholar
[CFKSV]Coates, J., Fukaya, T., Kato, K., Sujatha, R. and Venjakob, O.The GL 2 main conjecture for elliptic curves without complex multiplication. Inst. Hautes Etud. Sci. Publ. Math. 101 (2005), 163–208.CrossRefGoogle Scholar
[EL]Etesse, J. and Le Stum, B.Fonctions L associées aux F-isocristaux surconvergents II: Zéros et pôles unités. Invent. Math. 127, No.1 (1997), 1–31.CrossRefGoogle Scholar
[K]Kato, K.p-adic Hodge theory and values of zeta functions of modular forms. Astérisque 295 (2004), 117–290.Google Scholar
[K–T]Kato, K. and Trihan, F.On the conjecture of Birch and Swinnerton-Dyer in characteristic p > 0. Invent. Math. 153 (2003), 537–592.CrossRefGoogle Scholar
[L]Lang, S.Algebraic groups over finite fields. Ameri. J. Math. 78 (1956), 555–563.CrossRefGoogle Scholar
[Mn]Manin, Y.Cyclotomic fields and modular curves (Russian). Uspehi Mat. Nauk 26, no. 6(162) (1971), 7–71.Google Scholar
[Mz]Mazur, B.Rational points of abelian varieties with values in towers of number fields. Invent. Math. 18 (1972), 183–266.CrossRefGoogle Scholar
[MW]Mazur, B. and Wiles, A.Class fields of abelian extensions of Q. Invent. Math. 76, no. 2 (1984), 179–330.CrossRefGoogle Scholar
[Mi1]Milne, J. S.Etale Cohomology. Princeton Mathematical Series, 33 (Princeton University Press) 1980.CrossRefGoogle Scholar
[Mi2]Milne, J. S.Arithmetic Duality Theorems. Perspectives in Mathematics, 1 (Academic Press, 1986).Google Scholar
[Mu]Mumford, D.Abelian Varieties. Tata Institute of Fundamental Research Studies in Mathematics, No. 5 (1970).Google Scholar
[R]Rubin, K.The “main conjectures” of Iwasawa theory for imaginary quadratic fields. Invent. Math. 103 (1991), 25–68.CrossRefGoogle Scholar
[S]Schneider, P.p-adic height pairing. II. Invent. Math. 79 (1985), 329–374.CrossRefGoogle Scholar
[T]Tsuji, T.Poincaré duality for logarithmic crystalline cohomology. Compositio Math. 118, no. 1 (1999), 11–41.CrossRefGoogle Scholar
[We]Weil, A.Basic Number Theory (Springer-Verlag, 1973).CrossRefGoogle Scholar
[W]Wiles, A.The Iwasawa conjecture for totally real fields. Ann. of Math. (2) 131, no. 3 (1990), 493–540.CrossRefGoogle Scholar