• Corpus ID: 14138370

Notes on the Generalized Ramanujan Conjectures

  title={Notes on the Generalized Ramanujan Conjectures},
  author={Peter Sarnak},
1. GLn Ramanujan's original conjecture is concerned with the estimation of Fourier coe! cients of the weight 12 holomorphic cusp form " for SL(2,Z) on the upper half plane H. The conjecture may be reformulated in terms of the size of the eigen- values of the corresponding Hecke operators or equivalently in terms of the local representations which are components of the automorphic representation associated with " . This spectral reformulation of the problem of estimation of Fourier coef… 

Figures from this paper


To the memory of Armand Borel 1. Introduction. Early experiences with classical (holomorphic) cusp forms, which initially started with the Ramanujan τ –function, and later extended to even Maass cusp

Fourier Coefficients and Cuspidal Spectrum for Symplectic Groups

J. Arthur (The endoscopic classification of representations: orthogonal and Symplectic groups. Colloquium Publication, vol 61. American Mathematical Society, 2013) classifies the automorphic discrete

On the zeros of automorphic forms

The subject of this thesis is the zeros of automorphic forms. In the first part, we study the asymptotic behavior of nodal lines of Maass (cusp) forms on hyperbolic surfaces via taking intersection

Beyond Endoscopy via the trace formula, II: Asymptotic expansions of Fourier transforms and bounds towards the Ramanujan conjecture

abstract:We continue the analysis of the elliptic part of the trace formula for $GL(2)$ initiated in the earlier paper of the author with the same title. In that paper Poisson summation was applied

Sums of Fourier coefficients of a Maass form for SL3(ℤ) twisted by exponential functions

Abstract. Let f be a Maass cusp form for SL3(ℤ) with Fourier coefficients . We consider the sum , where . A bound better than is proved to be valid for certain transcendental numbers . This bound

Generalization of Selberg ’ s 316 theorem and affine sieve by

A celebrated theorem of Selberg [33] states that for congruence subgroups of SL2(Z) there are no exceptional eigenvalues below 3 16 . We prove a generalization of Selberg’s theorem for infinite index

The Siegel variance formula for quadratic forms

We introduce a smooth variance sum associated to a pair of positive definite symmetric integral matrices $A_{m\times m}$ and $B_{n\times n}$, where $m\geq n$. By using the oscillator representation,


We report on some recent striking advances on the quantum unique ergodicity, or QUE conjecture, concerning the distribution of large frequency eigenfunctions of the Laplacian on a negatively curved

Resonance of automorphic forms for GL(3)

Let f be a Maass form for SL3(Z) with Fourier coefficients Af (m,n). A smoothly weighted sum of Af (m,n) against an exponential function e(αnβ) of fractional power nβ for X ≤ n ≤ 2X is proved to have



Harmonic analysis and Hecke operators

We first construct new uniform pointwise bounds for the matrix coefficients of infinite dimensional unitary representations of a reductive algebraic group over a local field k with semisimple k-rank

On Selberg's eigenvalue conjecture

Let Γ ⊂ SL 2(Z) be a congruence subgroup, and λ0 = 0 3/16. Iwaniec ([I]) showed that for almost all Hecke congruence groups Γ0(p) with a certain multiplier χ p , one has λ1(Γ0(p), χ p ) ≥ 44/225 =

Simple algebras, base change, and the advanced theory of the trace formula

A general principle, discovered by Robert Langlands and named by him the "functoriality principle," predicts relations between automorphic forms on arithmetic subgroups of different reductive groups.

Isolated Unitary Representations

In this paper we collect some facts about the topology on the space of irreducible unitary representations of a real reductive group. The main goal is Theorem 10, which asserts that most of the

On the Selberg trace formula in the case of compact quotient

It is a standard fact (see §2) that 7rr(4>) is of trace class. In particular, 7Tr(4>) is completely continuous for 4>eC7(G). This implies that L(r\G) decomposes into an orthogonal direct sum of

Functoriality for the classical groups

Functoriality is one of the most central questions in the theory of automorphic forms and representations [1,2,35,36]. Locally and globally, it is a manifestation of Langlands’ formulation of a

Problems in the theory of automorphic forms to Salomon Bochner in gratitude

1. There has recently been much interest, if not a tremendous amount of progress, in the arithmetic theory of automorphic forms. In this lecture I would like to present the views not of a number

Combinatorial consequences of Arthur's conjectures and the Burger-Sarnak method

The Burger-Sarnak principle states that the restriction to a reductive subgroup of an automorphic representation of a reductive group has automorphic support. Arthur's conjectures parametrize

Functorial products for $\mathrm{GL}_2 \times \mathrm{GL}_3$ and the symmetric cube for $\mathrm{GL}_2$

In this paper we prove two new cases of Langlands functoriality. The first is a functorial product for cusp forms on GL2 × GL3 as automorphic forms on GL6, from which we obtain our second case, the