Automorphic lifts of prescribed types

We prove a variety of results on the existence of automorphic Galois representations lifting a residual automorphic Galois representation. We prove a result on the structure of deformation rings of local Galois representations, and deduce from this and the method of Khare and Wintenberger a result on the existence of modular lifts of specified type for Galois representations corresponding to Hilbert modular forms of parallel weight 2. We discuss some conjectures on the weights of n-dimensional mod p Galois representations. Finally, we use recent work of Taylor to prove level raising and lowering results for n-dimensional automorphic Galois representations.     

Modular forms

In this survey there are included results of recent years, concerning the theory of modular forms and representations connected with them of adele groups and Galois groups. There is discussed the hypothetical principle of functoriality of automorphic forms and other conjectures of Langlands concerning automorphic forms and the L-functions connected with them.     

Crystalline liftings and the weight part of Serre’s conjecture

We prove some new cases of the weight part of Serre’s conjectures for mod p Galois representation associated to automorphic representations on unitary groups U(d). The approach is a generalization of the work of Gee–Liu–Savitt, namely, we study reductions of certain crystalline representations, as well as crystalline lifts of these reductions.     

Crystalline cohomology and GL(2, ℚ)

This paper applies recent advances in crystalline cohomology to the classical case of open elliptic modular curves. In so doing control is gained over the action of inertia in the Galois representations attached to modular forms. Our aim is to study the modular Galois representations attached to automorphic forms modp of weightk≥2. We generalize to higher weightk several results which were previously accessible only in the case of weight 2 where jacobian varieties can be invoked. Additionally we reconsider Gross’s theorem on companion forms in a crystalline context. Partially supported by NSF grant DMS 90-02744. Partially supported by NSA grant MDA904-90-H-4020 and by a PSC-CUNY grant.     

Epipelagic representations and rigid local systems

We construct automorphic representations for quasi-split groups G over the function field \(F=k(t)\) one of whose local components is an epipelagic representation in the sense of Reeder and Yu. We also construct the attached Galois representations under the Langlands correspondence. These Galois representations give new classes of conjecturally rigid, wildly ramified \({}^{L}{G}\)-local systems over \(\mathbb {P}^{1}-\{0,\infty \}\) that generalize the Kloosterman sheaves constructed earlier by Heinloth, Ngô and the author. We study the monodromy of these local systems and compute all examples when G is a classical group.     

Automorphy for some l-adic lifts of automorphic mod l Galois representations

We extend the methods of Wiles and of Taylor and Wiles from GL₂ to higher rank unitary groups and establish the automorphy of suitable conjugate self-dual, regular (de Rham with distinct Hodge–Tate numbers), minimally ramified, l-adic lifts of certain automorphic mod l Galois representations of any dimension. We also make a conjecture about the structure of mod l automorphic forms on definite unitary groups, which would generalise a lemma of Ihara for GL₂. Following Wiles’ method we show that this conjecture implies that our automorphy lifting theorem could be extended to cover lifts that are not minimally ramified.     

Central Pairs, Galois Theory and Automorphic Forms

A central pair over a field k of characteristic 0 consists of a finite Abelian group which is equipped with a central 2-cocycle with values in the multiplicative group k ^* of k. In this paper we use specific central pairs to construct a class of projective representations of the absolute Galois group G _k of k and if k is a number field we investigate the liftings of these projective representations to linear representations of G _k . In particular we relate these linear representations to automorphic representations. It turns out that some of these automorphic representations correspond to certain indefinite modular forms already constructed by E. Hecke.     

The Artin conjecture for some S_5-extensions

We establish some new cases of Artin’s conjecture. Our results apply to Galois representations over $\mathbf{Q }$ with image $S_5$ satisfying certain local hypotheses, the most important of which is that complex conjugation is conjugate to $(12)(34)$ . In fact, we prove the stronger claim conjectured by Langlands that these representations are automorphic. For the irreducible representations of dimensions 4 and 6, our result follows from known 2-dimensional cases of Artin’s conjecture (proved by Sasaki) as well as the functorial properties of the Asai transfer proved by Ramakrishnan. For the irreducible representations of dimension 5, we encounter the problem of descending an automorphic form from a quadratic extension compatibly with the Galois representation. This problem is partly solved by working instead with a four dimensional representation of some central extension of $S_5$ . Our modularity results in this case are contingent on the non-vanishing of a certain Dedekind zeta function on the real line in the critical strip. A result of Booker show that one can (in principle) explicitly verify this non-vanishing, and with Booker’s help we give an example, verifying Artin’s conjecture for representations coming from the (Galois closure) of the quintic field $K$ of smallest discriminant (1609).     

Automorphy for some <Emphasis Type="Italic">l</Emphasis>-adic lifts of automorphic mod <Emphasis Type="Italic">l</Emphasis> Galois representations. II

We extend the results of [CHT] by removing the ‘minimal ramification’ condition on the lifts. That is we establish the automorphy of suitable conjugate self-dual, regular (de Rham with distinct Hodge–Tate numbers), l-adic lifts of certain automorphic mod l Galois representations of any dimension. The main innovation is a new approach to the automorphy of non-minimal lifts which is closer in spirit to the methods of [TW] than to those of [W], which relied on Ihara’s lemma.     

Automorphic equivalence in the varieties of representations of Lie algebras

Abstract

In this paper, we consider the wide class of subvarieties of the variety of all representation of Lie algebras over a field k of characteristic 0. We study the relation between the geometric equivalence and automorphic equivalence of the representations from these subvarieties.     

Finite-Dimensional Representations of Hyper Loop Algebras over Non-algebraically Closed Fields

We study finite-dimensional representations of hyper loop algebras over non-algebraically closed fields. The main results concern the classification of the irreducible representations, the construction of the Weyl modules, base change, tensor products of irreducible and Weyl modules, and the block decomposition of the underlying abelian category. Several results are interestingly related to the study of irreducible representations of polynomial algebras and Galois theory.     

A connectedness criterion for ℓ-adic galois representations-adic galois representations

To every compatible system of Galois representations of a global fieldK, there is associated a natural invariantK ^conn, the smallest extension ofK over which the associated algebraic monodromy groups become connected. We present a purely field-theoretic construction ofK ^conn for all Galois representations arising from cohomology. Partially supported by the Sloan Foundation and by NSF Grant DMS94-00833.     

On Potentially Abelian Geometric Representations

The finiteness is proved of the set of isomorphism classes of potentially abelian geometric Galois representations with a given set of data. This is a special case of the finiteness conjecture of Fontaine and Mazur.     

Kontsevich–Zagier integrals for automorphic Green’s functions. II

We introduce interaction entropies, which can be represented as logarithmic couplings of certain cycles on a class of algebraic curves of arithmetic interest. In particular, via interaction entropies for Legendre–Ramanujan curves \( Y^n=(1-X)^{n-1}X(1-\alpha X)\) (\( n\in \{6,4,3,2\}\)), we reformulate the Kontsevich–Zagier integral representations of weight-4 automorphic Green’s functions \( G_2^{\mathfrak H/\overline{\varGamma }_0(N)}(z_1,z_2)\) (\(N=4\sin ^2(\pi /n )\in \{1,2,3,4\}\)), in a geometric context. These geometric entropies allow us to establish algebraic relations between certain weight-4 automorphic self-energies and special values of weight-6 automorphic Green’s functions.     

Galois representations attached to representations of GU(3)

For a fixed prime we compute the -adic Lie algebra of the image of the -adic Galois representation attached to a stable cuspidal automorphic representation of the unitary similitude group GU(3). This result depends on whether admits extra twists in the sense defined below. Two cases emerge: orthogonal image and non-orthogonal image. We show that in the orthogonal case there exists a character such that is the Galois representation attached to the unitary adjoint lift of a cuspidal representation of GL(2). Received: 22 November 1999 / Accepted: 25 July 2000 / Published online: 23 July 2001     

On CAP Representations for Even Orthogonal Groups Ⅰ:A Correspondence of Unramified Representations

The authors prove the local unramified correspondence for a new type of construction of CAP representations of even orthogonal groups by a generalized automorphic descent method. This method is expected to work for all classical groups.     

On CAP Representations for Even Orthogonal Groups I:A Correspondence of Unramified Representations

The authors prove the local unramified correspondence for a new type of construction of CAP representations of even orthogonal groups by a generalized automorphic descent method. This method is expected to work for all classical groups.     

ℒ-invariants and logarithm derivatives of eigenvalues of Frobenius

Let K be a p-adic local field. We study a special kind of p-adic Galois representations of it. These representations are similar to the Galois representations occurred in the exceptional zero conjecture for modular forms. In particular, we verify that a formula of Colmez can be generalized to our case. We also include a degenerated version of Colmez’s formula.