Geodesic
On the geometric Langlands conjecture
Frenkel, E. 1   ; Gaitsgory, D. 2   ; Vilonen, K. 3
1 Department of Mathematics, University of California, Berkeley, California 94720
2 Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
3 Department of Mathematics, Northwestern University, Evanston, Illinois 60208
Journal of the American Mathematical Society, Tome 15 (2002) no. 2, pp. 367-417 Cet article a éte moissonné depuis la source American Mathematical Society

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Let $X$ be a smooth, complete, geometrically connected curve over a field of characteristic $p$. The geometric Langlands conjecture states that to each irreducible rank $n$ local system $E$ on $X$ one can attach a perverse sheaf on the moduli stack of rank $n$ bundles on $X$ (irreducible on each connected component), which is a Hecke eigensheaf with respect to $E$. In this paper we derive the geometric Langlands conjecture from a certain vanishing conjecture. Furthermore, using recent results of Lafforgue, we prove this vanishing conjecture, and hence the geometric Langlands conjecture, in the case when the ground field is finite.
DOI : 10.1090/S0894-0347-01-00388-5

Frenkel, E.  1   ; Gaitsgory, D.  2   ; Vilonen, K.  3

1 Department of Mathematics, University of California, Berkeley, California 94720
2 Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
3 Department of Mathematics, Northwestern University, Evanston, Illinois 60208 
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Frenkel, E.; Gaitsgory, D.; Vilonen, K. On the geometric Langlands conjecture. Journal of the American Mathematical Society, Tome 15 (2002) no. 2, pp. 367-417. doi: 10.1090/S0894-0347-01-00388-5

[1] Arthur, James, Clozel, Laurent Simple algebras, base change, and the advanced theory of the trace formula 1989

[2] Beĭlinson, A. A., Bernstein, J., Deligne, P. Faisceaux pervers 1982 5 171

[3] Borho, Walter, Macpherson, Robert Représentations des groupes de Weyl et homologie d’intersection pour les variétés nilpotentes C. R. Acad. Sci. Paris Sér. I Math. 1981 707 710

[4] Casselman, W., Shalika, J. The unramified principal series of 𝑝-adic groups. II. The Whittaker function Compositio Math. 1980 207 231

[5] Cogdell, J. W., Piatetski-Shapiro, I. I. Converse theorems for 𝐺𝐿_{𝑛} Inst. Hautes Études Sci. Publ. Math. 1994 157 214

[6] Deligne, Pierre La conjecture de Weil. II Inst. Hautes Études Sci. Publ. Math. 1980 137 252

[7] Drinfel′D, V. G. Two-dimensional 𝑙-adic representations of the fundamental group of a curve over a finite field and automorphic forms on 𝐺𝐿(2) Amer. J. Math. 1983 85 114

[8] Frenkel, E., Gaitsgory, D., Kazhdan, D., Vilonen, K. Geometric realization of Whittaker functions and the Langlands conjecture J. Amer. Math. Soc. 1998 451 484

[9] Fulton, William Young tableaux 1997

[10] Illusie, Luc Théorie de Brauer et caractéristique d’Euler-Poincaré (d’après P. Deligne) 1981 161 172

[11] Kazhdan, David On lifting 1984 209 249

[12] Laumon, Gérard Correspondance de Langlands géométrique pour les corps de fonctions Duke Math. J. 1987 309 359

[13] Laumon, Gérard, Moret-Bailly, Laurent Champs algébriques 2000

[14] Lusztig, George Singularities, character formulas, and a 𝑞-analog of weight multiplicities 1983 208 229

[15] Mirković, Ivan, Vilonen, Kari Perverse sheaves on affine Grassmannians and Langlands duality Math. Res. Lett. 2000 13 24

[16] Pjateckij-Šapiro, I. I. Euler subgroups 1975 597 620

[17] Rothstein, Mitchell Connections on the total Picard sheaf and the KP hierarchy Acta Appl. Math. 1996 297 308

[18] Shalika, J. A. The multiplicity one theorem for 𝐺𝐿_{𝑛} Ann. of Math. (2) 1974 171 193

[19] Shintani, Takuro On an explicit formula for class-1 “Whittaker functions” on 𝐺𝐿_{𝑛} over 𝑃-adic fields Proc. Japan Acad. 1976 180 182

[20] Brylinski, Jean-Luc (Co)-homologie d’intersection et faisceaux pervers 1982 129 157

[21] Towber, Jacob Young symmetry, the flag manifold, and representations of 𝐺𝐿(𝑛) J. Algebra 1979 414 462

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