On De Jong’s conjecture

By D. Gaitsgory. 2004

87 citations recorded by the source corpus.

AbstractLet X be a smooth projective curve over a finite field Fq. Let ρ be a continuous representation π(X) → GLn(F), where F = Fl((t)) with Fl being another finite field of order prime to q.Assume that $$ \rho \left| {_{\pi (\bar X)} } \right. $$ is irreducible. De Jong’s conjecture says that in this case $$ \rho (\pi (\bar X)) $$ is finite. As was shown in the original paper of de Jong, this conjecture follows from an existence of an F-valued automorphic form corresponding to ρ is the sense of Langlands. The latter follows, in turn, from a version of the Geometric Langlands conjecture.In this paper we sketch a proof of the required version of the geometric conjecture, assuming that char(F) ≠ 2, thereby proving de Jong’s conjecture in this case.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b418d12bb25d3c29c6cd8c3ea00b607b41df7976

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In Discrete Mathematics and Combinatorics

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Citations in the corpus

Unit Interval Orders and the Dot Action on the Cohomology of Regular Semisimple Hessenberg Varieties (2015, unknown): “de Jong, Drinfeld, Gaitsgory, and Böckle–Khare [31, 32, 9, 13, 15, 5].”
Coherent sheaves on the stack of Langlands parameters (2020, background): “ be as above, and let us equip Awith the m-adic topology. Let ˆ: W F;S!GL m(A) be a continuous homomorphism. Then ˆ(ˇ 1(U)) is nite. Proof. If A’[[t]], this is de Jong’s conjecture [dJ01], proved in [Ga07] when ‘>2 (see also [BK06]). Next, we assume that Ais reduced (in particular Ais an F ‘-alge”
Kloosterman sheaves for reductive groups (2010, background): “One defines the geometric Hecke operators as a functor (see [17]):”
Wild Harmonic Bundles and Wild Pure Twistor D-modules (2008, methodology): “It was also established by the works of Drinfeld, BoeckleKhare and Gaitsgory ([24], [11], [27]) via the method of arithmetic geometry based on the work of Lafforgue.”