Hostname: page-component-745bb68f8f-s22k5 Total loading time: 0 Render date: 2025-01-10T19:47:16.753Z Has data issue: false hasContentIssue false
  • English
  • Français

Homomorphisms of abelian varieties over geometric fields of finite characteristic

Published online by Cambridge University Press:  16 May 2012

Yuri G. Zarhin
Yuri G. Zarhin*
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We study analogues of Tate’s conjecture on homomorphisms for abelian varieties when the ground field is finitely generated over an algebraic closure of a finite field. Our results cover the case of abelian varieties without non-trivial endomorphisms.

Keywords

Abelian varietiesTate modulescyclotomic extensions
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 12 , Issue 2 , April 2013 , pp. 225 - 236
Copyright
©Cambridge University Press 2012

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.)

References

Arias-de-Reyna, S., Gajda, W. and Petersen, S., Abelian varieties over finitely generated fields and the conjecture of Geyer and Jarden on torsion (arXiv:1010.2444 [math.AG]).Google Scholar
Curtis, Ch. W. and Reiner, I., Representation theory of finite groups and associative algebras (Interscience Publishers, New York, London, 1962).Google Scholar
Deligne, P., Théorie de Hodge. II, Publ. Math. IHES 40 (1971), 5–58.Google Scholar
Faltings, G., Endlichkeitssätze für abelsche Varietäten über Zählkorpern, Invent. Math. 73 (1983), 349–366 (erratum: Invent. Math. 75 (1984), 381).Google Scholar
Faltings, G., Complements to Mordell, in Rational points (ed. Faltings, G. and Wüstholz, G. et al. ), Aspects of Mathematics, Volume E6, Chapter VI (Friedr. Vieweg & Sohn, Braunschweig, 1984).Google Scholar
Hindry, M. and Ratazzi, N., Points de torsion sur les variétés abéliennes de type GSp. J. Institut Math. Jussieu (available on CJO 05 Aug 2010 ) (arXiv:0911.5505 [math.NT]).Google Scholar
Moret-Bailly, L., Pinceaux de variétés abéliennes, Astérisque 129 (1985).Google Scholar
Mumford, D., Abelian varieties, 2nd edn (Oxford University Press, London, 1974).Google Scholar
Schappacher, N., Tate’s conjecture on the homomorphisms of abelian varieties, in Rational points (ed. Faltings, G. and Wüstholz, G. et al. ). Aspects of Mathematics, Volume E6, Chapter IV (Friedr. Vieweg & Sohn, Braunschweig, 1984).Google Scholar
Serre, J.-P., Sur les groupes des congruence des variétés abéliennes, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1) (1964), 3–18 Oeuvres II, pp. 230–245 (Springer-Verlag, Berlin, 1986).Google Scholar
Serre, J.-P., Abelian ℓ-adic representations and elliptic curves, 2nd edn (Addison-Wesley, New York, 1989).Google Scholar
Skorobogatov, A. N. and Zarhin, Y. G., A finiteness theorem for Brauer groups of abelian varieties and K3 surfaces, J. Algebraic Geom. 17 (3) (2008), 481–502.Google Scholar
Tate, J., Endomorphisms of Abelian varieties over finite fields, Invent. Math. 2 (1966), 134–144.Google Scholar
Zarhin, Y. G., Endomorphisms of Abelian varieties over fields of finite characteristic, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975) 272–277; Math. USSR Izv. 9 (1975) 255–260.Google Scholar
Zarhin, Y. G., Abelian varieties in characteristic $P$, Mat. Zametki 19 (1976) 393–400; Math. Notes 19 (1976) 240–244.Google Scholar
Zarhin, Y. G., Endomorphisms of Abelian varieties and points of finite order in characteristic $P$, Mat. Zametki 21 (1977) 737–744; Math. Notes 21 (1978) 415–419.Google Scholar
Zarhin, Y. G., Torsion of abelian varieties in finite characteristic, Mat. Zametki 22 (1977), 1–11; Math. Notes 22 (1978), 493–498.Google Scholar
Zarhin, Y. G., A finiteness theorem for unpolarized Abelian varieties over number fields with prescribed places of bad reduction, Invent. Math. 79 (1985), 309–321.Google Scholar
Zarhin, Y. G. and Parshin, A. N., Finiteness problems in diophantine geometry, Amer. Math. Soc. Transl. 143 (2) (1989), 35–102 (arXiv:0912.4325 [math.NT]).Google Scholar
Zarhin, Y. G., Hyperelliptic Jacobians without complex multiplication, Math. Res. Lett. 7 (2000), 123–132.Google Scholar
Zarhin, Y. G., Hyperelliptic Jacobians without complex multiplication in positive characteristic, Math. Res. Lett. 8 (2001), 429–435.Google Scholar
Zarhin, Y. G., Non-supersingular hyperelliptic Jacobians, Bull. Soc. Math. France 132 (2004), 617–634.Google Scholar
Zarhin, Y. G., Homomorphisms of abelian varieties over finite fields, in Higher-dimensional geometry over finite fields (ed. Kaledin, D. and Tschinkel, Yu. ), pp. 315–343 (IOS Press, Amsterdam, 2008).Google Scholar
Zarhin, Y. G., Endomorphisms of abelian varieties, cyclotomic extensions and Lie algebras. Math. Sb. 201(12) (2010), 93–102; Sb. Math. 201(12) (2010), 1801–1810.Google Scholar