Annotated bibliography

Le lemme fondamental de Jacquet et Ye en caractéristiques positives [JY.pdf]: Duke Mathematical Journal 96 (1999), 473-520
Le lemme fondamental de Jacquet et Ye en caractéristiques égales [CRAS.pdf]: Comptes Rendus de l'Académie des Sciences-Series I-Mathematics 325 (1997), 307-312

The paper published in Duke is almost identical to my PhD thesis, which is written under the supervision of G. Laumon and defended in Orsay in June 1997. The note aux Comptes Rendus is a shortened version of my thesis.

The Jacquet-Ye fundamental lemma is an identity between orbital integrals appearing in the Kuznetsov trace formula for GL(n) and their quadratic twists. After Jacquet and Ye, this identity implies the quadratic base change for GL(n) via a comparison of relative trace formulas and, at the same time, provides a characterization of the functorial image by the property of being distinguished by an unitary group. In GL(2) case, these orbital integrals look like Kloosterman sums and the conjectured identity can be reduced to the Hasse-Davenport identity between Gauss sums by means of the Mellin transform.

The method of studying local orbital integrals by the geometry of families of global orbital integrals is introduced here for the first time, as well as the use of perverse sheaves. This method has been later used successfully for other types of fundamental lemmas of which the Langlands-Shelstad conjecture in the theory of endoscopy is the most well-known [LF.pdf]. The general method has been formulated in my lectures in the PCMI 2015 [PCMI.pdf].

Note that in this work the proof of perversity is performed in only one special case with the help of some miraculous system of coordinates. This is the content of Theorem C in both [CRAS.pdf] and [JY.pdf]. It is desirable to find a conceptual way to prove the more general perversity conjecture 3.5.1 of [PCMI.pdf].

The geometric approach to fundamental lemmas initiated in this paper only treats the case of local field of positive characteristic. Using model theory, Cluckers and Loeser proved in Constructible exponential functions, motivic Fourier transform and transfer principle, Annals of Mathematics, 171 (2010), 1011–1065, that the identity in positive characteristic implies the same identity over the p-adic numbers for large prime p. In the mean time, Jacquet proved his conjecture by a completely different method in Kloosterman identities over a quadratic extension, Annals of Mathematics, 160 (2004), 755–779.

Note that this approach has been adopted in the paper of Do, Viet Cuong Le lemme fondamental métaplectique de Jacquet et Mao en égales caractéristiques. Bull. Soc. Math. France 143 (2015), no. 1, 125–196. To treat the Jacquet-Mao fundamental lemma, Do Viet Cuong needs to develop a l-adic sheaf theoretic interpretation of a natural 2-cocycle occurring in metaplectic groups. (updated on March 1, 2018)

Faisceaux pervers, homomorphisme de changement de base et lemme fondamental de Jacquet et Ye [ENS.pdf], Annales Scientifiques de l’École Normale Supérieure 32 (5), 1999, 619-679

This paper, which is the continuation of [JY.pdf], contains the proof of the Jacquet-Ye fundamental lemma for an arbitrary element of the unramified Hecke algebra.

Compared to the previous paper, the main novelty is the sheaf theoretic interpretation of the base change homomorphism in Hecke algebras. For split extension of degree r, the base change homomorphism would be given by the convolution power to the degree r. For non split extension, the base change of a spherical perverse sheaf would be its convolution power equipped with the Frobenius twisted by the commutativity constraint.

The proof of this result requires dealing with constant terms of spherical perverse sheaves, which is a part of the geometric Satake equivalence developed in Mirkovic and Vilonen's paper Perverse sheaves on affine Grassmannians and Langlands duality, Mathematical Research Letters 7(2000), 13–24, after an earlier unpublished work of Ginzburg. Our treatment of constant terms is based on the observation that every simple spherical perverse sheaf is a direct factor of a convolution product of minuscule spherical perverse sheaves and constant terms of a convolution product is can be conveniently expressed on constant terms of the factors.

The theory of perverse sheaves on affine Grassmannian fits miraculously well with the miraculous system of coordinates in Theorem C of [JY.pdf]. This follows a line of thoughts initiated by Lusztig in Green polynomials and singularities of unipotent classes, Advances in Mathematics 42(1981), 169-178. In that paper, Lusztig points out that for GL(n) singularities in closure of unipotent conjugacy classes are equivalent to singularities in closed Schubert cells in the affine Grassmannian. What makes our task easy is that the coordinate system in C of [JY.pdf] is basically the same as Lusztig's.

The Jacquet-Ye fundemental lemma for Hecke algebra proved in this paper in positive characteristic, was later proved in the p-adic case by a completely method by Jacquet in Kloosterman identities over a quadratic extension. Ann. Scient. Éc. Norm. Sup. II, 38(2005), 609-669.
with A. Bouthier and Y. Sakellaridis: On the formal arc space of a reductive monoid [BNY.pdf]

2014 with Le Hung and Ho in MRL: Average size of 2-Selmer groups of elliptic curves over function fields [bcq.pdf]

2014 Cont. Math. volume in honor of Piatetski-Shapiro: On a certain sum of L-functions [PS.pdf]

2013 with Heinloth and Yun in Ann. of Math.: Kloosterman sheaves for reductive groups [Kloosterman.pdf]

2011 with Frenkel in Bull. Math. Sci.: Geometrization of the trace formula [geom.pdf]

2011 with Frenkel and Langlands in Ann. Math. Quebec: Formule des traces et fonctorialité: le début d'un programme [quebec.pdf]

2010 ICM in Hyderabad: Endoscopy theory of automorphic forms [ICM.pdf]

2009 in Pub. IHES: Le lemme fondamental pour les algèbres de Lie [LF.pdf]

2009 chapter in M. Harris' book: Decomposition theorem and abelian fibration [support.pdf]

2009 with Genestier in Panorama et Synthèse : Lectures on Shimura varieties [Shimura.pdf]

2009 CRM Harvard: Survey on the fundamental lemma [survey.pdf]

2008 lecture in Qui Nhon: Report on the proof of of some conjectures on orbital integrals in Langlands' program [superseded]

2008 with Laumon in Ann. of Math. : Le lemme fondamental pour les groupes unitaires [LF-unitaire.pdf]

2006 ICM in Madrid: Fibration de Hitchin et structure endoscopique de la formule des traces [ICM-Madrid.pdf]

2006 in Inventiones: Fibration de Hitchin et endoscopie [Exupery.pdf]

2006 with Ngô Đắc Tuấn in JIMJ: Comptage de G-chtoucas : la partie elliptique régulière [comptage.pdf]

2002 in Annales ENS: D-chtoucas de Drinfeld à modifications symétriques et identité de changement de base [chbase.pdf]

2002 with Genestier in Annales Fourier: Alcôves et p-rang des variétés abéliennes [p-rang.pdf]

2002 with Haines in American JM: Alcoves associated to special fibers of local models [alcoves.pdf]

2002 with Haines in Compositio: Nearby cycles for local models of some Shimura varieties [nearby-cycle.pdf]

2001 with Polo in JAG: Résolutions de Demazure affines et formule de Casselman-Shalika géométrique  [ngo-polo.pdf]

2000 in Israel JM: Preuve d'une conjecture de Frenkel-Gaitsgory-Kazhdan-Vilonen pour les groupes linéaires généraux [fgkv.pdf]

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updated on November 23, 2015