Shtukas and the Langlands program: A bird’s eye view

Cohomology of Drinfeld Modular Varieties provides an introduction, in two volumes, both to this subject and to the Langlands correspondence for function fields. These varieties are the analogues for function fields of the Shimura varieties over number fields. The Langlands correspondence is a conjectured link between automorphic forms and Galois representations over a global field. By analogy with the number-theoretic case, one expects to establish the conjecture for function fields by studying the cohomology of Drinfeld modular varieties, which has been done by Drinfeld himself for the rank two case. This second volume is concerned with the Arthur-Selberg trace formula, and with the proof in some cases of the Rmamanujan-Petersson conjecture and the global Langlands conjecture for function fields. It is based on graduate courses taught by the author, who uses techniques which are extensions of those used to study Shimura varieties. Though the author considers only the simpler case of function rather than number fields, many important features of the number field case can be illustrated. Several appendices on background material keep the work reasonably self-contained. It is the first book on this subject and will be of much interest to all researchers in algebraic number theory and representation theory.

Automorphic forms and Galois representations have played a central role in the development of modern number theory, with the former coming to prominence via the celebrated Langlands program and Wiles' proof of Fermat's Last Theorem. This two-volume collection arose from the 94th LMS-EPSRC Durham Symposium on 'Automorphic Forms and Galois Representations' in July 2011, the aim of which was to explore recent developments in this area. The expository articles and research papers across the two volumes reflect recent interest in p-adic methods in number theory and representation theory, as well as recent progress on topics from anabelian geometry to p-adic Hodge theory and the Langlands program. The topics covered in volume one include the Shafarevich Conjecture, effective local Langlands correspondence, p-adic L-functions, the fundamental lemma, and other topics of contemporary interest.

For cuspidal automorphic forms on function fields and for cohomological cuspidal automorphic forms on number fields we give estimates for the p-adic valuations of eigenvalues of Hecke operators. In the case of number fields, these estimates correspond to Katz-Mazur estimates by the Langlands conjectures.

We prove a companion forms theorem for ordinary n-dimensional automorphic Galois representations, by use of automorphy lifting theorems developed by the second author, and a technique for deducing companion forms theorems due to the first author. We deduce results about the possible Serre weights of mod l Galois representations corresponding to automorphic representations on unitary groups. We then use functoriality to prove similar results for automorphic representations of GSp4 over totally real fields.

Let E / F E/F be a cyclic extension of p p -adic fields and n n a positive integer. Arthur and Clozel constructed a base change process π ↦ π E \pi \mapsto \pi _E which associates to a smooth irreducible representation of G L n ( F ) GL_n(F) a smooth irreducible representation of G L n ( E ) GL_n(E) , invariant under G a l ( E / F ) Gal(E/F) . When π \pi is tempered, π E \pi _E is tempered and is characterized by an identity (the Shintani character relation) relating the character of π \pi to the character of π E \pi _E twisted by the action of G a l ( E / F ) Gal(E/F) . In this paper we show that the Shintani relation also holds when π \pi is unitary or elliptic. We prove similar results for the extension C / R \mathbb {C}/\mathbb {R} . As a corollary we show that for a cyclic extension E / F \mathbf {E}/\mathbf {F} of number fields the base change for automorphic residual representations of G L n ( A F ) GL_n(\mathbb {A}_{\mathbf {F}}) respects the Shintani relation at each place of F \mathbf {F} .

It has been known for many years that the stabilization of the Arthur-Selberg trace formula would, or perhaps we should write will, have important consequences for the Langlands functoriality program as well as for the study of the Galois representations on the l-adic cohomology of Shimura varieties. At present, full stabilization is still only known for SL(2) and U(3) and their inner forms [LL,R]. The automorphic and arithmetic consequences of stabilization for U(3) form the subject of the influential volume [LR]. Under somewhat restrictive hypotheses, one can sometimes derive the expected corollaries of the stable trace formula. Examples of such pseudo-stabilization include Kottwitz' analysis in [K2] of the zeta functions of certain Shimura varieties attached to twisted forms of unitary groups over totally real fields, and the proof in [L1] of stable cyclic base change of automorphic representations which are locally Steinberg at at least two places. These conditional results have been used successfully to provide non-trivial examples of compatible systems of l-adic representations attached to certain classes of automorphic representations of GL(n) [C3], and of non-trivial classes of cohomology of S-arithmetic groups [BLS, L1]. Conditional results also suffice for important local applications, such as the local Langlands conjecture for GL(n) [HT, He]. The present article develops a technique for obtaining conditional base change and functorial transfer. Let Un be a unitary group over a number field F attached to a quadratic extension E/F. The technique applies to quadratic base change from Un to GL(n)E, and to transfer between inner forms of unitary groups. Roughly speaking, if p is an automorphic representation of U which is locally supercuspidal at two places of F split in E, then the expected consequences of the stable trace formula hold for p; in particular p admits a base change to a cuspidal automorphic representation of GL(n)E (Theorem 2.2.2). Slightly more general results are available when F is totally real and E is totally imaginary, and when p is of cohomological type. Automorphic descent from GL(n)E to Un can be proved under analogous hypotheses (Theorem 2.4.1, Theorem 3.1.2). Finally, we prove transfer between distinct inner forms of unitary groups (Jacquet-Langlands transfer) under quite general local hypotheses (Theorem 2.1.2 and, in a more precise form, Theorem 3.1.6 and Proposition 3.1.7). As in [L1], all results are obtained from the simple version of the Arthur-Selberg trace formula, in which non-elliptic and non-cuspidal terms are absent.

This expository article is an introduction to the Langlands functoriality conjectures and their applications to the arithmetic of representations of Galois groups of number fields. Thanks to the work of a great many people, the stable trace formula is now largely established in a version adequate for proving Langlands functoriality in the setting of endoscopy. These developments are discussed in the first two sections of the article. The final section describes the compatible families of l-adic Galois representations that can be attached to automorphic forms with the help of Shimura varieties. To illustrate the relevance of Langlands functoriality to number theory, the article concludes with a description of the Sato–Tate conjecture for elliptic modular forms, recently proved in joint work of Barnet-Lamb, Geraghty, Taylor, and the author.

Let I K { \rm I\!K} be an algebraically closed field of characteristic 0 0 , complete with respect to an ultrametric absolute value | . | |\ . \ | , consider an ”open” disk d ( 0 , R − ) d(0, R^-) of I K { \rm I\!K} , let M ( I K ) {\mathcal M}({ \rm I\!K}) be the field of meromorphic functions on I K { \rm I\!K} and let M u ( d ( 0 , R − ) ) {\mathcal M }_u(d(0,R^-)) be the field of meromorphic functions on d ( 0 , R − ) d(0,R^-) . All results previously proven about the Hayman conjecture on a p p -adic field are recalled, with help of the p p -adic Nevanlinna theory. Let f ∈ M ( I K ) ∖ I K ( x ) f\in {\mathcal M}({ \rm I\!K})\setminus { \rm I\!K}(x) or f ∈ M u ( d ( 0 , R − ) ) f\in {\mathcal M}_u(d(0,R^-)) and let b ∈ I K ∗ b\in { \rm I\!K}^* . For every n ≥ 3 n\geq 3 , f ′ f n − b f’f^n-b admits infinitely many zeros. If n = 2 n=2 and f ∈ M ( I K ) f\in {\mathcal M}({ \rm I\!K}) , we prove that f ′ f 2 − b f’f^2-b admits infinitely many zeros. Let f ∈ M ( I K ) f\in {\mathcal M}({ \rm I\!K}) . Then, f ′ + b f f’+bf has infinitely many zeros that are not zeros of f f and if f f has finitely many multiple zeros and finitely many poles of order 1 1 , then f ′ + b f 2 f’+bf^2 has infinitely many zeros that are not zeros of f f . If its number of poles inside disks d ( 0 , r − ) d(0,r^-) has an upper bound of the form r q r^q with q > 0 q>0 , then f ′ f − b f’f-b has infnitely many zeros again. Moreover, f ′ + b f f’+bf has infinitely many zeros that are not zeros of f f and if f f has finitely many multiple zeros and finitely many poles of order 1 1 , then f ′ + b f 2 f’+bf^2 has infinitely many zeros that are not zeros of f f .

The Langlands Program relates Galois representations and automorphic representations of reductive algebraic groups. The trace formula is a powerful tool in the study of this connection and the Langlands Functoriality Conjecture. After giving an introduction to the Langlands Program and its geometric version, which applies to curves over finite fields and over the complex field, I give a survey of my recent joint work with Robert Langlands and Ngô Bao Châu on a new approach to proving the Functoriality Conjecture using the trace formulas, and on the geometrization of the trace formulas. In particular, I discuss the connection of the latter to the categorification of the Langlands correspondence.

Let $F$ be a $p$--adic field, i.e., a finite extension of $\mathbb Q_p$ for some prime $p$. The local Langlands correspondence attaches to each continuous $n$--dimensional $\Phi$-semisimple representation $\rho$ of $W'_F$, the Weil--Deligne group for $\bar F/F$, an irreducible admissible representation $\pi(\rho)$ of $GL_n(F)$ such that, among other things, the local $L$- and $\varepsilon$-factors of pairs are preserved. This correspondence should be robust and preserve various parallel operations on the arithmetic and analytic sides, such as taking the exterior square or symmetric square. In this paper, we show that this is the case for the local arithmetic and analytic symmetric square and exterior square $\varepsilon$--factors, that is, that $\varepsilon(s,\Lambda^2\rho,\psi)=\varepsilon(s,\pi(\rho),\Lambda^2,\psi)$ and $\varepsilon(s,Sym^2\rho,\psi)=\varepsilon(s,\pi(\rho),Sym^2,\psi)$. The agreement of the $L$-functions also follows by our methods, but this was already known by Henniart. The proof is a robust deformation argument, combined with local/global techniques, which reduces the problem to the stability of the analytic $\gamma$-factor $\gamma(s,\pi,\Lambda^2,\psi)$ under highly ramified twists when $\pi$ is supercuspidal. This last step is achieved by relating the $\gamma$-factor to a Mellin transform of a partial Bessel function attached to the representation and then analyzing the asymptotics of the partial Bessel function, inspired in part by the theory of Shalika germs for Bessel integrals. The stability for every irreducible admissible representation $\pi$ then follows from those of the corresponding arithmetic $\gamma$--factors as a corollary.

We prove the Arithmetic Fundamental Lemma conjecture over a general p -adic field with odd residue cardinality q\geq \dim V . Our strategy is similar to the one used by the second author in his proof of the AFL over \mathbb{Q}_p , but only requires the modularity of divisor generating series on the Shimura variety (as opposed to its integral model). The resulting increase in flexibility allows us to work over an arbitrary base field. To carry out our strategy, we also generalize results of Howard (2012) on CM cycle intersection and of Ehlen–Sankaran (2018) on Green function comparison from \mathbb{Q} to general totally real base fields.

We report on the results and techniques of the author’s recent joint work with Richard Taylor, which analyzes in detail the bad reduction of certain Shimura varieties in order to prove the compatibility of local and global Langlands correspondences, obtaining as a consequence the local Langlands conjecture for GL(n) of a p-adic field. These Shimura varieties have natural models over p-adic integer rings, as moduli spaces for abelian varieties with additional structure. The starting point of the work with Taylor is the stratification of the special fiber of an integral model in minimal level, according to the isogeny type of the universal family of p-divisible groups attached to these abelian varieties Similar stratifications can conjecturally be constructed for any Shimura variety, and indeed are known to exist for most PEL types. We discuss a series of conjectures regarding the behavior of vanishing cycles along these strata, with the aim of extending Kottwitz’ conjectures on the cohomology of Shimura varieties to the case of bad reduction.KeywordsParabolic SubgroupSpecial FiberLevi FactorShimura VarietyDiscrete Series RepresentationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The study of special values of adjoint L-functions and congruence ideals is gradually becoming a classical theme in number theory, driven by the Bloch-Kato conjecture and generalisations of Wiles-Lenstra’s numerical criterion. In this paper, we relate L(1,pi ,{{,textrm{Ad},}}^circ ) to the congruence ideals for cohomological cuspidal automorphic representations pi of textrm{GL}_n over any number field. We then use this result to deduce relationships between the congruences of automorphic forms and adjoint L-functions. For CM fields, using the existence of Galois representations, we apply the result to obtain a lower bound on the cardinality of certain Selmer groups in terms of L(1,pi ,{{,textrm{Ad},}}^circ ). This can be viewed as partial progress on the Bloch-Kato conjecture. The main technical ingredients are a careful study of the cohomology associated with the locally symmetric space of textrm{GL}_n, its relation to automorphic representations, and the establishment of some algebraic properties of the congruence ideals. We anticipate that the methods developed here will find further applications in related problems, particularly in the study of congruence modules and their relation to the arithmetic of automorphic forms.

We investigate sections of the arithmetic fundamental groups of hyperbolic curves over function fields. As a consequence we prove that the anabelian section conjecture of Grothendieck holds over all finitely generated fields over \mathbb Q if it holds over all number fields, under the condition of finiteness (of the \mathcal l -primary parts) of certain Shafarevich–Tate groups. We also prove that if the section conjecture holds over all number fields then it holds over all finitely generated fields for curves which are defined over a number field.

We study the minimal resultant divisor of self-maps of the projective line over a number field or a function field and its relation to the conductor. The guiding focus is the exploration of a dynamical analog to Theorem 1.1, which bounds the degree of the minimal discriminant of an elliptic surface in terms of the conductor. The main theorems of this paper (5.5 and 5.6) establish that, for a degree 2 map, semi-stability in the Geometric Invariant Theory sense on the space of self maps, implies minimality of the resultant. We prove the singular reduction of a semi-stable presentation coincides with the simple bad reduction (Theorem 4.1). Given an elliptic curve over a function field with semi-stable bad reduction, we show the associated Lattes map has unstable bad reduction (Proposition 4.6). Degree 2 maps in normal form with semi-stable bad reduction are used to construct a counterexample (Example 3.1) to a simple dynamical analog to Theorem 1.1.