Geometric Applications of the Langlands Correspondence: Reading Rack

This is an annotated reading list for a working seminar at the Institute for Advanced Study during winter/spring 2019. Let me know if you have trouble with any of the links; some of them require access from within Princeton or IAS (possibly via a VPN).

The beginning of the story

• Pierre Deligne, La conjecture de Weil, II: open access.
  We will be focused on Conjecture 1.2.10, which concerns the properties of an irreducible lisse Weil sheaf with finite-order determinant on a smooth variety over a finite field.
• Anna Cadoret, La conjecture des compagnons [d'après Deligne, Drinfeld, L. Lafforgue, T. Abe, ...]: preprint not yet available (email me for the text).
  This is the text of a Seminaire Bourbaki lecture from January 2019 on the topic of Deligne's conjecture.

The (étale) Langlands correspondence for GL(n)

• Vladimir Drinfeld, Langlands' Conjecture for GL(2) over functional Fields: ICM proceedings.
  This paper introduces the concept of a shtuka and uses the moduli spaces of shtukas to make the automorphic-to-Galois construction for GL(2).
• Laurent Lafforgue, Chtoucas de Drinfeld et correspondance de Langlands: DOI.
  This paper extends the argument of Drinfeld to GL(n); in the process, one obtains most of Deligne's conjecture for curves. We will mostly treat this paper as a "black box". (Warning: there is a key error in the last section of this paper which does not affect the main result, but is relevant for our applications.)

p-adic Weil cohomology

• Kiran S. Kedlaya, Notes on isocrystals: arXiv.
  This is a high-level overview of the properties of p-adic Weil cohomology (rigid cohomology), and of the analogues of lisse étale/Weil sheaves in this theory (convergent and overconvergent F-isocrystals).
• Tomoyuki Abe, Langlands correspondence for isocrystals and existence of crystalline companion for curves: arXiv.
  This paper reproduces L. Lafforgue's work in p-adic cohomology, thus extending the Langlands correspondence for GL(n) to overconvergent F-isocrystals; this fills in more of Deligne's conjecture for curves. As with the paper of Lafforgue, we will mostly treat this paper as a "black box".

Bertini-type results over finite fields

• Nicholas M. Katz, Space filling curves over finite fields: DOI.
  This paper is an early example of a construction that fits smooth curves through points in an ambient space.
• Kiran S. Kedlaya, More étale covers of affine spaces in positive characteristic: link.
  This paper shows that every smooth variety over a finite field is covered by affine opens, each of which is finite étale over an affine space. (This is a purely positive-characteristic phenomenon; in characteristic 0, one has to allow étale morphisms which are not finite.)
• Götz Wiesend, A construction of covers of arithmetic schemes: DOI.
  This paper is referenced by Drinfeld, but for our purposes it is largely supplanted by the paper of Poonen.
• Bjorn Poonen, Bertini theorems over finite fields: DOI.
  This paper introduces a general method for establishing "genericity" statements over finite fields by sieving over sections of large powers of an ample line bundle.
• Alina Bucur and Kiran S. Kedlaya, The probability that a complete intersection is smooth: link.
  This paper extends Poonen's theorem by allowing the intersection of multiple sections of different powers of the same ample line bundle.
• Daniel Erman and Melanie Matchett Wood, Semiample Bertini theorems over finite fields: DOI.
  This paper extends Poonen's theorem by allowing the ample divisor to be replaced with a semiample divisor, at the expense of losing complete independence of the local conditions.

Bounding and counting coefficient objects

• Drinfeld, The number of two-dimensional irreducible representations of the fundamental group of a curve over a finite field: link.
  This brief paper (2 pages) uses the Langlands correspondence for GL(2) to count irreducible two-dimensional étale coefficient objects on a proper curve.
• Deligne, Comptage des faisceaux: open link.
  This paper takes Drinfeld's formula as the starting point to formulate some far-reaching conjectures about counting étale coefficient objects of prescribed shape on a curve.
• Pierre Deligne: Finitude de l'extension de Q engendrée par des traces de Frobenius, en caractéristique finie: link.
  This paper uses the Langlands correspondence to resolve part (ii) of Deligne's conjecture (uniform algebraicity of Frobenius traces).
• Hélène Esnault and Moritz Kerz, A finiteness theorem for Galois representations of function fields over finite fields (after Deligne): open access.
  This paper records an argument of Deligne establishing a finiteness result for étale coefficient objects, which also includes the uniform algebraicity of Frobenius traces.
• H. Yu, Comptage des systèmes locaux l-adiques sur une courbe: arXiv.
  This paper proves one of the conjectures of Deligne about counting local systems, by showing that these counts behave formally like the terms in a Lefschetz trace formula.

Newton polygons and p-adic valuations

• Vincent Lafforgue, Estimées pour les valuations p-adiques des valeurs propres des opérateurs de Hecke: DOI.
  This paper resolves part (iv) of Deligne's conjecture (bounding p-adic valuations of eigenvalues).
• Vladimir Drinfeld and Kiran S. Kedlaya, Slopes of indecomposable $F$-isocrystals: DOI.
  This paper sharpens V. Lafforgue's result using a totally different approach involving Cartier operators.
• Joseph Kramer-Miller, The monodromy of unit-root F-isocrystals with geometric origin: arXiv.
  This paper gives a totally different proof of the theorem of Drinfeld-Kedlaya paper in the case of curves, based on ramification in p-adic towers.

Étale companions of étale/crystalline coefficients

• Vladimir Drinfeld, On a conjecture of Deligne: link.
  This paper resolves part (v) of Deligne's conjecture (existence of étale companions).
• Tomoyuki Abe and Hélène Esnault, A Lefschetz theorem for overconvergent isocrystals with Frobenius structure: arXiv.
  This paper analogizes Drinfeld's result to the case where one starts with a crystalline coefficient object.
• Kiran S. Kedlaya, Étale and crystalline companions, I: arXiv.
  This paper reproves the Abe-Esnault result and includes some relevant associated results.

Vector bundles on curves

• Robin Hartshorne, Ample vector bundles on curves: open link.
  A good reference for some basic facts about vector bundles on curves.
• David Gieseker, Stable vector bundles and the Frobenius morphism: open link.
  This paper introduces some basic examples of "bad behavior" of vector bundles in positive characteristic, including failure of preservation of semistability under tensor product and Frobenius pullback.

Crystalline companions of étale coefficients

• Kiran S. Kedlaya, Étale and crystalline companions, II: pdf.
  This paper resolves part (vi) of Deligne's conjecture (existence of crystalline companions).

Applications of companions

• Nicholas M. Katz, Rigid local systems: link.
  Background for the next item.
• Hélène Esnault and Michael Groechenig, Cohomologically rigid local systems and integrality, arXiv.
  This paper uses the existence of étale companions to prove an integrality statement for the monodromy representation of a rigid local system.
• Raju Krishnamoorthy and Ambrus Pál, Rank 2 local systems and abelian varieties, arXiv.
  This paper uses companions to study a question about geometric origins of coefficient objects, inspired by work of Corlette-Simpson over the complex numbers.