Abstract
Following our joint work with Robert Langlands, we make the first steps toward developing geometric methods for analyzing trace formulas in the case of the function field of a curve defined over a finite field. We also suggest a conjectural framework of geometric trace formulas for curves defined over the complex field, which exploits the categorical version of the geometric Langlands correspondence.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Arbarello, E., Cornalba, M., Griffiths, P.A., Harris, J.: Geometry of algebraic curves, vol. I. Grundlehren der Mathematischen Wissenschaften, 267. Springer, New York (1985)
Atiyah M., Bott R.: A Lefschetz fixed point formula for elliptic complexes. I. Ann. Math. (2) 86, 374–407 (1967)
Atiyah M., Bott R.: A Lefschetz fixed point formula for elliptic complexes. II. Ann. Math. (2) 88, 451–491 (1968)
Beauville A., Laszlo Y.: Un lemme de descente. C. R. Acad. Sci. Paris, Sér. I Math. 320, 335–340 (1995)
Behrend, K.: Derived l-adic categories for algebraic stacks. Mem. Am. Math. Soc. 163(774) (2003)
Behrend K., Dhillon A.: Connected components of moduli stacks of torsors via Tamagawa numbers. Can. J. Math. 61, 3–28 (2009)
Ben-Zvi D., Francis J., Nadler D.: Integral transforms and Drinfeld centers in derived algebraic geometry. J. Amer. Math. Soc. 23, 909–966 (2010) arXiv:0805.0157
Ben-Zvi, D., Nadler, D.: Loop Spaces and Connections. Preprint. arXiv:1002.3636
Beilinson, A., Drinfeld, V.: Quantization of Hitchin’s integrable system and Hecke eigensheaves. Preprint. http://www.math.uchicago.edu/~mitya/langlands
Bourbaki N.: Groupes et algèbres de Lie Chapitres IV, V, VI. Hermann, Paris (1968)
Chaudouard P.-H., Laumon G.: Le lemme fondamental pondéré I: constructions géométriques. Compositio Math. 146, 1416–1506 (2010) arXiv:0902.2684
Drinfeld V.G.: Two-dimensional ℓ-adic representations of the fundamental group of a curve over a finite field and automorphic forms on GL(2). Am. J. Math. 105, 85–114 (1983)
Drinfeld, V.G.: Langlands conjecture for GL(2) over function field. In: Proceedings of International Congress of Mathematics (Helsinki, 1978), pp. 565–574 (1978)
Drinfeld V.G.: Moduli varieties of F-sheaves. Funct. Anal. Appl. 21, 107–122 (1987)
Drinfeld V.G.: The proof of Petersson’s conjecture for GL(2) over a global field of characteristic p. Funct. Anal. Appl. 22, 28–43 (1988)
Frenkel E. et al.: Lectures on the Langlands program and conformal field theory. In: Cartier, P. (eds) Frontiers in Number Theory, Physics and Geometry II., pp. 387–536. Springer, New York (2007) hep-th/0512172
Frenkel E.: Gauge theory and Langlands duality Séminaire Bourbaki, Juin 2009. Astérisque 332, 369–403 (2010) arXiv:0906.2747
Frenkel E., Gaitsgory D., Kazhdan D., Vilonen K.: Geometric realization of Whittaker functions and the Langlands conjecture. J. AMS 11, 451–484 (1998)
Frenkel E., Gaitsgory D., Vilonen K.: Whittaker patterns in the geometry of moduli spaces of bundles on curves. Ann. Math. 153, 699–748 (2001) arXiv:math/9907133
Frenkel E., Gaitsgory D., Vilonen K.: On the geometric Langlands conjecture. J. AMS 15, 367–417 (2001) arXiv:math/0012255
Frenkel E., Langlands R., Ngô B.C.: La formule des traces et la functorialité. Le début d’un Programme. Ann. Sci. Math. Québec 34, 199–243 (2010) arXiv:1003.4578
Frenkel E., Witten E.: Geometric endoscopy and mirror symmetry. Commun. Number Theory Phys. 2, 113–283 (2008) arXiv:0710.5939
Gross B.H., Prasad D.: On the decomposition of a representation of SO n when restricted to SOn-1. Can. J. Math. 44, 974–1002 (1992)
Heinloth, J., Schmitt, A.: The Cohomology Ring of Moduli Stacks of Principal Bundles over Curves. Preprint. http://www.uni-due.de/~hm0002/HS_v1.pdf
Hitchin N.: The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc. (3) 55, 59–126 (1987)
Hitchin N.: Stable bundles and integrable systems. Duke Math. J. 54, 91–114 (1987)
Hitchin N.: Lie groups and Teichmüller space. Topology 31, 449–473 (1992)
Illusie, L.: Formule de Lefschetz, SGA 5. Lecture Notes in Mathematics, vol. 589, pp. 73–137. Springer, New York (1977)
Ichino, A.: On critical values of adjoint L-functions for GSp(4). Preprint. http://www.math.ias.edu/~ichino/ad.pdf
Ichino A., Ikeda T.: On the periods of automorphic forms on special orthogonal groups and the Gross-Prasad conjecture. Geom. Funct. Anal. 19, 1378–1425 (2010)
Jacquet, H.: A guide to the relative trace formula. In: Automorphic Representations, L-functions and Applications: Progress and Prospects, vol. 11, pp. 257–272. Ohio State University Mathematical Research Institute Publications. De Gruyter, Berlin (2005)
Kapustin A., Witten E.: Electric-magnetic duality and the geometric Langlands program. Comm. Number Theory Phys. 1, 1–236 (2007) hep-th/0604151
Kontsevich, M.: Notes on motives in finite characteristic. Preprint. arXiv:math/0702206
Lafforgue L.: Chtoucas de Drinfeld et correspondance de Langlands. Invent. Math. 147, 1–241 (2002)
Lafforgue, V.: Quelques calculs reliés à la correspondance de Langlands géométrique pour \({{\mathbb P}^1}\).http://people.math.jussieu.fr/~vlafforg/geom.pdf
Lafforgue, V., Lysenko, S.: Compatibility of the Theta correspondence with the Whittaker functors. Preprint. arXiv:0902.0051
Langlands, R.: Problems in the theory of automorphic forms. In: Lecture Notes in Mathematics, vol. 170, pp. 18–61. Springer, New York (1970)
Langlands R.: Beyond endoscopy. In: Contributions to Automorphic Forms, Geometry, and Number Theory, pp. 611–697. Johns Hopkins University Press, Baltimore (2004)
Langlands R.: Un nouveau point de repère dans la théorie des formes automorphes. Can. Math. Bull. 50(2), 243–267 (2007)
Langlands, E.: Reflexions on receiving the Shaw Prize, 2007. http://publications.ias.edu/rpl
Laszlo Y., Olsson M.: The six operations for sheaves on Artin stacks. I. Finite coefficients. Publ. Math. IHES 107, 109–168 (2008)
Laumon, G.: Transformation de Fourier généralisée. Preprint. arXiv:alg-geom/9603004
Laumon G.: Correspondance de Langlands géométrique pour les corps de fonctions. Duke Math. J. 54, 309–359 (1987)
Lysenko, S.: Global geometrised Rankin-Selberg method for GL(n). Preprint. arXiv:math/0108208
Mirković I., Vilonen K.: Geometric Langlands duality and representations of algebraic groups over commutative rings. Annal. Math. 166, 95–143 (2007) arXiv:math.RT/0401222
Neumann, F., Stuhler, U.: Moduli stacks of vector bundles and Frobenius morphisms. Preprint
Ngô B.C.: Faisceaux pervers, homomorphisme de changement de base et lemme fondamental de Jacquet et Ye. Ann. Sci. Ec. Norm. Sup. 32, 619–679 (1999)
Ngô B.C.: Fibration de Hitchin et endoscopie. Invent. Math. 164, 399–453 (2006)
Ngô B.C.: Le lemme fondamental pour les algebres de Lie. Publications IHES 111, 1–169 (2010) arXiv:0801.0446
Rothstein M.: Connections on the total Picard sheaf and the KP hierarchy. Acta Appl. Math. 42, 297–308 (1996)
Sarnak, P.: Comments on Robert Langland’s Lecture: “Endoscopy and Beyond”. http://www.math.princeton.edu/sarnak/SarnakLectureNotes-1.pdf
Simpson C.: Higgs bundles and local systems. Publ. Math. IHES 75, 5–95 (1992)
Teleman C.: Borel-Weil-Bott theory on the moduli stack of G-bundles over a curve. Invent. Math. 134, 1–57 (1998)
Shafarevich, I.R. (ed.): Algebraic Geometry I: Algebraic Curves. Algebraic Manifolds and Schemes. Encyclopaedia of Mathematical Sciences, vol. 1. Springer, New York (1994)
Venkatesh A.: “Beyond endoscopy” and special forms on GL(2). J. Reine Angew. Math. 577, 23–80 (2004)
Vinberg, E.B.: On reductive algebraic semigroups. In: Gindikin, S., Vinberg, E. (eds.) Lie Groups and Lie Algebras. E.B. Dynkin Seminar. AMS Transl. (2) 169, 145–182 (1995)
Acknowledgments
This paper was conceived as part of a joint project with Robert Langlands initiated in [21]. We are deeply grateful to him for his collaboration and for generously sharing with us his ideas and insights on this subject. We thank Peter Sarnak for drawing our attention to the relative trace formula, Atsushi Ichino for communicating his and T. Ikeda’s unpublished conjecture on the norm of the Whittaker functional, and David Nadler for helpful discussions of the conjectural generalization of the Atiyah–Bott–Lefschetz fixed point formula for stacks presented in Sect. 6.4. E.F. is grateful to David Ben-Zvi, David Eisenbud, Alexander Givental, Dick Gross, Michael Harris, Atsushi Ichino, David Kazhdan, Vincent Lafforgue, Sergey Lysenko, and Martin Olsson for valuable discussions.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution,and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by E. Zelmanov.
E. Frenkel was supported by DARPA under the grant HR0011-09-1-0015 and B. C. Ngô was supported by the Simonyi Foundation and NSF under the grant DMS-1000356.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Frenkel, E., Ngô, B.C. Geometrization of trace formulas. Bull. Math. Sci. 1, 129–199 (2011). https://doi.org/10.1007/s13373-011-0009-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13373-011-0009-0