Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 042, 31 pages      arXiv:1812.08160      https://doi.org/10.3842/SIGMA.2020.042
Contribution to the Special Issue on Algebra, Topology, and Dynamics in Interaction in honor of Dmitry Fuchs

Is There an Analytic Theory of Automorphic Functions for Complex Algebraic Curves?

Edward Frenkel
Department of Mathematics, University of California, Berkeley, CA 94720, USA

Received September 30, 2019, in final form April 27, 2020; Published online May 16, 2020

Abstract
The geometric Langlands correspondence for complex algebraic curves differs from the original Langlands correspondence for number fields in that it is formulated in terms of sheaves rather than functions (in the intermediate case of curves over finite fields, both formulations are possible). In a recent preprint, Robert Langlands made a proposal for developing an analytic theory of automorphic forms on the moduli space of $G$-bundles on a complex algebraic curve. Langlands envisioned these forms as eigenfunctions of some analogues of Hecke operators. In these notes I show that if $G$ is an abelian group then there are well-defined Hecke operators, and I give a complete description of their eigenfunctions and eigenvalues. For non-abelian $G$, Hecke operators involve integration, which presents some difficulties. However, there is an alternative approach to developing an analytic theory of automorphic forms, based on the existence of a large commutative algebra of global differential operators acting on half-densities on the moduli stack of $G$-bundles. This approach (which implements some ideas of Joerg Teschner) is outlined here, as a preview of a joint work with Pavel Etingof and David Kazhdan.

Key words: Langlands Program; automorphic function; complex algebraic curve; principal $G$-bundle; Jacobian variety; differential operator; oper.

pdf (593 kb)   tex (47 kb)  

References

  1. Alvarenga R., Hall algebra and graphs of Hecke operators for elliptic curves, arXiv:1805.00567.
  2. Arinkin D., Gaitsgory D., Singular support of coherent sheaves and the geometric Langlands conjecture, Selecta Math. (N.S.) 21 (2015), 1-199, arXiv:1201.6343.
  3. Atiyah M.F., Vector bundles over an elliptic curve, Proc. London Math. Soc. 7 (1957), 414-452.
  4. Atiyah M.F., Bott R., The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), 523-615.
  5. Beilinson A., Bernstein J., A proof of Jantzen conjectures, in I.M. Gel'fand Seminar, Adv. Soviet Math., Vol. 16, Amer. Math. Soc., Providence, RI, 1993, 1-50.
  6. Beilinson A., Drinfeld V., Quantization of Hitchin's integrable system and Hecke eigensheaves, available at http://math.uchicago.edu/~drinfeld/langlands/QuantizationHitchin.pdf.
  7. Boozer D., Moduli spaces of Hecke modifications for rational and elliptic curves, arXiv:1805.11184.
  8. Braverman A., Kazhdan D., Some examples of Hecke algebras for two-dimensional local fields, Nagoya Math. J. 184 (2006), 57-84, arXiv:math.RT/0510538.
  9. Braverman A., Kazhdan D., The spherical Hecke algebra for affine Kac-Moody groups I, Ann. of Math. 174 (2011), 1603-1642, arXiv:0809.1461.
  10. Casselman W., Cely J.E., Hales T., The spherical Hecke algebra, partition functions, and motivic integration, Trans. Amer. Math. Soc. 371 (2019), 6169-6212, arXiv:1611.05773.
  11. Cluckers R., Loeser F., Constructible motivic functions and motivic integration, Invent. Math. 173 (2008), 23-121, arXiv:math.AG/0410203.
  12. Drinfeld V.G., Langlands' conjecture for ${\rm GL}(2)$ over functional fields, in Proceedings of the International Congress of Mathematicians (Helsinki, 1978), Acad. Sci. Fennica, Helsinki, 1980, 565-574.
  13. Drinfeld V.G., Two-dimensional $l$-adic representations of the fundamental group of a curve over a finite field and automorphic forms on ${\rm GL}(2)$, Amer. J. Math. 105 (1983), 85-114.
  14. Drinfeld V.G., Moduli varieties of $F$-sheaves, Funct. Anal. Appl. 21 (1987), 107-122.
  15. Drinfeld V.G., The proof of Petersson's conjecture for ${\rm GL}(2)$ over a global field of characteristic $p$, Funct. Anal. Appl. 22 (1988), 28-43.
  16. Etingof P., Frenkel E., Kazhdan D., An analytic version of the Langlands correspondence for complex curves, arXiv:1908.09677.
  17. Etingof P., Frenkel E., Kazhdan D., Hecke operators for curves over local fields, in preparation.
  18. Feigin B., Frenkel E., Affine Kac-Moody algebras at the critical level and Gelfand-Dikii algebras, Internat. J. Modern Phys. A 7 (1992), suppl. 1A, 197-215.
  19. Fesenko I., Analysis on arithmetic schemes. I, 2003, extra vol., 261-284.
  20. Fesenko I., Measure, integration and elements of harmonic analysis on generalized loop spaces, in Proceedings of the St. Petersburg Mathematical Society, Vol. XII, Amer. Math. Soc. Transl. Ser. 2, Vol. 219, Amer. Math. Soc., Providence, RI, 2006, 149-165.
  21. Frenkel E., Wakimoto modules, opers and the center at the critical level, Adv. Math. 195 (2005), 297-404, arXiv:math.QA/0210029.
  22. Frenkel E., Langlands correspondence for loop groups, Cambridge Studies in Advanced Mathematics, Vol. 103, Cambridge University Press, Cambridge, 2007.
  23. Frenkel E., Lectures on the Langlands program and conformal field theory, in Frontiers in Number Theory, Physics, and Geometry. II, Springer, Berlin, 2007, 387-533, arXiv:hep-th/0512172.
  24. Frenkel E., Gauge theory and Langlands duality, Talk given at MSRI, Berkeley, September 2014, available at https://youtu.be/NQfeBeRKMiw?t=4190.
  25. Frenkel E., Gaitsgory D., Vilonen K., On the geometric Langlands conjecture, J. Amer. Math. Soc. 15 (2002), 367-417, arXiv:math.AG/0012255.
  26. Gaitsgory D., On a vanishing conjecture appearing in the geometric Langlands correspondence, Ann. of Math. 160 (2004), 617-682, arXiv:math.AG/0204081.
  27. Gaitsgory D., Outline of the proof of the geometric Langlands conjecture for ${\rm GL}_2$, Astérisque 370 (2015), 1-112, arXiv:1302.2506.
  28. Gaitsgory D., Progrès récents dans la théorie de Langlands géométrique, Astérisque 390 (2017), exp. no. 1109, 139-168, arXiv:1606.09462.
  29. Gawędzki K., Lectures on conformal field theory, in Quantum Fields and Strings: a Course for Mathematicians, Vols. 1, 2 (Princeton, NJ, 1996/1997), Amer. Math. Soc., Providence, RI, 1999, 727-805.
  30. Ginzburg V., Perverse sheaves on a loop group and Langlands duality, arXiv:alg-geom/9511007.
  31. Goldman W.M., Projective structures with Fuchsian holonomy, J. Differential Geom. 25 (1987), 297-326.
  32. Gordon J., Motivic Haar measure on reductive groups, Canad. J. Math. 58 (2006), 93-114, arXiv:math.AG/0203106.
  33. Griffiths P., Harris J., Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience, New York, 1978.
  34. Hrushovski E., Kazhdan D., Integration in valued fields, in Algebraic geometry and number theory, Progr. Math., Vol. 253, Birkhäuser Boston, Boston, MA, 2006, 261-405, arXiv:math.AG/0510133.
  35. Hrushovski E., Kazhdan D., The value ring of geometric motivic integration, and the Iwahori Hecke algebra of ${\rm SL}_2$, Geom. Funct. Anal. 17 (2008), 1924-1967, arXiv:math.AG/0609115.
  36. Kapranov M., Double affine Hecke algebras and 2-dimensional local fields, J. Amer. Math. Soc. 14 (2001), 239-262, arXiv:math.AG/9812021.
  37. Kim H.H., Lee K.H., Spherical Hecke algebras of $\rm {\rm SL}_2$ over 2-dimensional local fields, Amer. J. Math. 126 (2004), 1381-1399.
  38. Knapp A.W., Vogan Jr. D.A., Cohomological induction and unitary representations, Princeton Mathematical Series, Vol. 45, Princeton University Press, Princeton, NJ, 1995.
  39. Kontsevich M., Notes on motives in finite characteristic, in Algebra, Arithmetic, and Geometry: in Honor of Yu.I. Manin, Vol. II, Progr. Math., Vol. 270, Birkhäuser Boston, Boston, MA, 2009, 213-247, arXiv:math.AG/0702206.
  40. Lafforgue L., Chtoucas de Drinfeld et correspondance de Langlands, Invent. Math. 147 (2002), 1-241.
  41. Langlands R.P., Problems in the theory of automorphic forms, in Lectures in Modern Analysis and Applications, III, Lecture Notes in Math., Vol. 170, Springer, Berlin, 1970, 18-61.
  42. Langlands R.P., On the analytical form of the geometric theory of automorphic forms, in Russian, available at http://publications.ias.edu/rpl/paper/2678.
  43. Laumon G., Correspondance de Langlands géométrique pour les corps de fonctions, Duke Math. J. 54 (1987), 309-359.
  44. Laumon G., Transformation de Fourier, constantes d'équations fonctionnelles et conjecture de Weil, Inst. Hautes Études Sci. Publ. Math. 65 (1987), 131-210.
  45. Lorscheid O., Automorphic forms for elliptic function fields, Math. Z. 272 (2012), 885-911, arXiv:1012.4825.
  46. Lusztig G., Singularities, character formulas, and a $q$-analog of weight multiplicities, Astérisque 101 (1983), 208-229.
  47. Mirković I., Vilonen K., Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. of Math. 166 (2007), 95-143, arXiv:math.RT/0401222.
  48. Morrow M., Integration on product spaces and ${\rm GL}_n$ of a valuation field over a local field, Commun. Number Theory Phys. 2 (2008), 563-592, arXiv:0712.2175.
  49. Morrow M., Integration on valuation fields over local fields, Tokyo J. Math. 33 (2010), 235-281, arXiv:0712.2172.
  50. Teschner J., Quantisation conditions of the quantum Hitchin system and the real geometric Langlands correspondence, arXiv:1707.07873.
  51. Waller R., Private communication, November 2018.
  52. Waller R., Measure and integration on non-locally compact spaces and representation theory, Ph.D. Thesis, University of Nottingham, 2019.
  53. Weil A., A 1940 letter of André Weil on analogy in mathematics (translated by Martin H. Krieger), Notices Amer. Math. Soc. 52 (2005), 334-341.

Previous article  Next article  Contents of Volume 16 (2020)