Skip to main content
SpringerLink
Log in

Explicit forms of the trace formula for GL(2)

  • Published:
Journal d'Analyse Mathématique Aims and scope

Abstract

There has recently been renewed interest in the trace formula–in particular, that of the initial case of GL(2)–due to counting applications in the function field case. For these applications, one needs a very precise form of the trace formula, with all terms computed explicitly. Our aim in this work is to compute the trace formula for GL(2) over a number field in as full detail as was done for the function field case and to give an accessible exposition, being motivated by these applications to counting, but also by pure curiosity as to the optimal form of this plastic formula. We also explain a correction argument in our context here of GL(2). The idea is to introduce a global summand which does not change the formula globally but changes the local weighted orbital integrals at the hyperbolic terms, so that their limit at the identity becomes a unipotent contribution to the trace formula. This gives a harmonious and pleasing form to the formula. Finally, we put the trace formula in an invariant form; thus all its terms are distributions whose value at a test function f y(x) = f(y −1 xy) is independent of y in GL(2,A).

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. Arthur, A trace formula for reductive groups I: Terms associated to classes in G(Q), Duke Math. J. 45 (1978), 911–952.

    Article  MathSciNet  Google Scholar 

  2. J. Arthur, A trace formula for reductive groups II: Applications of a truncation operator, Compositio Math. 40 (1980), 87–121.

    MathSciNet  MATH  Google Scholar 

  3. J. Arthur, The trace formula in invariant form, Ann. of Math. (2) 114 (1981), 1–74.

    Article  MathSciNet  MATH  Google Scholar 

  4. J. Arthur, The local behaviour of weighted orbital integrals, Duke Math. J. 56 (1988), 223–293.

    Article  MathSciNet  MATH  Google Scholar 

  5. J. Arthur, The invariant trace formula I. Local theory, J. Amer.Math. Soc. 1 (1988), 323–383.

    Article  MathSciNet  MATH  Google Scholar 

  6. J. Arthur, The invariant trace formula II. Global theory, J. Amer. Math. Soc. 1 (1988), 501–554.

    Article  MathSciNet  MATH  Google Scholar 

  7. J. Arthur, An introduction to the trace formula, in Harmonic Analysis, the Trace Formula, and Shimura Varieties, Amer. Math. Soc., Providence, RI, 2005, pp. 1–263.

    Google Scholar 

  8. A. Borel and H. Jacquet, Automorphic forms and automorphic representations, in Automorphic Forms, Representations and L-Functions, Part 1, Amer. Math. Soc., Providence, RI, 1979, pp. 189–202.

    Chapter  Google Scholar 

  9. J. N. Bernstein and A. V. Zelevinsky, Representations of the group GL(n, F), where F is a local non-Archimedean field, Uspehi Mat. Nauk 10 (1976), 5–70.

    MathSciNet  Google Scholar 

  10. P. Deligne, The Abel prize and curiosity driven research, https://www.ias.edu/ideas/2013/deligne-abel-curiosity.

  11. P. Deligne and Y. Z. Flicker, Counting local systems with principal unipotent local monodromy, Ann. of Math. (2) 178 (2013), 921–982.

    Article  MathSciNet  MATH  Google Scholar 

  12. J. Dixmier, Von Neumann Algebras, North-Holland Publishing Co., Amsterdam-New York, 1981.

    MATH  Google Scholar 

  13. V. G. Drinfeld, The number of two-dimensional irreducible representations of the fundamental group of a curve over a finite field, Funktsional. Anal. i Prilozhen. 15 (1981), 75–76.

    MathSciNet  Google Scholar 

  14. Y. Z. Flicker, Automorphic forms on covering groups of GL(2), Invent. Math. 57 (1980), 119–182.

    Article  MathSciNet  MATH  Google Scholar 

  15. Y. Z. Flicker, The Trace Formula and Base Change for GL(3), Springer-Verlag, Berlin-New York, 1982.

    Book  MATH  Google Scholar 

  16. Y. Z. Flicker, Regular trace formula and base change lifting, Amer. J. Math. 110 (1988), 739–764.

    Article  MathSciNet  MATH  Google Scholar 

  17. Y. Z. Flicker, Transfer of orbital integrals and division algebras, J. Ramanujan Math. Soc. 5 (1990), 107–121.

    MathSciNet  MATH  Google Scholar 

  18. Y. Z. Flicker, Automorphic Representations of Low Rank Groups, World Scientific, New-Jersey, London, Singapore, 2006.

    Book  MATH  Google Scholar 

  19. Y. Z. Flicker, Eisenstein series and the trace formula for GL(2) over a function field, Doc. Math. 19 (2014), 1–62.

    MathSciNet  MATH  Google Scholar 

  20. Y. Z. Flicker, Counting rank two local systems with at most one, unipotent, monodromy, Amer. J. Math. 137 (2015), 739–763.

    Article  MathSciNet  MATH  Google Scholar 

  21. Y. Z. Flicker, Arthur’s Invariant Trace Formula and Comparison of Inner Forms, Birkhäuser, Springer-Verlag, 2016.

    Book  MATH  Google Scholar 

  22. Y. Z. Flicker, Counting local systems via the trace formula, J. reine angew. Math., to appear.

  23. Y. Z. Flicker and D. Kazhdan, A simple trace formula, J. Analyse Math. 50 (1988), 189–200.

    Article  MathSciNet  MATH  Google Scholar 

  24. S. Gelbart, Lectures on the Arthur-Selberg Trace Formula, Amer. Math. Soc., Providence, RI, 1996.

    Book  MATH  Google Scholar 

  25. S. Gelbart and H. Jacquet, Forms of GL(2) from the analytic point of view, in Automorphic Forms, Representations and L-Functions, Part 1, Amer. Math. Soc., Providence, RI, 1979, pp. 213–251.

    Chapter  Google Scholar 

  26. R. Godement, The spectral decomposition of cusp forms, in Algebraic Groups and Discontiuous Subgroups, Amer. Math. Soc., Providence, RI, 1966, pp. 225–234.

    Chapter  Google Scholar 

  27. H. Jacquet and P. R. Langlands, Automorphic Forms on GL(2), Springer-Verlag, Berlin-New York, 1970.

    Book  MATH  Google Scholar 

  28. H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic representations I, Amer. J. Math. 103 (1981), 499–558.

    Article  MathSciNet  MATH  Google Scholar 

  29. H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic forms II, Amer. J. Math. 103 (1981), 777–815.

    Article  MathSciNet  MATH  Google Scholar 

  30. R. E. Kottwitz, Tamagawa numbers, Ann. of Math. (2) 127 (1988), 629–646.

    Article  MathSciNet  MATH  Google Scholar 

  31. A. W. Knapp, Theoretical aspects of the trace formula for GL(2), in Representation Theory and Automorphic Forms, Amer. Math. Soc., Providence, RI, 1997, pp. 355–405.

    Chapter  Google Scholar 

  32. S. Lang, Algebraic Number Theory, second edition, Springer-Verlag, New York, 1994.

    Book  MATH  Google Scholar 

  33. R. P. Langlands, On the Functional Equations Satisfied by Eisenstein series, Springer-Verlag, Berlin-New York, 1976.

    Book  MATH  Google Scholar 

  34. C. Moeglin and J.-L. Waldspurger, Spectral Decomposition and Eisenstein series, Cambridge University Press, Cambridge, 1995.

    Book  MATH  Google Scholar 

  35. V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Academic Press, Inc., Boston, MA, 1994.

    MATH  Google Scholar 

  36. E. C. Titchmarsh, The Theory of Functions, Oxford Science Publications, 1976.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Science Faculty, Atatürk University, Erzurum, Turkey

    A. Aydoğdu & R. Öztürk

  2. Ariel University, Ariel, 40700, Israel

    Y. Z. Flicker

  3. The Ohio State University, Columbus, OH, 43210, USA

    Y. Z. Flicker

  4. Department of Mathematics, Science and Art Faculty, Erzincan University, Erzincan, Turkey

    E. Özkan

Authors
  1. A. Aydoğdu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Y. Z. Flicker
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. E. Özkan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. R. Öztürk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Y. Z. Flicker.

Additional information

Based on lectures at Erzincan University, Department of Mathematics, Summer 2013.

Work on this paper was done in Turkey. Support of TUBITAK is gratefully acknowledged.

The second author was partially supported by a grant from the Simons Foundation (#267097 to Yuval Flicker). During the preparation of this work the second author was a Schonbrunn Professor at the Hebrew University. Support of ERC AdG Grant 247049 is gratefully acknowledged.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aydoğdu, A., Flicker, Y.Z., Özkan, E. et al. Explicit forms of the trace formula for GL(2). JAMA 131, 1–71 (2017). https://doi.org/10.1007/s11854-017-0001-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11854-017-0001-z

Search

Navigation