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.     

On the modularity of the GL<Subscript>2</Subscript>-twisted spinor L-function

In modern number theory there are famous theorems on the modularity of Dirichlet series attached to geometric or arithmetic objects. There is Hecke’s converse theorem, Wiles proof of the Taniyama-Shimura conjecture, and Fermat’s Last Theorem to name a few. In this article in the spirit of the Langlands philosophy we raise the question on the modularity of the GL₂-twisted spinor L-function Z _{G ⊗ h} (s) related to automorphic forms G,h on the symplectic group GSp₂ and GL₂. This leads to several promising results and finally culminates into a precise very general conjecture. This gives new insights into the Miyawaki conjecture on spinor L-functions of modular forms. We indicate how this topic is related to Ramakrishnan’s work on the modularity of the Rankin-Selberg L-series.     

Langlands-shahidi method and poles of automorphicL-functions II

We use Langlands-Shahidi method and the observation that the local components of residual automorphic representations are unitary representations, to study the Rankin-SelbergL-functions of GL _k × classical groups. Especially we prove thatL(s, σ ×τ) is holomorphic, except possibly ats=0, 1/2, 1, whereσ is a cuspidal representation of GL _k which satisfies weak Ramanujan property in the sense of Cogdell and Piatetski-Shapiro andτ is any generic cuspidal representation of SO_2l+1. Also we study the twisted symmetric cubeL-functions, twisted by cuspidal representations of GL₂. Partially supported by NSF grant DMS9610387.     

Global Jacquet–Langlands correspondence,multiplicity one and classification of automorphic representations

In this paper we generalize the local Jacquet-Langlands correspondence to all unitary irreducible representations. We prove the global Jacquet-Langlands correspondence in characteristic zero. As consequences we obtain the multiplicity one and strong multiplicity one Theorems for inner forms of GL(n) as well as a classification of the residual spectrum and automorphic representations in analogy with results proved by Mœglin–Waldspurger and Jacquet–Shalika for GL(n).     

A proof of Selberg's orthogonality for automorphic L-functions

Let π and π′ be automorphic irreducible cuspidal representations of GL_m(Q_A) and GL_m′(Q_A), respectively. Assume that π and π′ are unitary and at least one of them is self-contragredient. In this article we will give an unconditional proof of an orthogonality for π and π′, weighted by the von Mangoldt function Λ(n) and 1−n/x. We then remove the weighting factor 1−n/x and prove the Selberg orthogonality conjecture for automorphic L-functions L(s,π) and L(s,π′), unconditionally for m≤4 and m′≤4, and under the Hypothesis H of Rudnick and Sarnak [20] in other cases. This proof of Selberg's orthogonality removes such an assumption in the computation of superposition distribution of normalized nontrivial zeros of distinct automorphic L-functions by Liu and Ye [12].     

Local Shalika models and functoriality

We prove, over a p-adic local field F, that an irreducible supercuspidal representation of GL_2n (F) is a local Langlands functorial transfer from SO_2n+1(F) if and only if it has a nonzero Shalika model (Corollary 5.2, Proposition 5.4 and Theorem 5.5). Based on this, we verify (Sect. 6) in our cases a conjecture of Jacquet and Martin, a conjecture of Kim, and a conjecture of Speh in the theory of automorphic forms.     

A spectral identity between symmetric spaces

Let π be a cuspidal automorphic representation ofGL _2n . We prove an identity between two spectral distributions onSp _2n andGL _2n respectively. The first is the spherical distribution with respect toSp _n×Sp _nof the residual Eisenstein series induced from π. The second is the weighted spherical distribution of π with respect toGL _n×GL _nand a certain degenerate Eisenstein series. A similar identity relates the pair (U _2n ,Sp _n) and (GL _n/E,GL _n/F) whereE/F is the quadratic extension defining the quasi-split unitary groupU _2n . We also have a Whittaker version of these trace identities. First-named author partially supported by NSF grant DMS 0070611. Second-named author partially supported by NSF grant DMS 9970342.     

Bessel functions for GL 2

In classical analytic number theory there are several trace formulas or summation formulas for modular forms that involve integral transformations of test functions against classical Bessel functions. Two prominent such are the Kuznetsov trace formula and the Voronoi summation formula. With the paradigm shift from classical automorphic forms to automorphic representations, one is led to ask whether the Bessel functions that arise in the classical summation formulas have a representation theoretic interpretation. We introduce Bessel functions for representations of GL ₂ over a finite field first to develop their formal properties and introduce the idea that the γ-factor that appears in local functional equations for L-functions should be the Mellin transform of a Bessel function. We then proceed to Bessel functions for representations of GL ₂(?) and explain their occurrence in the Voronoi summation formula from this point of view. We briefly discuss Bessel functions for GL ₂ over a p-adic field and the relation between γ-factors and Bessel functions in that context. We conclude with a brief discussion of Bessel functions for other groups and their application to the question of stability of γ-factors under highly ramified twists.     

On De Jong’s conjecture

Let X be a smooth projective curve over a finite field F _q . Let ρ be a continuous representation π(X) → GL _n (F), where F = F _l ((t)) with F _l being another finite field of order prime to q. Assume that is irreducible. De Jong’s conjecture says that in this case 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.     

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.     

Fractional Moments of Automorphic L-Functions on GL（m）

Let π be an irreducible unitary cuspidal representation of GLm(AQ), m ≥ 2. Assume that π is self-contragredient. The author gets upper and lower bounds of the same order for fractional moments of automorphic L-function L(s, π) on the critical line under Generalized Ramanujan Conjecture; the upper bound being conditionally subject to the truth of Generalized Riemann Hypothesis.     

Overgroups of Unitary Groups

This paper classifies under a local stable rank condition for rings with form parameter, subgroups of the general linear group GL _2l which contain the elementary unitary subgroup EU _2l .     

Distinction by the quasi-split unitary group

In earlier work, we proved that any quadratic base change automorphic cuspidal representation of GL(n) is distinguished by a unitary group. Here we prove that we can take the unitary group to be quasi-split     

The image of Colmez’s Montreal functor

We prove a conjecture of Colmez concerning the reduction modulo p of invariant lattices in irreducible admissible unitary p-adic Banach space representations of GL₂(Q _{?? p} ) with p≥5. This enables us to restate nicely the p-adic local Langlands correspondence for GL₂(Q _{?? p} ) and deduce a conjecture of Breuil on irreducible admissible unitary completions of locally algebraic representations.     

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.     

Normalized local theta correspondence and the duality of inner product formulas

We conjecture that local theta correspondence can be normalized by the leading coefficient of a weighted local period integral, and that there exists a duality of local and global inner product formulas. The conjecture is verified for the pair (, PGL ₂) and (SL ₂, SO(2, 2)). As an application, global inner product formulas are obtained for liftings in the directions PGL ₂ → , GSO(2, 2) → GL ₂.     

Fourier–Jacobi periods and the central value of Rankin–Selberg L-functions

In this paper, we propose a conjectural formula, relating the Fourier–Jacobi periods of automorphic forms on U(n)×U(n) and the central value of some Rankin–Selberg L-function. This can be viewed as a refinement of the Gan–Gross–Prasad conjecture for unitary groups. We then use the relative trace formula technique to prove this conjectural formula in some cases. We also have give applications to the conjecture of Ichino–Ikeda and N. Harris on the Bessel period of automorphic forms on unitary groups.     

On symplectic periods of the discrete spectrum ofGL 2n

We provide a formula for the symplectic period of an automorphic form in the discrete spectrum ofGL _2n.It is a generalization of a formula of Jacquet and Rallis.