On a vanishing conjecture appearing in the geometric Langlands correspondence

By D. Gaitsgory. 2002

72 citations recorded by the source corpus.

0.1. This paper should be regarded as a sequel to [7]. There it was shown that the geometric Langlands conjecture for GLn follows from a certain vanishing conjecture. The goal of the present paper is to prove this vanishing conjecture. Let X be a smooth projective curve over a ground field k. Let E be an m-dimensional local system on X, and let Bunm be the moduli stack of rank m vector bundles on X. The geometric Langlands conjecture says that to E we can associate a perverse sheaf FE on Bunm, which is a Hecke eigensheaf with respect to E. The vanishing conjecture of [7] says that for all integers n < m, a certain functor AvE , depending on E and a parameter d ∈ Z+, which maps the category D(Bunn) to itself, vanishes identically, when d is large enough. The fact that the vanishing conjecture implies the geometric Langlands conjecture may be regarded as a geometric version of the converse theorem. Moreover, as will be explained in the sequel, the vanishing of the functor AvE is analogous to the condition that the Rankin-Selberg convolution of E, viewed as an m-dimensional Galois representation, and an automorphic form on GLn with n < m is well-behaved. Both the geometric Langlands conjecture and the vanishing conjecture can be formulated in any of the sheaf-theoretic situations, e.g., Q -adic sheaves (when char(k) = ), D-modules (when char(k) = 0), and sheaves with coefficients in a finite field F (again, when char(k) = ). When the ground field is the finite field Fq and we are working with -adic coefficients, it was shown in [7] that the vanishing conjecture can be deduced from Lafforgue’s theorem that establishes the full Langlands correspondence for global fields of positive characteristic; cf. [9].

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8b8008f21e5104294c09a6f1d3a9499859eb0ffe

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In Algebra and Number Theory

1. E. Witten
2. E. Frenkel
3. A. Kapustin
4. D. Nadler
5. D. Gaitsgory
6. D. Arinkin
7. T. Arakawa
8. R. Bezrukavnikov

In Geometry and Topology

1. E. Witten
2. D. Gaitsgory
3. E. Frenkel
4. D. Nadler
5. A. Kapustin
6. D. Arinkin
7. Sergei Gukov
8. D. Gaiotto

In Mathematical Physics

1. E. Witten
2. D. Nadler
3. D. Gaitsgory
4. A. Kapustin
5. E. Frenkel
6. Sergei Gukov
7. D. Arinkin
8. P. Deligne

Citations in the corpus

• Spherical varieties and Langlands duality (2006, background): “r I G, and the corresponding middle-extension functor j!∗: P(Gr I G|C˚I ) →P(Gr I). SPHERICAL VARIETIES AND LANGLANDS DUALITY 27 The following fundamental fact is directly implied by [MV04] (see also [Gai04]). Proposition 6.2.4. For any A ∈PG(O)(GrG)⊗I the perverse sheaf j!∗(ρI(A)) is ULA with respe”
• Twisted Whittaker model and factorizable sheaves (2007, background): “ponding groups Nreg y′ , N reg y′′ and N reg y and torsors with respect to them over fW n good at y . Let us consider the corresponding three subcategories of D-modc fW n good at y . As in [Ga], Corollary 4.14 one shows that the three categories coincide (as long as y′ and y′′ are non-empty). T”
• Autoduality of compactified Jacobians for curves with plane singularities (2010, background): “1) is inspired by results in the area of geometric Langlands conjecture, it is not clear whether the proofs from [12, 16, 17] can be adapted to our settings.”
• Electric-Magnetic Duality And The Geometric Langlands Program (2006, methodology): “In addition, a geometric version of the Langlands program for curves has been much developed [4-8], both for curves over a field of characteristic p and for ordinary complex Riemann surfaces.”
• Lectures on the Langlands program and conformal field theory (2005, result): “These works motivated the conjecture in the case of GLn, which has been proved in [55, 56] (these works were”
• Recent advances in the Langlands Program (2003, methodology): “The geometric Langlands conjecture for GLn has recently been proved in [FGV], [ Ga ] following earlier works of P. Deligne, V. Drinfeld [Dr4] and G. Laumon [La2], [La3].”
• On De Jong’s conjecture (2004, background): “ed functor on De(Bunn). (2) If E is such that the functor Av1 E is exact on De(Bunn), then so is Avd E for any d. Once Proposition 4.6 is proved, we finish the proof of Theorem 4.5 as in Sect. 2.16 of [6]. 4.7. Let De(Bun n ×S) be the system of quotient categories of D(Bunn ×S), satisfying Properties”
• Outline of the proof of the geometric Langlands conjecture for $GL_2$ (2013, background): “out actually lifting our function on G(A)/G(O). Here is how we will do this. The trick explained below stands behind the definition of the corresponding verions of the Whittaker category in [FGV1] and [Ga1]. 5.3.2. Let x be a finite collection of points on X, and let Axdenote the corresponding product”