Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-09T01:35:57.192Z Has data issue: false hasContentIssue false
  • English
  • Français

Beyond endoscopy via the trace formula: 1. Poisson summation and isolation of special representations

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  01 June 2015

Salim Ali Altuğ
Salim Ali Altuğ*
Affiliation:
Mathematics Department, Columbia University, 2990 Broadway, New York, NY 10027, USA 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.

With analytic applications in mind, in particular beyond endoscopy, we initiate the study of the elliptic part of the trace formula. Incorporating the approximate functional equation into the elliptic part, we control the analytic behavior of the volumes of tori that appear in the elliptic part. Furthermore, by carefully choosing the truncation parameter in the approximate functional equation, we smooth out the singularities of orbital integrals. Finally, by an application of Poisson summation we rewrite the elliptic part so that it is ready to be used in analytic applications, and in particular in beyond endoscopy. As a by product we also isolate the contributions of special representations as pointed out in [Beyond endoscopy, in Contributions to automorphic forms, geometry and number theory (Johns Hopkins University Press, Baltimore, MD, 2004), 611–697].

Keywords

beyond endoscopytrace formula

MSC classification

Primary: 11F72: Spectral theory; Selberg trace formula
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 10 , October 2015 , pp. 1791 - 1820
Copyright
© The Author 2015 

References

Arthur, J., The endoscopic classification of representations, available athttp://www.claymath.org/cw/arthur/pdf/arthur-endoscopic-tifr.pdf.Google Scholar
Arthur, J., An introduction to the trace formula, in Harmonic analysis, the trace formula and Shimura varieties, Clay Mathematics Institute Proceedings (American Mathematical Society, Providence, RI, 2005), 1–265.Google Scholar
Borel, A. and Jacquet, H., Automorphic forms and automorphic representations, in Automorphic forms, representations and L-functions, Proceedings of Symposia in Pure Mathematics (part I), vol. 33 (American Mathematical Society, Providence, RI, 1979), 189–202.CrossRefGoogle Scholar
Bykovskii, V. A., Density theorems and the mean value of arithmetic functions on short intervals (Russian), Zap. Nauchn. Sem. St.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 212 (1994), Anal. Teor. Chisel i Teor. Funktsii. 12, 5670, 196; Engl. transl. J. Math. Sci. (N.Y.) 83 (1997), 720730.Google Scholar
Frenkel, E., Langlands, R. P. and Ngô, B. C., La formule des traces et la functorialité. Le debut d’un programme, Ann. Sci. Math. Québec 34 (2010), 199–243.Google Scholar
Herman, P. E., Quadratic base change and the analytic continuation of the Asai $L$-function: a new trace formula approach, Preprint (2010), arXiv:1008.3921v2.Google Scholar
Iwaniec, H. and Kowalski, E., Analytic number theory, AMS Colloquium Publications, vol. 53 (American Mathematical Society, Providence, RI, 2004).Google Scholar
Jacquet, H. and Langlands, R. P., Automorphic forms on GL(2), Lecture Notes in Mathematics, vol. 114 (Springer, Berlin, 1970).CrossRefGoogle Scholar
Knapp, A., Representation theory of semisimple groups: an overview based on examples (Princeton University Press, Princeton, NJ, 1986).CrossRefGoogle Scholar
Labesse, J.-P., Introduction to endoscopy, notes for Snowbird lectures, in Representation theory of real reductive groups, Contemporary Mathematics, vol. 472 (American Mathematical Society, Providence, RI, 2008), 175.Google Scholar
Langlands, R. P., Beyond endoscopy, in Contributions to automorphic forms, geometry and number theory (Johns Hopkins University Press, Baltimore, MD, 2004), 611–697.Google Scholar
Sarnak, P., Comments on Langlands’ Lecture ‘Endoscopy and Beyond’, available athttp://publications.ias.edu/sites/default/files/SarnakLectureNotes-1.pdf.Google Scholar
Sarnak, P., Notes on the Generalized Ramanujan Conjectures, in Harmonic analysis, the trace formula and Shimura varieties, Clay Mathematics Institute Proceedings, vol. 4 (American Mathematical Society, Providence, RI, 2005), 659–687.Google Scholar
Shakarchi, R. and Stein, E., Complex analysis, Princeton Lectures in Analysis, vol. II (Princeton University Press, Princeton, NJ, 2003).Google Scholar
Shealsted, D., Orbital integrals for GL2(ℝ), in Automorphic forms, representations and L-functions, Proceedings of Symposia in Pure Mathematics (part I), vol. 33 (American Mathematical Society, Providence, RI, 1979), 189–202.Google Scholar
Soundararajan, K. and Young, M., The prime geodesic theorem, J. Reine Angew. Math. 676 (2013), 105–120.Google Scholar
Venkatesh, A., Beyond endoscopy and special forms on GL(2), J. Reine Angew. Math. 577 (2004), 23–80.Google Scholar
Zagier, D., Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields, in Modular functions of one variable, VI (Proceedings of Second International Conference, University of Bonn, Bonn, 1976), Lecture Notes in Mathematics, vol. 627 (Springer, Berlin, 1977), 105–169.CrossRefGoogle Scholar