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6 - Motives

Published online by Cambridge University Press:  28 April 2020

Bruno Kahn
Affiliation:
Institut de Mathématiques de Jussieu-Paris Rive Gauche
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Summary

This last chapter is dedicated to motives and their zeta functions. We consider only an elementary case: that of pure motives of Grothendieck associated to smooth projective varieties over a finite field. One can go much further using the triangulated categories of motives introduced by Voevodsky and developed by Ivorra, Ayoub, and Cisinski–D’eglise, but this would go beyond the scope of the book. Nonetheless, it is explained how this viewpoint considerably clarifies how Weil cohomologies are used to prove rationality and the functional equation, and this theory is applied to prove a somewhat forgotten theorem of Weil: Artin’s conjecture for non-abelian L-functions in positive characteristic.

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Publisher: Cambridge University Press
Print publication year: 2020

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  • Motives
  • Bruno Kahn
  • Book: Zeta and L-Functions of Varieties and Motives
  • Online publication: 28 April 2020
  • Chapter DOI: https://doi.org/10.1017/9781108691536.007
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  • Motives
  • Bruno Kahn
  • Book: Zeta and L-Functions of Varieties and Motives
  • Online publication: 28 April 2020
  • Chapter DOI: https://doi.org/10.1017/9781108691536.007
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Motives
  • Bruno Kahn
  • Book: Zeta and L-Functions of Varieties and Motives
  • Online publication: 28 April 2020
  • Chapter DOI: https://doi.org/10.1017/9781108691536.007
Available formats
×