Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-08T14:26:18.558Z Has data issue: false hasContentIssue false
  • English
  • Français

The semisimplicity conjecture for A-motives

Part of: Arithmetic algebraic geometry General commutative ring theory Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  18 March 2010

Nicolas Stalder
Nicolas Stalder*
Affiliation:
Department of Mathematics, ETH Zurich, Rämistrasse 101, 8092 Zürich, Switzerland (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove the semisimplicity conjecture for A-motives over finitely generated fields K. This conjecture states that the rational Tate modules V𝔭(M) of a semisimple A-motive M are semisimple as representations of the absolute Galois group of K. This theorem is in analogy with known results for abelian varieties and Drinfeld modules, and has been sketched previously by Tamagawa. We deduce two consequences of the theorem for the algebraic monodromy groups G𝔭(M) associated to an A-motive M by Tannakian duality. The first requires no semisimplicity condition on M and states that G𝔭(M) may be identified naturally with the Zariski closure of the image of the absolute Galois group of K in the automorphism group of V𝔭(M). The second states that the connected component of G𝔭(M) is reductive if M is semisimple and has a separable endomorphism algebra.

Keywords

A-motivet-motivesemisimpleGalois representationmonodromy groupsemilinear algebra

MSC classification

Secondary: 11G09: Drinfel'd modules; higher-dimensional motives, etc. 11F80: Galois representations 13A35: Characteristic $p$ methods (Frobenius endomorphism) and reduction to characteristic $p$; tight closure
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 3 , May 2010 , pp. 561 - 598
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Anderson, G. W., t-motives, Duke Math. J. 53 (1986), 457–502.CrossRefGoogle Scholar
[2]Bourbaki, N., Élements de mathématique, Algèbre, Chapitre 8: Modules et anneaux semi-semiples (Hermann, Paris, 1958).Google Scholar
[3]Faltings, G., Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), 349–366.CrossRefGoogle Scholar
[4]Goss, D., Basic structures in function field arithmetic (Springer, Berlin, 1996).CrossRefGoogle Scholar
[5]Jacobson, N., Basic algebra II, second edition (W. H. Freeman and Company, San Francisco, CA, 1990).Google Scholar
[6]Pink, R., The isogeny conjecture for A-motives, Preprint (2008).Google Scholar
[7]Pink, R. and Traulsen, M., The isogeny conjecture for t-motives associated to direct sums of Drinfeld modules, J. Number Theory 117 (2006), 355–375.CrossRefGoogle Scholar
[8]Stalder, N., Scalar extension of abelian and Tannakian categories, Preprint (2008), arXiv:0806.0308.Google Scholar
[9]Taelman, L., Artin t-motifs, J. Number Theory 129 (2009), 141–157.CrossRefGoogle Scholar
[10]Taguchi, Y., Semisimplicity of the Galois representations attached to Drinfeld modules over fields of ‘finite characteristics’, Duke Math. J. 62 (1991), 593–599.CrossRefGoogle Scholar
[11]Taguchi, Y., Semi-simplicity of the Galois representations attached to Drinfeld modules over fields of ‘infinite characteristics’, J. Number Theory 44 (1993), 292–314.CrossRefGoogle Scholar
[12]Taguchi, Y., The Tate conjecture for t-motives, Proc. Amer. Math. Soc. 123 (1995), 3285–3287.Google Scholar
[13]Taguchi, Y., On φ-modules, J. Number Theory 60 (1996), 124–141.CrossRefGoogle Scholar
[14]Tamagawa, A., Generalization of Anderson’s t-motives and Tate conjecture, RIMS Kōkyūroku 884 (1994), 154–159.Google Scholar
[15]Tamagawa, A., The Tate conjecture for A-premotives, Preprint (1994).Google Scholar
[16]Tamagawa, A., The Tate conjecture and the semisimplicity conjecture for t-modules, RIMS Kōkyūroku 925 (1995), 89–94.Google Scholar
[17]Tamagawa, A., Private communication (2004).Google Scholar
[18]Thakur, D., Function field arithmetic (World Scientific, Singapore, 2004).CrossRefGoogle Scholar
[19]Zarhin, Y. G., Abelian varieties in characteristic p, Math. Notes 19 (1976), 240–244.CrossRefGoogle Scholar