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Simons Symposium on the Trace Formula

SSTF 2016: Geometric Aspects of the Trace Formula pp 1–21Cite as

Functoriality and the Trace Formula

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Abstract

We shall summarize two different lectures that were presented on Beyond Endoscopy, the proposal of Langlands to apply the trace formula to the principle of functoriality. We also include an elementary description of functoriality, and in the last section, some general reflections on where the study of Beyond Endoscopy might be leading.

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References

  1. Salim Ali Altuğ, Beyond Endoscopy via the Trace Formula, ProQuest LLC, Ann Arbor, MI, 2013, Thesis (Ph.D.)–Princeton University.

    Google Scholar 

  2. _________ , Beyond Endoscopy via the trace formula – I: Poisson summation and isolation of special representations, Compos. Math. 151 (2015), no. 10, 1791–1820.

    Google Scholar 

  3. _________ , Beyond Endoscopy via the trace formula – II: Asymptotic expansions of Fourier transforms and bounds towards the Ramanujan conjecture, 2015, Amer. J. Math. 139 (2017), 863–913.

    Google Scholar 

  4. _________ , Beyond Endoscopy via the trace formula – III: The standard representation, To appear in Journal of the Institute of Mathematics at Jussieu.

    Google Scholar 

  5. James Arthur, Problems beyond endoscopy, in Representation Theory, Number Theory, and Invariant Theory, Progress in Mathematics 323 (2018), 23–46.

    Google Scholar 

  6. _________ , A stratification related to characteristic polynomials, Advances in Math. 327 (2018), 425–469.

    Article  MathSciNet  Google Scholar 

  7. _________ , A note on the automorphic Langlands group, Canad. Math. Bull. 45 (2002), no. 4, 466–482, Dedicated to Robert V. Moody. MR 1941222

    Article  MathSciNet  Google Scholar 

  8. _________ , An introduction to the trace formula, Harmonic Analysis, the Trace Formula, and Shimura Varieties, Clay Math. Proc., vol. 4, Amer. Math. Soc., Providence, RI, 2005, pp. 1–263.

    Google Scholar 

  9. _________ , The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups, American Mathematical Society Colloquium Publications, vol. 61, American Mathematical Society, Providence, RI, 2013.

    Google Scholar 

  10. Jinpeng An, Jiu-Kang Yu, and Jun Yu, On the dimension datum of a subgroup and its application to isospectral manifolds, J. Differential Geom. 94 (2013), no. 1, 59–85. MR 3031860

    Article  MathSciNet  Google Scholar 

  11. A. Braverman and D. Kazhdan, γ-functions of representations and lifting, Geom. Funct. Anal. (2000), no. Special Volume, Part I, 237–278, With an appendix by V. Vologodsky, GAFA 2000 (Tel Aviv, 1999).

    Google Scholar 

  12. Alexander Braverman and David Kazhdan, γ-sheaves on reductive groups, Studies in Memory of Issai Schur (Chevaleret/Rehovot, 2000), Progr. Math., vol. 210, Birkhäuser Boston, Boston, MA, 2003, pp. 27–47.

    Chapter  Google Scholar 

  13. Pierre Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. (1974), no. 43, 273–307. MR 0340258

    Google Scholar 

  14. E. Frenkel, R. Langlands, and BC. Ngo, Formule des traces et fonctorialité: Le début d’un programme, Ann. Sci. Math. Québec 34 (2010), 199–243.

    MathSciNet  MATH  Google Scholar 

  15. Roger Godement and Hervé Jacquet, Zeta functions of simple algebras, Lecture Notes in Mathematics, Vol. 260, Springer-Verlag, Berlin-New York, 1972. MR 0342495

    Google Scholar 

  16. Hervé Jacquet and Robert P Langlands, Automorphic forms on gl (2), vol. 114, Springer, 2006.

    Google Scholar 

  17. Robert E. Kottwitz, Stable trace formula: cuspidal tempered terms, Duke Math. J. 51 (1984), no. 3, 611–650. MR 757954

    Article  MathSciNet  Google Scholar 

  18. Robert E. Kottwitz and Diana Shelstad, Foundations of twisted endoscopy, Astérisque (1999), no. 255, vi+190. MR 1687096

    Google Scholar 

  19. R. P. Langlands, Problems in the theory of automorphic forms, Lectures in Modern Analysis and Applications, III, Lecture Notes in Math., vol. 170, Springer, Berlin, 1970, pp. 18–61.

    Google Scholar 

  20. _________ , Automorphic representations, Shimura varieties, and motives. Ein Märchen, Automorphic Forms, Representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 205–246.

    Google Scholar 

  21. _________ , On the notion of an automorphic representation. a supplement to the preceding paper, Automorphic Forms, Representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 203–207.

    Google Scholar 

  22. Robert P. Langlands, Beyond endoscopy, Contributions to Automorphic Forms, Geometry, and Number Theory, Johns Hopkins Univ. Press, Baltimore, MD, 2004, pp. 611–697.

    Google Scholar 

  23. _________ , Singularités et transfert, Ann. Math. Qué. 37 (2013), no. 2, 173–253.

    Article  MathSciNet  Google Scholar 

  24. Chung Pang Mok, Endoscopic classification of representations of quasi-split unitary groups, Mem. Amer. Math. Soc. 235 (2015), no. 1108.

    Google Scholar 

  25. C. Mœglin and J.-L. Waldspurger, Le spectre résiduel de GL(n), Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 4, 605–674.

    Article  MathSciNet  Google Scholar 

  26. Bao Châu Ngô, Le lemme fondamental pour les algèbres de Lie, Publ. Math. Inst. Hautes Études Sci. (2010), no. 111, 1–169.

    Google Scholar 

  27. _________ , On a certain sum of automorphic L-functions, Automorphic Forms and Related Geometry: Assessing the Legacy of I. I. Piatetski-Shapiro, Contemp. Math., vol. 614, Amer. Math. Soc., Providence, RI, 2014, pp. 337–343.

    Google Scholar 

  28. Yiannis Sakellaridis, Beyond endoscopy for the relative trace formula I: Local theory, Automorphic Representations and L-Functions, Tata Inst. Fundam. Res. Stud. Math., vol. 22, Tata Inst. Fund. Res., Mumbai, 2013, pp. 521–590.

    Google Scholar 

  29. Jean-Pierre Serre, Abelian l-adic representations and elliptic curves, McGill University lecture notes written with the collaboration of Willem Kuyk and John Labute, W. A. Benjamin, Inc., New York-Amsterdam, 1968.

    MATH  Google Scholar 

  30. Richard Taylor, Automorphy for some l-adic lifts of automorphic mod l Galois representations. II, Publ. Math. Inst. Hautes Études Sci. (2008), no. 108, 183–239.

    Article  MathSciNet  Google Scholar 

  31. Akshay Venkatesh, “Beyond endoscopy” and special forms on GL(2), J. Reine Angew. Math. 577 (2004), 23–80.

    MathSciNet  MATH  Google Scholar 

  32. Jun Yu, On the dimension datum of a subgroup, Duke Math. J. 165 (2016), no. 14, 2683–2736.

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Toronto, Toronto, ON, Canada

    James Arthur

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  1. James Arthur
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Correspondence to James Arthur .

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Editors and Affiliations

  1. Mathematical Institute, University of Bonn, Bonn, Germany

    Werner Müller

  2. Department of Mathematics, University of California, Berkeley, Berkeley, CA, USA

    Sug Woo Shin

  3. Department of Mathematics, Cornell University, Ithaca, NY, USA

    Nicolas Templier

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Arthur, J. (2018). Functoriality and the Trace Formula. In: Müller, W., Shin, S., Templier, N. (eds) Geometric Aspects of the Trace Formula. SSTF 2016. Simons Symposia. Springer, Cham. https://doi.org/10.1007/978-3-319-94833-1_1

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