Skip to main content
SpringerLink
Log in

Noncommutative L-Functions for Varieties over Finite Fields

  • Conference paper
  • First Online:
Iwasawa Theory 2012

Part of the book series: Contributions in Mathematical and Computational Sciences ((CMCS,volume 7))

  • 1257 Accesses

Abstract

We consider certain perfect complexes of adic sheaves on varieties over finite fields that take values in modules over noncommutative rings Λ. To each such complex we associate an L-function living in the first \(\mathop{\text{K}}\nolimits\)-group of the power series ring over Λ. We then show that these L-functions satisfy a suitably generalised multiplicative Grothendieck trace formula.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Artin, M., Grothendieck, A., Verdier, J.L.: Théorie des topos et cohomologie étale des schémas (SGA 4–3). Lecture Notes in Mathematics, vol. 305. Springer, Berlin (1972)

    Google Scholar 

  • Avramov, L.L., Foxby, H.-B.: Homological dimensions of unbounded complexes. J. Pure Appl. Algebra 71, 129–155 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  • Chinburg, T., Pappas, G., Taylor, M.J.: K 1 of a p-adic group ring II. The determinantal kernel SK 1 (2013, preprint). arXiv:1303.5337v1

    Google Scholar 

  • Chinburg, T., Pappas, G., Taylor, M.J.: Higher adeles and non-abelian Riemann-Roch (2012, preprint). arXiv:1204.4520v2

    Google Scholar 

  • Deligne, P.: Cohomologie étale (SGA 4\(\frac{1} {2}\)). Lecture Notes in Mathematics, vol. 569. Springer, Berlin (1977)

    Google Scholar 

  • Emerton, M., Kisin, M.: Unit L-functions and a conjecture of Katz. Ann. Math. 153(2), 329–354 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Fukaya,T., Kato, K.: A formulation of conjectures on p-adic zeta functions in non-commutative Iwasawa theory. In: Proceedings of the St. Petersburg Mathematical Society, vol. XII, American Mathematical Society, Series 2, vol. 219, pp. 1–85. American mathematical society, Providence (2006)

    Google Scholar 

  • Grothendieck, A.: Cohomologie ℓ-adique et fonctions L (SGA 5). Lecture Notes in Mathematics, vol. 589. Springer, Berlin (1977)

    Google Scholar 

  • Lam, T.Y.: A First Course in Noncommutative Rings. Graduate Texts in Mathematics, vol. 131. Springer, Berlin (1991)

    Google Scholar 

  • Milne, J.S.: Etale Cohomology. Princeton Mathematical Series, vol. 33. Princeton University Press, Princeton (1980)

    Google Scholar 

  • Muro, F., Tonks, A.: The 1-type of a Waldhausen K-theory spectrum. Adv. Math. 216(1), 178–211 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Muro, F., Tonks, A.: On K 1 of a Waldhausen category. K-theory and noncommutative geometry. EMS Series of Congress Reports, pp. 91–116 (2008)

    Google Scholar 

  • Nagata, M.: A generalization of the imbedding problem of an abstract variety in a complete variety. J. Math. Kyoto Univ. 3, 89–102 (1963)

    MathSciNet  MATH  Google Scholar 

  • Neukirch, J., Schmidt, A., Wingberg, K.: Cohomology of Number Fields. Grundlehren der mathematischen Wissenschaften, vol. 323. Springer, Berlin/Heidelberg (2000)

    Google Scholar 

  • Oliver, R.: Whitehead Groups of Finite Groups. London Mathematical Society lecture notes series, vol. 132. Cambridge University Press, Cambridge (1988)

    Google Scholar 

  • Schlichting, M.: A note on K-theory and triangulated categories. Invent. Math. 150(1), 111–116 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Schneider, P.: Über gewisse Galoiscohomologiegruppen. Math. Z. 260, 181–205 (1979)

    Article  MATH  Google Scholar 

  • Thomason, R.W., Trobaugh, T.: Higher algebraic K-theory of schemes and derived categories. The Grothendieck Festschrift, vol. III, Progress in Mathematics, vol. 88, pp. 247–435. Birkhäuser, Boston (1990)

    Google Scholar 

  • Waldhausen, F.: Algebraic K-theory of spaces. In: Algebraic and Geometric Topology (Berlin Heidelberg). Lecture Notes in Mathematics, vol. 1126, pp. 318–419. Springer, Berlin (1985)

    Google Scholar 

  • Witte, M.: Noncommutative Iwasawa main conjectures for varieties over finite fields. Ph.D. thesis, Universität Leipzig (2008). Available at http://www.dart-europe.eu/full.php?id=162794

  • Witte, M.: Unit L-functions for étale sheaves of modules over noncommutative rings (2013, preprint). arXiv:1309.2493v1

    Google Scholar 

Download references

Acknowledgements

The author would like to thank Annette Huber and Alexander Schmidt for their encouragement and for valuable discussions. He also thanks the anonymous referee for various corrections.

Author information

Authors and Affiliations

  1. Mathematisches Institut, Universität Heidelberg, Im Neuenheimer Feld 288, Heidelberg, Germany

    Malte Witte

Authors
  1. Malte Witte
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Malte Witte .

Editor information

Editors and Affiliations

  1. Dept. of Mathematical Sciences, Durham University, Durham, United Kingdom

    Thanasis Bouganis

  2. Institute of Mathematics, University of Heidelberg, Heidelberg, Germany

    Otmar Venjakob

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Witte, M. (2014). Noncommutative L-Functions for Varieties over Finite Fields. In: Bouganis, T., Venjakob, O. (eds) Iwasawa Theory 2012. Contributions in Mathematical and Computational Sciences, vol 7. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55245-8_15

Download citation

Publish with us

Policies and ethics

Search

Navigation