Fourier Coefficients Of Automorphic Forms Research Articles

A reduction principle for Fourier coefficients of automorphic forms

We consider a special class of unipotent periods for automorphic forms on a finite cover of a reductive adelic group mathbf {G}(mathbb {A}_mathbb {K}), which we refer to as Fourier coefficients associated to the data of a ‘Whittaker pair’. We describe a quasi-order on Fourier coefficients, and an algorithm that gives an explicit formula for any coefficient in terms of integrals and sums involving higher coefficients. The maximal elements for the quasi-order are ‘Levi-distinguished’ Fourier coefficients, which correspond to taking the constant term along the unipotent radical of a parabolic subgroup, and then further taking a Fourier coefficient with respect to a {mathbb K}-distinguished nilpotent orbit in the Levi quotient. Thus one can express any Fourier coefficient, including the form itself, in terms of higher Levi-distinguished coefficients. In companion papers we use this result to determine explicit Fourier expansions of minimal and next-to-minimal automorphic forms on split simply-laced reductive groups, and to obtain Euler product decompositions of certain Fourier coefficients.

Eulerianity of Fourier coefficients of automorphic forms

We study the question of Eulerianity (factorizability) for Fourier coefficients of automorphic forms, and we prove a general transfer theorem that allows one to deduce the Eulerianity of certain coefficients from that of another coefficient. We also establish a ‘hidden’ invariance property of Fourier coefficients. We apply these results to minimal and next-to-minimal automorphic representations, and deduce Eulerianity for a large class of Fourier and Fourier–Jacobi coefficients. In particular, we prove Eulerianity for parabolic Fourier coefficients with characters of maximal rank for a class of Eisenstein series in minimal and next-to-minimal representations of groups of ADE-type that are of interest in string theory.

Fourier coefficients of minimal and next-to-minimal automorphic representations of simply-laced groups

AbstractIn this paper, we analyze Fourier coefficients of automorphic forms on a finite cover G of an adelic split simply-laced group. Let $\pi $ be a minimal or next-to-minimal automorphic representation of G. We prove that any $\eta \in \pi $ is completely determined by its Whittaker coefficients with respect to (possibly degenerate) characters of the unipotent radical of a fixed Borel subgroup, analogously to the Piatetski-Shapiro–Shalika formula for cusp forms on $\operatorname {GL}_n$ . We also derive explicit formulas expressing the form, as well as all its maximal parabolic Fourier coefficient, in terms of these Whittaker coefficients. A consequence of our results is the nonexistence of cusp forms in the minimal and next-to-minimal automorphic spectrum. We provide detailed examples for G of type $D_5$ and $E_8$ with a view toward applications to scattering amplitudes in string theory.

Fourier coefficients attached to small automorphic representations of [formula omitted

We show that Fourier coefficients of automorphic forms attached to minimal or next-to-minimal automorphic representations of SLn(A) are completely determined by certain highly degenerate Whittaker coefficients. We give an explicit formula for the Fourier expansion, analogously to the Piatetski-Shapiro–Shalika formula. In addition, we derive expressions for Fourier coefficients associated to all maximal parabolic subgroups. These results have potential applications for scattering amplitudes in string theory.

Exponential sums with Fourier coefficients of automorphic forms

Let $$\pi _1,\pi _2, \ldots , \pi _k$$ be cuspidal automorphic representations of $$\mathrm{GL}(2)$$ or $$\mathrm{GL}(3)$$ with trivial conductor and trivial central character, and $$\pi :=\pi _1\boxplus \pi _2\boxplus \cdots \boxplus \pi _k$$ be the isobaric representation associated to them. We establish that there exists a positive constant $$\vartheta _k <1$$ such that for any $$\alpha \in {\mathbb {R}}$$ , any $$x \ge 1$$ and any positive $$\varepsilon $$ , one has $$\begin{aligned} \sum _{n \le x} \lambda _{\pi _1\boxplus \pi _2\boxplus \cdots \boxplus \pi _k}(n)\mathrm{e}(\alpha n) \ll _{\pi _j,\varepsilon } x^{\vartheta _k+\varepsilon }, \end{aligned}$$ where the implied constant does not depend on $$\alpha $$ . We also consider some isobaric representations containing the trivial representation. As an application, we consider averages of shifted convolution sums of the type $$\begin{aligned} \sum _{h=1}^{H}\sum _{n=1}^{X}a(n)\lambda _{\pi _1\boxplus \pi _2\boxplus \cdots \boxplus \pi _k}(n+h), \end{aligned}$$ where a(n) is any arithmetic function.

On an analogue of the Ichino–Ikeda conjecture for Whittaker coefficients on the metaplectic group

In previous papers we formulated an analogue of the Ichino--Ikeda conjectures for Whittaker--Fourier coefficients of automorphic forms on classical group and the metaplectic group. In the latter case we reduced the conjecture to a local identity. In this paper we will prove the local identity in the $p$-adic case, and hence the global conjecture under simplifying conditions at the archimedean places.

Small automorphic representations and degenerate Whittaker vectors

We investigate Fourier coefficients of automorphic forms on split simply-laced Lie groups G. We show that for automorphic representations of small Gelfand–Kirillov dimension the Fourier coefficients are completely determined by certain degenerate Whittaker vectors on G. Although we expect our results to hold for arbitrary simply-laced groups, we give complete proofs only for G=SL(3) and G=SL(4). This is based on a method of Ginzburg that associates Fourier coefficients of automorphic forms with nilpotent orbits of G. Our results complement and extend recent results of Miller and Sahi. We also use our formalism to calculate various local (real and p-adic) spherical vectors of minimal representations of the exceptional groups E6, E7, E8 using global (adelic) degenerate Whittaker vectors, correctly reproducing existing results for such spherical vectors obtained by very different methods.

Fourier coefficients of automorphic forms and integrable discrete series

Let G be the group of R-points of a semisimple algebraic group G defined over Q. Assume that G is connected and noncompact. We study Fourier coefficients of Poincaré series attached to matrix coefficients of integrable discrete series. We use these results to construct explicit automorphic cuspidal realizations, which have appropriate Fourier coefficients ≠0, of integrable discrete series in families of congruence subgroups. In the case of G=Sp2n(R), we relate our work to that of Li [14]. For G quasi-split over Q, we relate our work to the result about Poincaré series due to Khare, Larsen, and Savin [12].

Arthur Parameters and Fourier coefficients for Automorphic Forms on Symplectic Groups

We study the structures of Fourier coefficients of automorphic forms on symplectic groups based on their local and global structures related to Arthur parameters. This is a first step towards the general conjecture on the relation between the structure of Fourier coefficients and Arthur parameters given by the first named author in [J14].

IMPROPER INTERSECTIONS OF KUDLA–RAPOPORT DIVISORS AND EISENSTEIN SERIES

We consider a certain family of Kudla–Rapoport cycles on an integral model of a Shimura variety attached to a unitary group of signature (1, 1), and prove that the arithmetic degrees of these cycles are Fourier coefficients of the central derivative of an Eisenstein series of genus 2. The integral model in question parameterizes abelian surfaces equipped with a non-principal polarization and an action of an imaginary quadratic number ring, and in this setting the cycles are degenerate: they may contain components of positive dimension. This result can be viewed as confirmation, in the degenerate setting and for dimension 2, of conjectures of Kudla and Kudla–Rapoport that predict relations between the intersection numbers of special cycles and the Fourier coefficients of automorphic forms.

Corrigendum: central limit theorems for classical cusp forms

A correction is made to the renormalising constants in an example relating the classical theorem of Erdős-Kac to the Fourier coefficients of automorphic forms.

Fourier coefficients of automorphic forms, character variety orbits, and small representations

We consider the Fourier expansions of automorphic forms on general Lie groups, with a particular emphasis on exceptional groups. After describing some principles underlying known results on GL ( n ) , Sp ( 4 ) , and G 2 , we perform an analysis of the expansions on split real forms of E 6 and E 7 where simplifications take place for automorphic realizations of real representations which have small Gelfand–Kirillov dimension. Though the character varieties are more complicated for exceptional groups, we explain how the nonvanishing Fourier coefficients for small representations behave analogously to Fourier coefficients on GL ( n ) . We use this mechanism, for example, to show that the minimal representation of either E 6 or E 7 never occurs in the cuspidal automorphic spectrum. We also give a complete description of the internal Chevalley modules of all complex Chevalley groups – that is, the orbit decomposition of the Levi factor of a maximal parabolic on its unipotent radical. This generalizes classical results on trivectors and in particular includes a full description of the complex character variety orbits for all maximal parabolics. The results of this paper have been applied in the string theory literature to the study of BPS instanton contributions to graviton scattering (Green et al., 2011, [12] ). For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=ELkyOT8c28I . Author Video Watch what authors say about their articles

Sparse equidistribution problems, period bounds and subconvexity

We introduce a geometric method to bound periods of automorphic forms. The key features of this method are the use of equidistribution results in place of mean value theorems, and the systematic use of mixing and the spectral gap. Applications are given to equidistribution of sparse subsets of horocycles and to equidistribution of CM points; to subconvexity of the triple product period in the level aspect over number fields, which implies subconvexity for certain standard and Rankin-Selberg L-functions; and to bounding Fourier coefficients of automorphic forms.

Lattice points in rational ellipsoids

We combine exponential sums, character sums and Fourier coefficients of automorphic forms to improve the best known upper bound for the lattice error term associated to rational ellipsoids.

On Fourier coefficients of automorphic forms of symplectic groups

In this paper we study certain properties of Fourier coefficients of cuspidal representations on symplectic groups. We prove that every cuspidal representation has a nontrivial Fourier coefficient with respect to a certain type of unipotent class.

Euler products and Fourier coefficients of automorphic forms on symplectic groups

Euler products and Fourier coefficients of automorphic forms on symplectic groups

Special Functions for the Symmetric Space of Positive Matrices

Section 1 of the paper gives three applications of a basic principle that eigenfunctions of invariant differential operators are eigenfunetions of invariant integral operators. The first application is a derivation of a noneuclidean analogue of the Poisson summation formula, the second is the evaluation of an integral of Muirhead which has been of interest in multivariate statistics, the third is the evaluation of gammaa type integrals arising in the theory of Eisenstein series for the general linear group. Section 2 of the paper concerns K-Bessel functions for the general linear group. It relates such functions with gamma functions for the symplectic group and applies the theory to the estimation of Fourier coefficients of automorphic forms for ${\operatorname{GL}}(3,\mathbb{Z})$.

On the Fourier coefficients of automorphic forms at various cusps and some applications to Rankin's convolution

On the Fourier coefficients of automorphic forms at various cusps and some applications to Rankin's convolution

Magnitude of the Fourier coefficients of automorphic forms of negative dimension

Magnitude of the Fourier coefficients of automorphic forms of negative dimension

The Fourier coefficients of automorphic forms of horocyclic groups. III.

The Fourier coefficients of automorphic forms of horocyclic groups. III.