Advertisement

The Subregular Unipotent Contribution to the Geometric Side of the Arthur Trace Formula for the Split Exceptional Group G2

  • Tobias Finis
  • Werner Hoffmann
  • Satoshi Wakatsuki
Conference paper
Part of the Simons Symposia book series (SISY)

Abstract

In this paper, a zeta integral for the space of binary cubic forms is associated with the subregular unipotent contribution to the geometric side of the Arthur trace formula for the split exceptional group G2.

Keywords

Arthur trace formula Zeta integrals of prehomogeneous vector spaces Exceptional group G2 Binary cubic forms 

Notes

Acknowledgements

The author “Werner Hoffmann” was partially supported by the Collaborative Research Center 701 of the DFG. The author “Satoshi Wakatsuki” was partially supported by JSPS Grant-in-Aid for Scientific Research (No. 26800006, 25247001, 15K04795).

References

  1. [Ar]
    J. Arthur, A measure on the unipotent variety, Canad. J. Math. 37 (1985), 1237–1274.MathSciNetCrossRefGoogle Scholar
  2. [BS]
    F. van der Blij, T. A. Springer, The arithmetics of octaves and of the group G2, Nederl. Akad. Wetensch. Proc. Ser. A 62 = Indag. Math. 21 (1959), 406–418.CrossRefGoogle Scholar
  3. [Ch1]
    P.-H. Chaudouard, Sur la contribution unipotente dans la formule des traces d’Arthur pour les groupes généraux linéaires, Israel J. Math. 218 (2017), 175–271.MathSciNetCrossRefGoogle Scholar
  4. [Ch2]
    P.-H. Chaudouard, Sur certaines contributions unipotentes dans la formule des traces d’Arthur, arXiv:1510.02783, to appear in Amer. J. Math.Google Scholar
  5. [CL]
    P.-H. Chaudouard, G. Laumon, Sur le comptage des fibrés de Hitchin nilpotents, J. Inst. Math. Jussieu 15 (2016), 91–164.MathSciNetCrossRefGoogle Scholar
  6. [CNP]
    A. M. Cohen, G. Nebe, W. Plesken, Maximal integral forms of the algebraic group G2 defined by finite subgroups, J. Number Theory 72 (1998), 282–308.MathSciNetCrossRefGoogle Scholar
  7. [CM]
    D. Collingwood, W. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993.zbMATHGoogle Scholar
  8. [DK]
    S. DeBacker, D. Kazhdan, Stable distributions supported on the nilpotent cone for the group G 2, The unity of mathematics, 205–262, Progr. Math., 244, Birkhäuser Boston, Boston, MA, 2006.Google Scholar
  9. [DW]
    B. Datskovsky, D. Wright, The adelic zeta function associated with the space of binary cubic forms II: Local theory, J. Reine Angew. Math. 367, 27–75.Google Scholar
  10. [FL]
    T. Finis, E. Lapid, On the continuity of the geometric side of the trace formula, Acta Math. Vietnam 41 (2016), 425–455.MathSciNetCrossRefGoogle Scholar
  11. [GGS]
    W.-T. Gan, B. Gross, G. Savin, Fourier coefficients of modular forms on G 2, Duke Math. J. 115 (2002), 105–169.MathSciNetCrossRefGoogle Scholar
  12. [GG]
    W.-T. Gan, N. Gurevich, Non-tempered Arthur packets of G 2, Automorphic representations, L-functions and applications: progress and prospects, 129–155, Ohio State Univ. Math. Res. Inst. Publ., 11, de Gruyter, Berlin, 2005.Google Scholar
  13. [Ho1]
    W. Hoffmann, The nonsemisimple term in the trace formula for rank one lattices, J. Reine Angew. Math. 379 (1987), 1–21.MathSciNetzbMATHGoogle Scholar
  14. [Ho2]
    W. Hoffmann, The trace formula and prehomogeneous vector spaces, Müller, Werner (ed.) et al., Families of automorphic forms and the trace formula. Proceedings of the Simons symposium, Puerto Rico, January 26–February 1, 2014. Simons Symposia, 175–215 (2016).Google Scholar
  15. [HW]
    W. Hoffmann, S. Wakatsuki, On the geometric side of the Arthur trace formula for the symplectic group of rank 2, arXiv:1310.0541, to appear in Mem. Amer. Math. Soc.Google Scholar
  16. [KWY]
    H. Kim, S. Wakatsuki, T. Yamauchi, An equidistribution theorem for holomorphic Siegel modular forms for GSp4, arXiv:1604.02036, to appear in J. Inst. Math. Jussieu.Google Scholar
  17. [Ko]
    T. Kogiso, Simple calculation of the residues of the adelic zeta function associated with the space of binary cubic forms, J. Number Theory 51 (1995), 233–248.MathSciNetCrossRefGoogle Scholar
  18. [Ma1]
    J. Matz, Arthur’s trace formula for GL(2) and GL(3) and non-compactly supported test functions, Dissertation, Universität Düsseldorf.Google Scholar
  19. [Ma2]
    J. Matz, Bounds for global coefficients in the fine geometric expansion of Arthur’s trace formula for GL(n), Israel J. Math. 205 (2015), 337–396.MathSciNetCrossRefGoogle Scholar
  20. [Ma3]
    J. Matz, Weyl’s law for Hecke operators on GL(n) over imaginary quadratic number fields, Amer. J. Math. 139 (2017), 57–145.MathSciNetCrossRefGoogle Scholar
  21. [MT]
    J. Matz, N. Templier, Sato-Tate equidistribution for families of Hecke-Maass forms on \({\mathrm {SL}}(n,\mathbb {R})/{\mathrm {SO}}(n)\), arXiv:1505.07285, 2015.Google Scholar
  22. [Sa1]
    H. Saito, Explicit form of the zeta functions of prehomogeneous vector spaces, Math. Ann. 315 (1999), 587–615.MathSciNetCrossRefGoogle Scholar
  23. [Sa2]
    H. Saito, Convergence of the zeta functions of prehomogeneous vector spaces, Nagoya Math. J. 170 (2003), 1–31.MathSciNetCrossRefGoogle Scholar
  24. [SS]
    M. Sato, T. Shintani, On zeta functions associated with prehomogeneous vector spaces, Ann. of Math. (2) 100 (1974), 131–170.MathSciNetCrossRefGoogle Scholar
  25. [Sh]
    T. Shintani, On Dirichlet series whose coefficients are class-numbers of integral binary cubic forms, J. Math. Soc. Japan 24 (1972), 132–188MathSciNetCrossRefGoogle Scholar
  26. [SV]
    T. A. Springer, F. D. Veldkamp, Octonions, Jordan algebras and exceptional groups, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2000.CrossRefGoogle Scholar
  27. [St]
    R. Steinberg, Lectures on Chevalley groups, Notes prepared by John Faulkner and Robert Wilson, Yale University, New Haven, Conn., 1968.zbMATHGoogle Scholar
  28. [Ta]
    T. Taniguchi, Distributions of discriminants of cubic algebras, math.NT/0606109, 2006.Google Scholar
  29. [Wr]
    D. Wright, The adelic zeta function associated to the space of binary cubic forms part I: Global theory, Math. Ann. 270 (1985), 503–534.MathSciNetCrossRefGoogle Scholar
  30. [Yu]
    A. Yukie, Shintani zeta functions, London Mathematical Society Lecture Note Series, 183, Cambridge University Press, Cambridge, 1993.Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Tobias Finis
    • 1
  • Werner Hoffmann
    • 2
  • Satoshi Wakatsuki
    • 3
  1. 1.Mathematisches InstitutUniversität LeipzigLeipzigGermany
  2. 2.Fakultät für MathematikUniversität BielefeldBielefeldGermany
  3. 3.Faculty of Mathematics and PhysicsInstitute of Science and Engineering, Kanazawa UniversityKanazawaJapan

Personalised recommendations