Automorphic Forms with Anisotropic Periods on a Unitary Group

Let H |r denote the upper half plane H with r additional odd (anticommuting) coordinates. It admits a transitive super action of a certain super Lie group 𝒢. First, we define the spaces of super automorphic and cusp forms on H |r for an ordinary lattice Γ of 𝒢, give an asymptotic formula for their dimensions for high weight and show how to embed Γ \\ H |r into the super projective space with the help of super automorphic forms. For also involving the odd directions of 𝒢 we introduce local super deformation of lattices in 𝒢 and show that for high weight the spaces of super automorphic and cusp forms are stable under such local super deformations.

The Tate conjecture on algebraic cycles is proved for any algebraic K3 surface over a number field. If the canonical representation of the Hodge group in the Q-lattice of transcendental cohomology classes is absolutely irreducible, then the Mumford-Tate conjecture is true for such a K3 surface.

L-packets for the quasi-split unitary group in three variables U(3). We shall give an explicit parametrization of these L-packets using theta liftings and describe some relations between the structure of L-packets and certain period integrals. We also prove that an endoscopic L-packet contains a unique generic representation in local and global cases. Suppose that G is a reductive algebraic group over a local or global field

The paper (Jiang, Automorphic forms: L-functions and related geometry: assessing the legacy of I.I. Piatetski-Shapiro. Contemporary mathematics, vol 614. American Mathematical Society, Providence, RI, 2014, pp 179–242) forms Part I of the theory of Automorphic Integral Transforms for Classical Groups, where the first named author made a conjecture on how the global Arthur parameters may govern the structure of the Fourier coefficients of the automorphic representations in the corresponding global Arthur packets. This leads to a better understanding of the automorphic kernel functions with which the integral transforms yield conjecturally the endoscopic correspondences for classical groups. In this paper, we discuss the Twisted Automorphic Descent method and its variants that construct concrete modules for irreducible cuspidal automorphic representations of general classical groups. When the global Arthur parameters are generic, the details of the theory are referred to Jiang et al. (2015, accepted by IMRN), Jiang and Zhang (2015, submitted; 2015, in preparation), which extend the automorphic descent method of Ginzburg-Rallis-Soudry (The descent map from automorphic representations of GL(n) to classical groups. World Scientific, Singapore, 2011) to great generality.

The Schur–Weyl construction of irreducible tensors by symmetrized Lth rank exterior products of a defining n-dimensional vector space establishes a duality between the coupling algebra of the unitary group Un and the symmetric group SL . The coupling coefficients shared by these two groups are real and possess symmetries due to complex conjugation in Un and association in SL . The unitary group is factored by its unitary unimodular subgroup SUn≂Un/U1 so it is usual to identify the coupling algebras of Un and SUn by choosing the trivial (phaseless) coupling for U1. This establishes a set of equivalence relations since all pseudoscalar irreducible representations (irreps) of Un subduce onto the scalar irrep of SUn. The question remains can this trivial identification with the coupling algebra of SUn be made consistent with the symmetries which follow from duality? Specifically the symmetries which follow from duality must be consistent with the equivalence relations. We argue such consistency may be obtained. The resulting symmetries place significant restrictions on the resolution of multiplicities and the choice of phase conventions for transformations which in the absence of duality are normally considered independently. We review the usual formulation of a coupling algebra and the identifications which follow from duality using a double coset decomposition of SL . Particular attention is given to the phase transformation introduced by a transposition in order of the two component irreps being coupled. Phase transformations needed for association and complex conjugation are examined to determine the symmetries they must exhibit to be consistent with duality. Two phase transformations needed for complex conjugation, the Derome–Sharp matrix and the 1jm factor, are shown to be related by association. A transposition phase convention and an association phase convention are proposed and shown to be consistent with all the symmetries required by duality.

Une suite spectrale de Hochschild–Serre pour l'uniformisation de Rapoport–Zink

Let E / F {E/F} be a quadratic extension of non-archimedean local fields of characteristic 0. In this paper, we investigate two approaches which attempt to describe the irreducible smooth representations of GL n ⁢ ( E ) {\mathrm{GL}_{n}(E)} that are distinguished by its subgroup GL n ⁢ ( F ) {\mathrm{GL}_{n}(F)} . One relates this class to representations which come as base change lifts from a quasi-split unitary group over F, while another deals with a certain symmetry condition. By characterizing the union of images of the base change maps, we show that these two approaches are closely related. Using this observation, we are able to prove a statement relating base change and distinction for ladder representations. We then produce a wide family of examples in which the symmetry condition does not impose GL n ⁢ ( F ) {\mathrm{GL}_{n}(F)} -distinction, and thus exhibit the limitations of these two approaches.

We present the basics of the local theory, which arises from global Rankin-Selberg integrals, attached to pairs of irreducible, automorphic, cuspidal and globally generic representations of U 2 k ( A F ) × G L m ( A E ) \mathrm {U}_{2k}(\mathbb {A}_F)\times \mathrm {GL}_m(\mathbb {A}_E) , k > m k>m , where U 2 k \mathrm {U}_{2k} is the quasi-split unitary group in 2 k 2k variables, attached to a quadratic extension of number fields E / F E/F .

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.

We extend the methods of Wiles and of Taylor and Wiles from GL 2 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 GL 2 . 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.

In this paper we explore some properties of periods attached to automorphic representations of unitary groups over CM fields and the critical values of their L -functions. We prove a formula expressing the critical values in the range of absolute convergence in terms of Petersson norms of holomorphic automorphic forms. On the other hand, we express the Deligne period of a related motive as a product of quadratic periods and compare the two expressions by means of Deligne's conjecture.

Refined forms of the local Langlands correspondence seek to relate representations of reductive groups over local fields with sheaves on stacks of Langlands parameters. But what kind of sheaves? Conjectures in the spirit of Kazhdan-Lusztig theory (due to Vogan and Soergel) describe representations of a group and its pure inner forms with fixed central character in terms of constructible sheaves. Conjectures in the spirit of geometric Langlands (due to Fargues, Zhu and Hellmann) describe representations with varying central character of a large family of groups associated to isocrystals in terms of coherent sheaves. The latter conjectures also take place on a larger parameter space, in which Frobenius (or complex conjugation) is allowed a unipotent part. In this article we propose a general mechanism that interpolates between these two settings. This mechanism derives from the theory of cyclic homology, as interpreted through circle actions in derived algebraic geometry. We apply this perspective to categorical forms of the local Langlands conjectures for both archimedean and non-archimedean local fields. In the nonarchimedean case, we describe how circle actions relate coherent and constructible realizations of affine Hecke algebras and of all smooth representations of G L n GL_n , and propose a mechanism to relate the two settings in general. In the archimedean case, we explain how to use circle actions to derive the constructible local Langlands correspondence (in the form due to Adams-Barbasch-Vogan and Soergel) from a coherent form (a real counterpart to Fargues’ conjecture): the tamely ramified geometric Langlands conjecture on the twistor line, which we survey.

<p>Consider either (i) <em>f </em>= <em>f</em><sub>1 </sub>⊠ <em>f</em><sub>2 </sub>for two Maass cusp forms for SL<sub>m</sub>(ℤ) and SL<sub>m′</sub>(ℤ), respectively, with 2 ≤ m ≤ m′, or (ii) <em>f</em>= <em>f</em><sub>1 </sub>⊠ <em>f</em><sub>2 </sub>⊠ <em>f</em><sub>3 </sub>for three weight 2<em>k</em> holomorphic cusp forms for SL<sub>2</sub>(ℤ). Let λ<sub>f</sub>(n) be the normalized coefficients of the associated L-function L(s, <em>f</em>), which is either (i) the Rankin-Selberg L-function L(s, <em>f</em>1 ×<em>f</em>2), or (ii) the Rankin triple product L-function L(s, <em>f</em>1 ×<em>f</em>2 ×<em>f</em>3). First, we derive a Voronoi-type summation formula for λ<sub><em>f</em></sub> (n) involving the Meijer G-function. As an application we obtain the asymptotics for the smoothly weighted average of λ<sub><em>f</em></sub> (n) against <em>e</em>(αn<sup>β</sup>), i.e. the asymptotics for the associated resonance sums. Let ℓ be the degree of L(s, <em>f</em>). When β = 1/ℓ and α is close or equal to ±ℓq 1/ℓ for a positive integer <em>q</em>, the average has a main term of size |λ<sub><em>f</em></sub> (<em>q</em>)|X <sup>1/2ℓ+1/2</sup> . Otherwise, when α is fixed and 0 < β < 1/ℓ it is shown that this average decays rapidly. Similar results have been established for individual SL<sub>m</sub>(ℤ) automorphic cusp forms and are due to the oscillatory nature of the coefficients λ<sub><em>f</em></sub> (n).</p>

These are the notes of a five-lecture course presented at the Computations with Modular Forms summer school, aimed at graduate students in number theory and related areas. Sections 1–4 give a sketch of the theory of reductive algebraic groups over Q, and of Gross’s purely algebraic definition of automorphic forms in the special case when G(R) is compact. Sections 5–9 describe how these automorphic forms can be explicitly computed, concentrating on the case of definite unitary groups; and Sects. 10 and 11 describe how to relate the results of these computations to Galois representations, and present some examples where the corresponding Galois representations can be identified, giving illustrations of various instances of Langlands’ functoriality conjectures.KeywordsModular FormAlgebraic GroupReductive GroupCompact SubgroupAutomorphic FormThese 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.

In this paper we prove the functoriality of the exterior square of cusp forms on G L 4 GL_{4} as automorphic forms on G L 6 GL_{6} and the symmetric fourth of cusp forms on G L 2 GL_{2} as automorphic forms on G L 5 GL_{5} . We prove these by applying a converse theorem of Cogdell and Piatetski-Shapiro to analytic properties of certain L L -functions obtained by the Langlands-Shahidi method. We give several applications: First, we prove the weak Ramanujan property of cuspidal representations of G L 4 GL_{4} and the absolute convergence of the exterior square L L -functions of G L 4 GL_{4} . Second, we prove that the fourth symmetric power L L -functions of cuspidal representations of G L 2 GL_{2} are entire, except for those of dihedral and tetrahedral type. Third, we prove the bound 3 26 \frac {3}{26} for Hecke eigenvalues of Maass forms over any number field.