Peter Sarnak

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Peter Sarnak is a renowned mathematician best known for his work in number theory, analysis, and related fields. He holds positions at Princeton University and the Institute for Advanced Study. Born in South Africa in 1953, Sarnak completed his Ph.D. at Stanford University under the supervision of Paul Cohen. His research includes fundamental contributions to automorphic forms, L-functions, and spectral theory on manifolds. He has received numerous honors, including the Wolf Prize in Mathematics and membership in the National Academy of Sciences. Sarnak’s work has had extensive influence on the development of both analytic and algebraic aspects of number theory.

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Ramanujan graphs

1988-09-01

Alexander Lubotzky , Ralph S. Phillips , Peter Sarnak

Random Matrices, Frobenius Eigenvalues, and Monodromy

1998-11-24

Nicholas M. Katz , Peter Sarnak

Statements of the main results Reformulation of the main results Reduction steps in proving the main theorems Test functions Haar measure Tail estimates Large $N$ limits and Fredholm determinants Several … Statements of the main results Reformulation of the main results Reduction steps in proving the main theorems Test functions Haar measure Tail estimates Large $N$ limits and Fredholm determinants Several variables Equidistribution Monodromy of families of curves Monodromy of some other families GUE discrepancies in various families Distribution of low-lying Frobenius eigenvalues in various families Appendix AD: Densities Appendix AG: Graphs References.

Zeros of principal L-functions and random matrix theory

1996-01-01

Zeév Rudnick , Peter Sarnak

Extremals of determinants of Laplacians

1988-09-01

Brad Osgood , Ralph S. Phillips , Peter Sarnak

Zeroes of zeta functions and symmetry

1999-01-01

Nicholas Katz , Peter Sarnak

Hilbert and Polya suggested that there might be a natural spectral interpretation of the zeroes of the Riemann Zeta function. While at the time there was little evidence for this, … Hilbert and Polya suggested that there might be a natural spectral interpretation of the zeroes of the Riemann Zeta function. While at the time there was little evidence for this, today the evidence is quite convincing. Firstly, there are the “function field” analogues, that is zeta functions of curves over finite fields and their generalizations. For these a spectral interpretation for their zeroes exists in terms of eigenvalues of Frobenius on cohomology. Secondly, the developments, both theoretical and numerical, on the local spacing distributions between the high zeroes of the zeta function and its generalizations give striking evidence for such a spectral connection. Moreover, the low-lying zeroes of various families of zeta functions follow laws for the eigenvalue distributions of members of the classical groups. In this paper we review these developments. In order to present the material fluently, we do not proceed in chronological order of discovery. Also, in concentrating entirely on the subject matter of the title, we are ignoring the standard body of important work that has been done on the zeta function and<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"><mml:semantics><mml:mi>L</mml:mi><mml:annotation encoding="application/x-tex">L</mml:annotation></mml:semantics></mml:math></inline-formula>-functions.

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Low lying zeros of families of L-functions

2000-12-01

Henryk Iwaniec , Wenzhi Luo , Peter Sarnak

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The behaviour of eigenstates of arithmetic hyperbolic manifolds

1994-03-01

Zeév Rudnick , Peter Sarnak

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Density of integer points on affine homogeneous varieties

1993-07-01

William Duke , Zeév Rudnick , Peter Sarnak

A basic problem of diophantine analysis is to investigate the asymptotics as T of (1.2) N(T, V)= {m V(Z): Ilmll T} where we denote by V(A), for any ring A, … A basic problem of diophantine analysis is to investigate the asymptotics as T of (1.2) N(T, V)= {m V(Z): Ilmll T} where we denote by V(A), for any ring A, the set of A-points of V. Hence I1" is some Euclidean norm on R".The only general method available for such problems is the Hardy-Littlewood circle method, which however has certain limitations, requiring roughly that the codimension of V in the ambient space A", as well as the degree of the equations (1.1), be small relative to n.Furthermore, there are restrictions on the size of the singular sets of the related varieties:V u {x e C": f(x) &, j 1,..., v}, u () e c".We refer to [Bi] and [Sch] for a discussion of the restriction.Regardless of these restrictions, one hopes that for many more cases N(T, V) can be given in the form predicted by the Hardy-Littlewood method, that is, as a product oflocal densities: (,) N(T, V) l--I l,(V)lUoo( T, v), p <oo where the "singular series" I-I,< #,(V) is given by p-adic densities:and/(T, V) is a real densitymthe "singular integral."Following Schmidt [Sch], we say that V is a Hardy-Littlewood system if the above asymptotics (,) is valid.

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Determinants of Laplacians

1987-03-01

Peter Sarnak

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Chebyshev's Bias

1994-01-01

Michael Rubinstein , Peter Sarnak

The title refers to the fact, noted by Chebyshev in 1853, that primes congruent to 3 modulo 4 seem to predominate over those congruent to 1. We study this phenomenon … The title refers to the fact, noted by Chebyshev in 1853, that primes congruent to 3 modulo 4 seem to predominate over those congruent to 1. We study this phenomenon and its generalizations. Assuming the Generalized Riemann Hypothesis and the Grand Simplicity Hypothesis (about the zeros of the Dirichlet L-function), we can characterize exactly those moduli and residue classes for which the bias is present. We also give results of numerical investigations on the prevalence of the bias for several moduli. Finally, we briefly discuss generalizations of the bias to the distribution to primes in ideal classes in number fields, and to prime geodesics in homology classes on hyperbolic surfaces.

Perspectives on the Analytic Theory of L-Functions

2000-01-01

Henryk Iwaniec , Peter Sarnak

To the general mathematician L-functions might appear to be an esoteric and special topic in number theory. We hope that the discussion below will convince the reader otherwise. Time and … To the general mathematician L-functions might appear to be an esoteric and special topic in number theory. We hope that the discussion below will convince the reader otherwise. Time and again it has turned out that the crux of a problem lies in the theory of these functions. At some level it is not entirely clear to us why L-functions should enter decisively, though in hindsight one can give reasons. Our plan is to introduce L-functions and describe the central problems connected with them. We give a sample (this is certainly not meant to be a survey) of results towards these conjectures as well as some problems that can be resolved by finessing these conjectures. We also mention briefly some of the successful present-day tools and the role they might play in the big picture.

L ∞ Norms of Eigenfunctions of Arithmetic Surfaces

1995-03-01

Henryk Iwaniec , Peter Sarnak

Here and elsewhere A ,l1o0 as A -xc is a basic problem of Quantum Chaos [S]. In any case almost nothing beyond (0.2) is known about 110,joc, when the curvature … Here and elsewhere A ,l1o0 as A -xc is a basic problem of Quantum Chaos [S]. In any case almost nothing beyond (0.2) is known about 110,joc, when the curvature is negative (one can push the standard techniques and replace A1/4 by A1/4/ log A in this case). In this paper we use arithmetic techniques, in particular modular correspondences to obtain the first improvement in the exponent (0.2). We also obtain lower bounds on the L' norms in these arithmetic cases. In more detail, let A = (,b) be a quaternion division algebra over Q. A is linearly generated by 1, w, Q, wQ over Q and w2 = a, Q2 = b, wQ + Qw = 0. Here a, b E Z are square free and we will assume that a > 0. As usual the norm and trace are defined by N(a) = a-d, tr(a) = a + oY where if a = Xo + x1W + x2Q + X3WQ,

On Selberg’s Eigenvalue Conjecture

1995-01-01

Wenzhi Luo , Zeév Rudnick , Peter Sarnak

Let Γ ⊂ SL 2(Z) be a congruence subgroup, and λ0 = 0 < λ1 < … be the eigenvalues of the non-euclidean Laplacian on L 2(Γ\H 2). A fundamental … Let Γ ⊂ SL 2(Z) be a congruence subgroup, and λ0 = 0 < λ1 < … be the eigenvalues of the non-euclidean Laplacian on L 2(Γ\H 2). A fundamental conjecture of Selberg ([Se]) asserts that the smallest nonzero eigenvalue λ1(Γ) ≥1/4 = 0.25. In the same paper Selberg proved that λ1(Γ) ≥3/16 = 0.1875. Gelbart and Jacquet ([GJ]), using very different methods, improved this to λ1(Γ) > 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 = 0.19555…. In [I], he also established a density theorem for possible exceptional eigenvalues as above, which while not giving any improvement on 3/16 for an individual Γ is sufficiently strong to substitute for Selberg's conjecture in many applications to number theory. Selberg's conjecture is the archimedean analogue of the "Ramanujan Conjectures" on the Fourier coefficients of Maass forms. For these, much progress has been made in improving the relevant estimates, beginning with Serre ([Ser]) and later on Shahidi ([Sh2]) and Bump-Duke-Hoffstein-Iwaniec ([BDHI]). In this paper we restore the balance and establish in part for the archimedean place what is known at the finite places. The method on the face of it is quite different, but the quality of the results coincide (the reason will be made clear later).

Compact isospectral sets of surfaces

1988-09-01

Brad Osgood , Ralph S. Phillips , Peter Sarnak

Hecke operators and distributing points on the sphere I

1986-01-01

Alexander Lubotzky , Ralph S. Phillips , Peter Sarnak

Affine linear sieve, expanders, and sum-product

2009-11-25

Jean Bourgain , Alex Gamburd , Peter Sarnak

Let $\mathcal{O}$ be an orbit in ℤ n of a finitely generated subgroup Λ of GL n (ℤ) whose Zariski closure Zcl(Λ) is suitably large (e.g. isomorphic to SL2). We … Let $\mathcal{O}$ be an orbit in ℤ n of a finitely generated subgroup Λ of GL n (ℤ) whose Zariski closure Zcl(Λ) is suitably large (e.g. isomorphic to SL2). We develop a Brun combinatorial sieve for estimating the number of points on $\mathcal{O}$ at which a fixed integral polynomial is prime or has few prime factors, and discuss applications to classical problems, including Pythagorean triangles and integral Apollonian packings. A fundamental role is played by the expansion property of the "congruence graphs" that we associate with $\mathcal{O}$ . This expansion property is established when Zcl(Λ)=SL2, using crucially sum-product theorem in ℤ/qℤ for q square-free.

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On the period matrix of a Riemann surface of large genus (with an Appendix by J.H. Conway and N.J.A. Sloane)

1994-12-01

Peter Buser , Peter Sarnak

Explicit expanders and the Ramanujan conjectures

1986-01-01

Alexander Lubotzky , Ralph S. Phillips , Peter Sarnak

Article Free Access Share on Explicit expanders and the Ramanujan conjectures Authors: A Lubotzky Institute of Mathematics and Computer Science, Hebrew University, Jerusalem, Israel Institute of Mathematics and Computer Science, … Article Free Access Share on Explicit expanders and the Ramanujan conjectures Authors: A Lubotzky Institute of Mathematics and Computer Science, Hebrew University, Jerusalem, Israel Institute of Mathematics and Computer Science, Hebrew University, Jerusalem, IsraelView Profile , R Phillips Department of Mathematics, Stanford University, Stanford, California Department of Mathematics, Stanford University, Stanford, CaliforniaView Profile , P Sarnak Department of Mathematics, Stanford University, Stanford, California Department of Mathematics, Stanford University, Stanford, CaliforniaView Profile Authors Info & Claims STOC '86: Proceedings of the eighteenth annual ACM symposium on Theory of computingNovember 1986 Pages 240–246https://doi.org/10.1145/12130.12154Online:01 November 1986Publication History 77citation1,237DownloadsMetricsTotal Citations77Total Downloads1,237Last 12 Months64Last 6 weeks9 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

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The non-vanishing of central values of automorphic L-functions and Landau-Siegel zeros

2000-12-01

Henryk Iwaniec , Peter Sarnak

The Pair Correlation Function of Fractional Parts of Polynomials

1998-05-01

Ze x E v Rudnick , Peter Sarnak

Bounds for multiplicities of automorphic representations

1991-10-01

Peter Sarnak , Xiaoxi Xue

Spectra of hyperbolic surfaces

2003-07-17

Peter Sarnak

These notes attempt to describe some aspects of the spectral theory of modular surfaces. They are by no means a complete survey. These notes attempt to describe some aspects of the spectral theory of modular surfaces. They are by no means a complete survey.

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Estimates for Rankin–Selberg L-Functions and Quantum Unique Ergodicity

2001-08-01

Peter Sarnak

Heegner points, cycles and Maass forms

1993-02-01

Svetlana Katok , Peter Sarnak

The arithmetic and geometry of some hyperbolic three manifolds

1983-01-01

Peter Sarnak

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Asymptotic behavior of periodic orbits of the horocycle flow and eisenstein series

1981-11-01

Peter Sarnak

On cusp forms for co-finite subgroups ofPSL(2,?)

1985-06-01

Ralph S. Phillips , Peter Sarnak

Class numbers of indefinite binary quadratic forms II

1985-12-01

Peter Sarnak

Sums of Kloosterman sums

1983-06-01

Dorian Goldfeld , Peter Sarnak

Class numbers of indefinite binary quadratic forms

1982-10-01

Peter Sarnak

The distribution of spacings between the fractional parts of n 2 α

2001-07-01

Zeév Rudnick , Peter Sarnak , Alexandru Zaharescu

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Geodesics in homology classes

1987-06-01

Ralph S. Phillips , Peter Sarnak

Ramanujan duals II

1991-12-01

M. Burger , Peter Sarnak

Minima of Epstein’s Zeta function and heights of flat tori

2006-03-13

Peter Sarnak , Andreas Strömbergsson

Hecke operators and distributing points on S2. II

1987-07-01

Alexander Lubotzky , Ralph S. Phillips , Peter Sarnak

Line-Graph Lattices: Euclidean and Non-Euclidean Flat Bands, and Implementations in Circuit Quantum Electrodynamics

2019-12-13

Alicia J. Kollár , Mattias Fitzpatrick , Peter Sarnak +1 more

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Disjointness of Moebius from Horocycle Flows

2012-07-30

Jean Bourgain , Peter Sarnak , Tamar Ziegler

Statistical Properties of Eigenvalues of the Hecke Operators

1987-01-01

Peter Sarnak

Two basic questions concerning the Ramanujan τ-function concern the size and variation of these numbers: Two basic questions concerning the Ramanujan τ-function concern the size and variation of these numbers:

Spectra of elements in the group ring of $\mathrm{SU}(2)$

1999-03-31

Alex Gamburd , Dmitry Jakobson , Peter Sarnak

We present a new method for establishing the “gap” property for finitely generated subgroups of \mathrm{SU}(2) , providing an elementary solution of Ruziewicz problem on S^2 as well as giving … We present a new method for establishing the “gap” property for finitely generated subgroups of \mathrm{SU}(2) , providing an elementary solution of Ruziewicz problem on S^2 as well as giving many new examples of finitely generated subgroups of \mathrm{SU}(2) with an explicit gap. The distribution of the eigenvalues of the elements of the group ring \mathbf{R}[\mathrm{SU}(2)] in the N -th irreducible representation of \mathrm{SU}(2) is also studied. Numerical experiments indicate that for a generic (in measure) element of \mathbf{R}[\mathrm{SU}(2)] , the “unfolded” consecutive spacings distribution approaches the GOE spacing law of random matrix theory (for N even) and the GSE spacing law (for N odd) as N\to \infty ; we establish several results in this direction. For certain special “arithmetic” (or Ramanujan ) elements of \mathbf{R}[\mathrm{SU}(2)] the experiments indicate that the unfolded consecutive spacing distribution follows Poisson statistics; we provide a sharp estimate in that direction.

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On the generalized Ramanujan conjecture for 𝐺𝐿(𝑛)

1999-01-01

Wenzhi Luo , Zeév Rudnick , Peter Sarnak

Moduli Space, Heights and Isospectral Sets of Plane Domains

1989-03-01

Brad Osgood , R. Phillips , Peter Sarnak

Let Q be a plane domain of finite connectivity n with smooth boundary and choose a fixed domain 2 of the same type. Then there exists a flat metric g … Let Q be a plane domain of finite connectivity n with smooth boundary and choose a fixed domain 2 of the same type. Then there exists a flat metric g on 2 such that Q is isometric with 29g. In what follows we do not distinguish between isometric domains. By the spectrum of 29 we mean the spectrum of the Laplace-Beltrami operator Ag on 29 with Dirichlet boundary conditions. The height h(Eg) = - log det Ag is a spectral invariant and plays a central role in this paper. Among all suitably normalized flat metrics on 2 conformal to a given metric g there is a unique flat metric for which the height is a minimum. This metric is characterized by the fact that d 2 has constant geodesic curvature; we call such a metric uniform and denote it by u. The set of all such metrics is denoted by u(2). We can therefore identify ( with the moduli space #(2) of conformal structures on E. For n ? 3 we introduce a special parametrization for Mu(2) by means of which we show that h(u) - oo as u approaches the boundary of fu(2). Using this along with the heat invariants for the Laplacian we then show that any isospectral set of plane domains is compact in the C' topology. Similar results hold for n = 1 and 2.

The Möbius function and distal flows

2015-05-14

Jianya Liu , Peter Sarnak

We prove that the M\"{o}bius function is linearly disjoint from an analytic skew product on the $2$-torus. These flows are distal and can be irregular in the sense that their … We prove that the M\"{o}bius function is linearly disjoint from an analytic skew product on the $2$-torus. These flows are distal and can be irregular in the sense that their ergodic averages need not exist for all points. The previous cases for which such disjointness has been proved are all regular. We also establish the linear disjointness of M\"{o}bius from various distal homogeneous flows.

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Recent progress on the quantum unique ergodicity conjecture

2011-01-10

Peter Sarnak

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 manifold. … 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 manifold. The account falls naturally into two categories. The first concerns the general conjecture where the tools are more or less limited to microlocal analysis and the dynamics of the geodesic flow. The second is concerned with arithmetic such manifolds where tools from number theory and ergodic theory of flows on homogeneous spaces can be combined with the general methods to resolve the basic conjecture as well as its holomorphic analogue. Our main emphasis is on the second category, especially where QUE has been proven. This note is not meant to be a survey of these topics, and the discussion is not chronological. Our aim is to expose these recent developments after introducing the necessary backround which places them in their proper context.

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Number variance for arithmetic hyperbolic surfaces

1994-03-01

Wenzhi Luo , Peter Sarnak

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Dirichlet L-functions at the central point

1999-12-31

Henryk Iwaniec , Peter Sarnak

Spectral behavior of quasi periodic potentials

1982-09-01

Peter Sarnak

Generalization of Selberg’s $ \frac{3}{{16}} $ theorem and affine sieve

2011-01-01

Jean Bourgain , Alex Gamburd , Peter Sarnak

An analogue of the well-known $ \frac{3}{{16}} $ lower bound for the first eigenvalue of the Laplacian for a congruence hyperbolic surface is proven for a congruence tower associated with … An analogue of the well-known $ \frac{3}{{16}} $ lower bound for the first eigenvalue of the Laplacian for a congruence hyperbolic surface is proven for a congruence tower associated with any non-elementary subgroup L of SL(2, Z). The proof in the case that the Hausdorff of the limit set of L is bigger than $ \frac{1}{2} $ is based on a general result which allows one to transfer such bounds from a combinatorial version to this archimedian setting. In the case that delta is less than $ \frac{1}{2} $ we formulate and prove a somewhat weaker version of this phenomenon in terms of poles of the corresponding dynamical zeta function. These "spectral gaps" are then applied to sieving problems on orbits of such groups.

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Nodal Domains of Maass Forms I

2013-07-08

Amit Ghosh , André Reznikov , Peter Sarnak

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Ramanujan duals and automorphic spectrum

1992-04-01

M. Burger , J.-S. Li , Peter Sarnak

We introduce the notion of the automorphic dual of a matrix algebraic group defined over Q .This is the part of the unitary dual that corresponds to arithmetic spectrum.Basic funcional … We introduce the notion of the automorphic dual of a matrix algebraic group defined over Q .This is the part of the unitary dual that corresponds to arithmetic spectrum.Basic funcional properties of this set are derived and used both to deduce arithmetic vanishing theorems of "Ramanujan" type as well as to give a new construction of automorphic forms.

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The laplacian for domains in hyperbolic space and limit sets of Kleinian groups

1985-01-01

Ralph S. Phillips , Peter Sarnak

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A universal lower bound for certain quadratic integrals of automorphic L–functions

2024-03-22

Laurent Clozel , Peter Sarnak

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Convergence to the Plancherel measure of Hecke eigenvalues

2024-01-01

Peter Sarnak , Nina Zubrilina

We give improved uniform estimates for the rate of convergence to Plancherel measure of Hecke eigenvalues of holomorphic forms of weight $2$ and level $N$. These are applied to determine … We give improved uniform estimates for the rate of convergence to Plancherel measure of Hecke eigenvalues of holomorphic forms of weight $2$ and level $N$. These are applied to determine the sharp cutoff for the non-backtracking random walk on arithmetic

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Bounded Cutoff Window for the Non-backtracking Random Walk on Ramanujan Graphs

2023-04-01

Evita Nestoridi , Peter Sarnak

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Integral points on Markoff type cubic surfaces

2022-05-30

Amit Ghosh , Peter Sarnak

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Commutators in $SL_2$ and Markoff surfaces I

2022-01-18

Amit Ghosh , Chen Meiri , Peter Sarnak

We show that the commutator equation over $\mathrm{SL}_2(\mathbb{Z})$ satisfies a profinite local to global principle, while it can fail with infinitely many exceptions for $ \mathrm{SL}_2(\mathbb{Z}[\frac{1}{p}])$. The source of the … We show that the commutator equation over $\mathrm{SL}_2(\mathbb{Z})$ satisfies a profinite local to global principle, while it can fail with infinitely many exceptions for $ \mathrm{SL}_2(\mathbb{Z}[\frac{1}{p}])$. The source of the failure is a reciprocity obstruction to the Hasse Principle for cubic Markoff surfaces.

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A universal lower bound for certain quadratic integrals of automorphic L-functions

2022-01-01

Laurent Clozel , Peter Sarnak

We obtain uniform lower bounds, true for all automorphic L-functions L(s) associated to cuspidal representations of GL(m,A) where A denotes the adeles of the rationals Q, of the integral on … We obtain uniform lower bounds, true for all automorphic L-functions L(s) associated to cuspidal representations of GL(m,A) where A denotes the adeles of the rationals Q, of the integral on the vertical line (Re(s)=1/2) of the absolute value squared of L(s)/s; and also of L(s)/(s-s_0) when s_0 is a zero of the L-function on the critical line. Several variants are also obtained in small degrees m, for the vertical integrals at different abscissas in the critical strip. For the estimates required to prove convergence, we are led to generalise a result of Friedlander-Iwaniec (Can. J. Math. 57,2005). We obtain new results on the abscissa of convergence of the L-series. Finally, a problem is posed about the behaviour of the quadratic integral when s_0 tends to infinity, in particular for the Riemann zeta function.

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Convergence to the Plancherel measure of Hecke Eigenvalues

2022-01-01

Peter Sarnak , Nina Zubrilina

We give improved uniform estimates for the rate of convergence to Plancherel measure of Hecke eigenvalues of holomorphic forms of weight 2 and level N. These are applied to determine … We give improved uniform estimates for the rate of convergence to Plancherel measure of Hecke eigenvalues of holomorphic forms of weight 2 and level N. These are applied to determine the sharp cutoff for the non-backtracking random walk on arithmetic Ramanujan graphs and to Serre's problem of bounding the multiplicities of modular forms whose coefficients lie in number fields of degree d.

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Commutators in $\mathrm{SL}_2$ and Markoff surfaces I

2021-10-21

Amit Ghosh , Chen Meiri , Peter Sarnak

Gap sets for the spectra of cubic graphs

2021-10-07

Alicia J. Kollár , Peter Sarnak

We study gaps in the spectra of the adjacency matrices of large finite cubic graphs. It is known that the gap intervals <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 2 StartRoot 2 … We study gaps in the spectra of the adjacency matrices of large finite cubic graphs. It is known that the gap intervals <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 2 StartRoot 2 EndRoot comma 3 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:msqrt> <mml:mn>2</mml:mn> </mml:msqrt> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(2 \sqrt {2},3)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket negative 3 comma negative 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mo>−</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[-3,-2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> achieved in cubic Ramanujan graphs and line graphs are maximal. We give constraints on spectra in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket negative 3 comma 3 right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mo>−</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[-3,3]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which are maximally gapped and construct examples which achieve these bounds. These graphs yield new instances of maximally gapped intervals. We also show that every point in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket negative 3 comma 3 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mo>−</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[-3,3)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be gapped by planar cubic graphs. Our results show that the study of spectra of cubic, and even planar cubic, graphs is subtle and very rich.

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An appreciation of Jean Bourgain’s work

2021-02-12

Peter Sarnak

Jean Bourgain viewed himself as an "analyst", and as the record shows he was uniquely gifted as such, and much more.Analytic, combinatorial, and probabilistic reasoning is at the heart of … Jean Bourgain viewed himself as an "analyst", and as the record shows he was uniquely gifted as such, and much more.Analytic, combinatorial, and probabilistic reasoning is at the heart of many central problems of modern mathematics and its applications, and these naturally attracted Jean's attention.The combination of his brilliance, his thirst to solve long-standing problems, and his many fruitful collaborations led him to transformative contributions in a striking number of areas.Jean's research and its impact remind one of the great Russian analyst Kolmogorov.It is said that Kolmogorov made major contributions to all fields except number theory.A list of areas to which Jean made decisive contributions include functional analysis, harmonic analysis, probability theory, ergodic theory, partial differential equations, mathematical physics, number theory, group theory, and theoretical computer science.It is impossible in a single volume, let alone an issue of the Bulletin of the American Mathematical Society, to give anything like a comprehensive account of Jean's mathematical achievements.Gathering his over 500 (and counting) publications in a collected works would be physically impossible.Fortunately, Jean was very purposeful and proactive in preparing his papers for publication, and almost all of these are in print and are accessible.He made sure that anyone committed to understanding and using his work would have it available.

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Bounded cutoff window for the non-backtracking random walk on Ramanujan Graphs

2021-01-01

Evita Nestoridi , Peter Sarnak

We prove that the non-backtracking random walk on Ramanujan graphs with large girth exhibits the fastest possible cutoff with a bounded window. We prove that the non-backtracking random walk on Ramanujan graphs with large girth exhibits the fastest possible cutoff with a bounded window.

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Stable polynomials and crystalline measures

2020-08-01

Pavel Kurasov , Peter Sarnak

Explicit examples of positive crystalline measures and Fourier quasicrystals are constructed using pairs of stable polynomials, answering several open questions in the area. Explicit examples of positive crystalline measures and Fourier quasicrystals are constructed using pairs of stable polynomials, answering several open questions in the area.

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Gap Sets for the Spectra of Cubic Graphs

2020-05-11

Alicia J. Kollár , Peter Sarnak

We study gaps in the spectra of the adjacency matrices of large finite cubic graphs. It is known that the gap intervals $(2 \sqrt{2},3)$ and $[-3,-2)$ achieved in cubic Ramanujan … We study gaps in the spectra of the adjacency matrices of large finite cubic graphs. It is known that the gap intervals $(2 \sqrt{2},3)$ and $[-3,-2)$ achieved in cubic Ramanujan graphs and line graphs are maximal. We give constraints on spectra in [-3,3] which are maximally gapped and construct examples which achieve these bounds. These graphs yield new instances of maximally gapped intervals. We also show that every point in $[-3,3)$ can be gapped by cubic graphs, even by planar ones. Our results show that the study of spectra of cubic, and even planar cubic, graphs is subtle and very rich.

Gap Sets for the Spectra of Cubic Graphs

2020-01-01

Alicia J. Kollár , Peter Sarnak

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Line-Graph Lattices: Euclidean and Non-Euclidean Flat Bands, and Implementations in Circuit Quantum Electrodynamics

2019-12-13

Alicia J. Kollár , Mattias Fitzpatrick , Peter Sarnak +1 more

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The Quantum variance of the Modular Surface

2019-01-01

Peter Sarnak , Peng Zhao

Topologies of Nodal Sets of Random Band‐Limited Functions

2018-11-10

Peter Sarnak , Igor Wigman

Abstract It is shown that the topologies and nestings of the zero and nodal sets of random (Gaussian) band‐limited functions have universal laws of distribution. Qualitative features of the supports … Abstract It is shown that the topologies and nestings of the zero and nodal sets of random (Gaussian) band‐limited functions have universal laws of distribution. Qualitative features of the supports of these distributions are determined. In particular, the results apply to random monochromatic waves and to random real algebraic hypersurfaces in projective space. © 2018 Wiley Periodicals, Inc.

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Topology and Nesting of the Zero Set Components of Monochromatic Random Waves

2018-10-15

Yaiza Canzani , Peter Sarnak

This paper is dedicated to the study of the topologies and nesting configurations of the components of the zero set of monochromatic random waves. We prove that the probability of … This paper is dedicated to the study of the topologies and nesting configurations of the components of the zero set of monochromatic random waves. We prove that the probability of observing any diffeomorphism type and any nesting arrangement among the zero set components is strictly positive for waves of large enough frequencies. Our results are a consequence of building Laplace eigenfunctions in euclidean space whose zero sets have a component with prescribed topological type or an arrangement of components with prescribed nesting configuration. © 2018 Wiley Periodicals, Inc.

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Book Review: Prime numbers and the Riemann hypothesis

2018-02-20

Peter Sarnak

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Spatial statistics for lattice points on the sphere I: Individual results

2017-08-01

Jean Bourgain , Zeév Rudnick , Peter Sarnak

Super-Golden-Gates for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mi>P</mml:mi><mml:mi>U</mml:mi><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:math>

2017-07-10

Ori Parzanchevski , Peter Sarnak

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Nodal domains of Maass forms, II

2017-01-01

Amit Ghosh , André Reznikov , Peter Sarnak

In Part I we gave a polynomial growth lower-bound for the number of nodal domains of a Hecke-Maass cuspform in a compact part of the modular surface, assuming a Lindel\"{o}f … In Part I we gave a polynomial growth lower-bound for the number of nodal domains of a Hecke-Maass cuspform in a compact part of the modular surface, assuming a Lindel\"{o}f hypothesis. That was a consequence of a topological argument and known subconvexity estimates, together with new sharp lower-bound restriction theorems for the Maass forms. This paper deals with the same question for general (compact or not) arithmetic surfaces which have a reflective symmetry. The topological argument is extended and representation theoretic methods are needed for the restriction theorems, together with results of Waldspurger. Various explicit examples are given and studied.

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Topology and nesting of the zero set components of monochromatic random waves

2017-01-01

Yaiza Canzani , Peter Sarnak

This paper is dedicated to the study of the topologies and nesting configurations of the components of the zero set of monochromatic random waves. We prove that the probability of … This paper is dedicated to the study of the topologies and nesting configurations of the components of the zero set of monochromatic random waves. We prove that the probability of observing any diffeomorphism type, and any nesting arrangement, among the zero set components is strictly positive for waves of large enough frequencies. Our results are a consequence of building Laplace eigenfunctions in Euclidean space whose zero sets have a component with prescribed topological type, or an arrangement of components with prescribed nesting configuration.

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Topology and nesting of the zero set components of monochromatic random waves

2016-12-30

Yaiza Canzani , Peter Sarnak

This paper is dedicated to the study of the topologies and nesting configurations of the components of the zero set of monochromatic random waves. We prove that the probability of … This paper is dedicated to the study of the topologies and nesting configurations of the components of the zero set of monochromatic random waves. We prove that the probability of observing any diffeomorphism type, and any nesting arrangement, among the zero set components is strictly positive for waves of large enough frequencies. Our results are a consequence of building Laplace eigenfunctions in Euclidean space whose zero sets have a component with prescribed topological type, or an arrangement of components with prescribed nesting configuration.

Spatial statistics for lattice points on the sphere I: Individual results

2016-06-19

Jean Bourgain , Zeév Rudnick , Peter Sarnak

We study the spatial distribution of point sets on the sphere obtained from the representation of a large integer as a sum of three integer squares. We examine several statistics … We study the spatial distribution of point sets on the sphere obtained from the representation of a large integer as a sum of three integer squares. We examine several statistics of these point sets, such as the electrostatic potential, Ripley's function, the variance of the number of points in random spherical caps, and the covering radius. Some of the results are conditional on the Generalized Riemann Hypothesis.

Markoff triples and strong approximation

2016-01-26

Jean Bourgain , Alexander Gamburd , Peter Sarnak

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Local Statistics of Lattice Points on the Sphere

2016-01-01

Jean Bourgain , Peter Sarnak , Zeév Rudnick

A celebrated result of Legendre and Gauss determines which inte- gers can be represented as a sum of three squares, and for those it is typically the case that there … A celebrated result of Legendre and Gauss determines which inte- gers can be represented as a sum of three squares, and for those it is typically the case that there are many ways of doing so. These different representations give collections of points on the unit sphere, and a fundamental result, con- jectured by Linnik, is that under a simple condition these become uniformly distributed on the sphere. In this note we survey some of our recent work, which explores what happens beyond uniform distribution, giving evidence to randomness on smaller scales. We treat the electrostatic energy, local statistics such as the point pair statistic (Ripley's function), nearest neighbour statis- tics, minimum spacing and covering radius. We briefly discuss the situation in other dimensions, which is very different. In an appendix we compute the corresponding quantities for random points.

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Families of L-Functions and Their Symmetry

2016-01-01

Peter Sarnak , Sug Woo Shin , Nicolas Templier

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Markoff Surfaces and Strong Approximation: 1

2016-01-01

Jean Bourgain , Alexander Gamburd , Peter Sarnak

We prove results pertaining to strong approximation for Markoff triples in the case of prime moduli. We prove results pertaining to strong approximation for Markoff triples in the case of prime moduli.

Topologies of nodal sets of random band limited functions

2016-01-01

Peter Sarnak , Igor Wigman

It is shown that the topologies and nestings of the zero and nodal sets of random (Gaussian) band limited functions have universal laws of distribution. Qualitative features of the supports … It is shown that the topologies and nestings of the zero and nodal sets of random (Gaussian) band limited functions have universal laws of distribution. Qualitative features of the supports of these distributions are determined. In particular the results apply to random monochromatic waves and to random real algebraic hyper-surfaces in projective space.

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Spatial statistics for lattice points on the sphere I: Individual results

2016-01-01

Jean Bourgain , Zeév Rudnick , Peter Sarnak

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Markoff Triples and Strong Approximation

2015-05-24

Jean Bourgain , Alex Gamburd , Peter Sarnak

We investigate the transitivity properties of the group of morphisms generated by Vieta involutions on the solutions in congruences to the Markoff equation as well as to other Markoff type … We investigate the transitivity properties of the group of morphisms generated by Vieta involutions on the solutions in congruences to the Markoff equation as well as to other Markoff type affine cubic surfaces. These are dictated by the finite $\bar{\mathbb Q}$ orbits of these actions and these can be determined effectively. The results are applied to give forms of strong approximation for integer points, and to sieving, on these surfaces

The Möbius function and distal flows

2015-05-14

Jianya Liu , Peter Sarnak

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Topologies of nodal sets of random band limited functions

2015-01-01

Peter Sarnak , Igor Wigman

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Markoff Triples and Strong Approximation

2015-01-01

Jean Bourgain , Alex Gamburd , Peter Sarnak

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Hyperbolic monodromy groups for the hypergeometric equation and Cartan involutions

2014-09-17

Elena Fuchs , Chen Meiri , Peter Sarnak

We give a criterion which ensures that a group generated by Cartan involutions in the automorph group of a rational quadratic form of signature (n-1,1) is "thin", namely it is … We give a criterion which ensures that a group generated by Cartan involutions in the automorph group of a rational quadratic form of signature (n-1,1) is "thin", namely it is of infinite index in the latter. It is based on a graph defined on the integral Cartan root vectors, as well as Vinberg's theory of hyperbolic reflection groups. The criterion is shown to be robust for showing that many hyperbolic hypergeometric groups for _nF_{n-1} are thin.

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The horocycle flow at prime times

2014-07-08

Peter Sarnak , Adrián Ubis

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The Mobius disjointness conjecture for distal flows

2014-01-01

Jianya Liu , Peter Sarnak

We summarize main results in our paper "The Mobius function and distal flows", and give a direct proof with rate of that the Mobius function is disjoint from Furstenberg's irregular … We summarize main results in our paper "The Mobius function and distal flows", and give a direct proof with rate of that the Mobius function is disjoint from Furstenberg's irregular system. This will be published in the Proceedings of the Sixth ICCM, held in Taipei in 2013.

On the topology of the zero sets of monochromatic random waves

2014-01-01

Yaiza Canzani , Peter Sarnak

This note concerns the topology of the connected components of the zero sets of monochromatic random waves on compact Riemannian manifolds without boundary. In [SW] it is shown that these … This note concerns the topology of the connected components of the zero sets of monochromatic random waves on compact Riemannian manifolds without boundary. In [SW] it is shown that these are distributed according to a universal measure on the space of smooth topological types. Our goal is to determine the support of this measure.

Families of L-functions and their Symmetry

2014-01-01

Peter Sarnak , Sug-Woo Shin , Nicolas Templier

In [90] the first-named author gave a working definition of a family of automorphic L-functions. Since then there have been a number of works [33], [107], [67] [47], [66] and … In [90] the first-named author gave a working definition of a family of automorphic L-functions. Since then there have been a number of works [33], [107], [67] [47], [66] and especially [98] by the second and third-named authors which make it possible to give a conjectural answer for the symmetry type of a family and in particular the universality class predicted in [64] for the distribution of the zeros near s=1/2. In this note we carry this out after introducing some basic invariants associated to a family.

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Nodal Domains of Maass Forms I

2013-07-08

Amit Ghosh , André Reznikov , Peter Sarnak

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Hyperbolic monodromy groups for the hypergeometric equation and Cartan involutions

2013-05-03

Elena Fuchs , Chen Meiri , Peter Sarnak

We give a criterion which ensures that a group generated by Cartan involutions in the automorph group of a rational quadratic form of signature (n-1,1) is thin, namely it is … We give a criterion which ensures that a group generated by Cartan involutions in the automorph group of a rational quadratic form of signature (n-1,1) is thin, namely it is of infinite index in the latter. It is based on a graph defined on the integral Cartan root vectors, as well as Vinberg's theory of hyperbolic reflection groups. The criterion is shown to be robust for showing that many hyperbolic hypergeometric groups for n_F_(n-1) are thin.

The affine sieve

2013-04-01

Alireza Salehi Golsefidy , Peter Sarnak

We establish the main saturation conjecture connected with executing a Brun sieve in the setting of an orbit of a group of affine linear transformations. This is carried out under … We establish the main saturation conjecture connected with executing a Brun sieve in the setting of an orbit of a group of affine linear transformations. This is carried out under the condition that the Zariski closure of the group is Levi-semisimple. It is likely that this condition is also necessary for such saturation to hold.

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The Quantum Variance of the Modular Surface

2013-01-01

Peter Sarnak , Peng Zhao , Appendix by Michael Woodbury

The variance of observables of quantum states of the Laplacian on the modular surface is calculated in the semiclassical limit. It is shown that this hermitian form is diagonalized by … The variance of observables of quantum states of the Laplacian on the modular surface is calculated in the semiclassical limit. It is shown that this hermitian form is diagonalized by the irreducible representations of the modular quotient and on each of these it is equal to the classical variance of the geodesic flow after the insertion of a subtle arithmetical special value of the corresponding $L$-function.

Hyperbolic monodromy groups for the hypergeometric equation and Cartan involutions

2013-01-01

Elena Fuchs , Chen Meiri , Peter Sarnak

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Topologies of nodal sets of random band limited functions

2013-01-01

Peter Sarnak , Igor Wigman

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Disjointness of Moebius from Horocycle Flows

2012-07-30

Jean Bourgain , Peter Sarnak , Tamar Ziegler

Real zeros of holomorphic Hecke cusp forms

2012-02-04

Amit Ghosh , Peter Sarnak

This note is concerned with the zeros of holomorphic Hecke cusp forms of large weight on the modular surface. The zeros of such forms are symmetric about three geodesic segments … This note is concerned with the zeros of holomorphic Hecke cusp forms of large weight on the modular surface. The zeros of such forms are symmetric about three geodesic segments and we call those zeros that lie on these segments, real. Our main results give estimates for the number of real zeros as the weight goes to infinity.

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Notes on Thin Matrix Groups

2012-01-01

Peter Sarnak

These notes were prepared for the MSRI hot topics workshop on superstrong approximation (2012). We give a brief overview of the developments in the theory, especially the fundamental expansion theorem. … These notes were prepared for the MSRI hot topics workshop on superstrong approximation (2012). We give a brief overview of the developments in the theory, especially the fundamental expansion theorem. Applications to diophantine problems on orbits of integer matrix groups, the affine sieve, group theory, gonality of curves and Heegaard genus of hyperbolic three manifolds, are given. We also discuss the ubiquity of thin matrix groups in various contexts, and in particular that of monodromy groups.

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Local statistics of lattice points on the sphere

2012-01-01

Jean Bourgain , Peter Sarnak , Zeév Rudnick

A celebrated result of Legendre and Gauss determines which integers can be represented as a sum of three squares, and for those it is typically the case that there are … A celebrated result of Legendre and Gauss determines which integers can be represented as a sum of three squares, and for those it is typically the case that there are many ways of doing so. These different representations give collections of points on the unit sphere, and a fundamental result, conjectured by Linnik, is that under a simple condition these become uniformly distributed on the sphere. In this note we survey some of our recent work, which explores what happens beyond uniform distribution, giving evidence to randomness on smaller scales. We treat the electrostatic energy, local statistics such as the point pair statistic (Ripley's function), nearest neighbour statistics, minimum spacing and covering radius. We briefly discuss the situation in other dimensions, which is very different. In an appendix we compute the corresponding quantities for random points

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Coauthor	Papers Together
Jean Bourgain	17
Ralph S. Phillips	16
Zeév Rudnick	15
Wenzhi Luo	9
Henryk Iwaniec	8
Alex Gamburd	7
Nicholas Katz	7
Amit Ghosh	6
Chen Meiri	5
Yaiza Canzani	4
Igor Wigman	4
Dubi Kelmer	4
Alexander Lubotzky	4
Alicia J. Kollár	4
Elena Fuchs	3
Jianya Liu	3
Alireza Salehi Golsefidy	3
Brad Osgood	3
Nicolas Templier	2
Jianshu Li	2
M. Burger	2
Alexandru Zaharescu	2
T. N. Shorey	2
Jan‐Hendrik Evertse	2
Adrián Ubis	2
Amos Nevo	2
André Reznikov	2
James Lee Hafner	2
Marc Burger	2
J. R. Quine	2
Tamar Ziegler	2
Alexander Gamburd	2
Nina Zubrilina	2
Ilya Piatetski-Shapiro	2
Laurent Clozel	2
Alex Kontorovich	2
Evita Nestoridi	2
Enrico Bombieri	2
Kevin S. McCurley	1
D. S. Lubinsky	1
Sergiù Klainerman	1
Jacob Tsimerman	1
Jean‐Marc Deshouillers	1
M. Ram Murty	1
Yoichi Motohashi	1
James Cogdell	1
Sug-Woo Shin	1
Yiannis N. Petridis	1
Andrzej Schinzel	1
Yu. V. Nesterenko	1

The Selberg Trace Formula for PSL(2,ℝ)

1983-01-01

Dennis A. Hejhal

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Harmonic Analysis and Discontinuous Groups in Weakly Symmetric Riemannian Spaces With Applications to Dirichlet Series

1956-09-01

Atle Selberg

In The following lectures we shall give a brief sketch of some representative parts of certain investigations that have been undertaken during the last five years. The center of these … In The following lectures we shall give a brief sketch of some representative parts of certain investigations that have been undertaken during the last five years. The center of these investigations is a general relation which can be considered as a generalization of the so-called Poisson summation formula (in one or more dimensions). This relation we here refer to as the trace-formula.

On the estimation of Fourier coefficients of modular forms

1965-01-01

Atle Selberg

The Asymptotic Distribution of Lattice Points in Euclidean and Non-Euclidean Spaces

1982-01-01

Peter D. Lax , Ralph S. Phillips

La conjecture de Weil. I

1974-12-01

Pierre Deligne

Automorphic Forms on GL (2)

1970-01-01

H. Jacquet , R. P. Langlands

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Coefficients of Maass Forms and the Siegel Zero

1994-07-01

Jeffrey Hoffstein , Paul Lockhart

Perspectives on the Analytic Theory of L-Functions

2000-01-01

Henryk Iwaniec , Peter Sarnak

The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces

1982-05-01

Peter D. Lax , Ralph S. Phillips

On Some Exponential Sums

1948-05-01

André Weil

Rational Quadratic Forms

1982-01-01

J. W. S. Cassels

Multiplicative Number Theory

1980-01-01

H. Davenport

A relation between automorphic representations of ${\rm GL}(2)$ and ${\rm GL}(3)$

1978-01-01

Stephen Gelbart , Hervé Jacquet

chier doit contenir la présente mention de copyright. chier doit contenir la présente mention de copyright.

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Low lying zeros of families of L-functions

2000-12-01

Henryk Iwaniec , Wenzhi Luo , Peter Sarnak

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Bounds for multiplicities of automorphic representations

1991-10-01

Peter Sarnak , Xiaoxi Xue

Small eigenvalues of the Laplace operator on compact Riemann surfaces

1974-01-01

Burton Randol

Let Sf be a compact Riemann surface, which we will assume to have curvature normalized to be — 1 , and let 0=A0 0) , and satisfying a growth condition … Let Sf be a compact Riemann surface, which we will assume to have curvature normalized to be — 1 , and let 0=A0 0) , and satisfying a growth condition of the form \h(z) = 0 ( l + |z|)~, uniformly in the strip. Associate with the sequence o(%)> hix)* ' ' • of eigenvalues, a sequence R, consisting of those numbers r(x) that satisfy the equations hn(%)=l+r (x) (n=0, 1, 2, • • •)• Apart

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On cusp forms for co-finite subgroups ofPSL(2,?)

1985-06-01

Ralph S. Phillips , Peter Sarnak

Functoriality for the exterior square of 𝐺𝐿₄ and the symmetric fourth of 𝐺𝐿₂

2002-10-30

Henry Kim

In this paper we prove the functoriality of the exterior square of cusp forms on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L 4"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow … In this paper we prove the functoriality of the exterior square of cusp forms on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L 4"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>4</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">GL_{4}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as automorphic forms on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L 6"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>6</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">GL_{6}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the symmetric fourth of cusp forms on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L 2"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">GL_{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as automorphic forms on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L 5"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>5</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">GL_{5}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We prove these by applying a converse theorem of Cogdell and Piatetski-Shapiro to analytic properties of certain <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-functions obtained by the Langlands-Shahidi method. We give several applications: First, we prove the weak Ramanujan property of cuspidal representations of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L 4"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>4</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">GL_{4}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the absolute convergence of the exterior square <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-functions of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L 4"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>4</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">GL_{4}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Second, we prove that the fourth symmetric power <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-functions of cuspidal representations of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L 2"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">GL_{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are entire, except for those of dihedral and tetrahedral type. Third, we prove the bound <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="three twenty-sixths"> <mml:semantics> <mml:mfrac> <mml:mn>3</mml:mn> <mml:mn>26</mml:mn> </mml:mfrac> <mml:annotation encoding="application/x-tex">\frac {3}{26}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for Hecke eigenvalues of Maass forms over any number field.

Uniform expansion bounds for Cayley graphs of SL2(𝔽p)

2008-03-01

Jean Bourgain , Alex Gamburd

We prove that Cayley graphs of SL 2 (F p ) are expanders with respect to the projection of any fixed elements in SL(2, Z) generating a non-elementary subgroup, and … We prove that Cayley graphs of SL 2 (F p ) are expanders with respect to the projection of any fixed elements in SL(2, Z) generating a non-elementary subgroup, and with respect to generators chosen at random in SL 2 (F p ).

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Zeros of principal L-functions and random matrix theory

1996-01-01

Zeév Rudnick , Peter Sarnak

Congruence Properties of Zariski-Dense Subgroups I

1984-05-01

Christopher R. Matthews , L. N. Vaserstein , Boris Weisfeiler

The arithmetic and geometry of some hyperbolic three manifolds

1983-01-01

Peter Sarnak

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The behaviour of eigenstates of arithmetic hyperbolic manifolds

1994-03-01

Zeév Rudnick , Peter Sarnak

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Discrete Subgroups of Semisimple Lie Groups

1991-01-01

Gregori Aleksandrovitch Margulis

�ber eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichlet scher Reihen durch Funktionalgleichungen

1949-12-01

Hans Maaß

Representation theory and automorphic functions

1969-01-01

М. И. Граев , Ilya Piatetski-Shapiro , I. M. Gel'fand +1 more

Estimates for Rankin–Selberg L-Functions and Quantum Unique Ergodicity

2001-08-01

Peter Sarnak

Algebraic groups and number theory

1992-04-30

В. П. Платонов , Andrei S. Rapinchuk

(Chapter Heading): Algebraic Number Theory. Algebraic Groups. Algebraic Groups over Locally Compact Fields. Arithmetic Groups and Reduction Theory. Adeles. Galois Cohomology. Approximation in Algebraic Groups. Class Numbers andClass Groups of … (Chapter Heading): Algebraic Number Theory. Algebraic Groups. Algebraic Groups over Locally Compact Fields. Arithmetic Groups and Reduction Theory. Adeles. Galois Cohomology. Approximation in Algebraic Groups. Class Numbers andClass Groups of Algebraic Groups. Normal Structure of Groups of Rational Points of Algebraic Groups. Appendix A. Appendix B: Basic Notation. Algebraic Number Theory: Algebraic Number Fields, Valuations, and Completions. Adeles and Ideles Strong and Weak Approximation The Local-Global Principle. Cohomology. Simple Algebras over Local Fields. Simple Algebras over Algebraic Number Fields. Algebraic Groups: Structural Properties of Algebraic Groups. Classification of K-Forms Using Galois Cohomology. The Classical Groups. Some Results from Algebraic Geometry. Algebraic Groups over Locally Compact Fields: Topology and Analytic Structure. The Archimedean Case. The Non-Archimedean Case. Elements of Bruhat-Tits Theory. Results Needed from Measure Theory. Arithmetic Groups and Reduction Theory: Arithmetic Groups. Overview of Reduction Theory: Reduction in GLn(R).Reduction in Arbitrary Groups. Group-Theoretic Properties of Arithmetic Groups. Compactness of GR/GZ. The Finiteness of the Volume of GR/GZ. Concluding Remarks on Reduction Theory. Finite Arithmetic Groups. Adeles: Basic Definitions. Reduction Theory for GA Relative to GK. Criteria for the Compactness and the Finiteness of Volume of GA/GK. Reduction Theory for S-Arithmetic Subgroups. Galois Cohomology: Statement of the Main Results. Cohomology of Algebraic Groups over Finite Fields. Galois Cohomology of Algebraic Tori. Finiteness Theorems for Galios Cohomology. Cohomology of Semisimple Algebraic Groups over Local Fields and Number Fields. Galois Cohomology and Quadratic, Hermitian, and Other Forms. Proof of Theorems 6.4 and 6.6: Classical Groups. Proof of Theorems 6.4 and 6.6: Exceptional Groups. Approximation in Algebraic Groups: Strong and Weak Approximation in Algebraic Varieties. The Kneser-Tits Conjecture. Weak Approximation in Algebraic Groups. The Strong Approximation Theorem. Generalization of the Strong Approximation Theorem. Class Numbers and Class Groups of Algebraic Groups: Class Numbers of Algebraic Groups and Number of Classes in a Genus. Class Numbers and Class Groups of Semisimple Groups of Noncompact Type The Realization Theorem. Class Numbers of Algebraic Groups of Compact Type. Estimating the Class Number for Reductive Groups. The Genus Problem. Normal Subgroup Structure of Groups of Rational Points of Algebraic Groups: Main Conjecture and Results. Groups of Type An. The Classical Groups. Groups Split over a Quadratic Extension. The Congruence Subgroup Problem (A Survey). Appendices: Basic Notation. Bibliography. Index.

On Modular Forms of Half Integral Weight

1973-05-01

Goro Shimura

The recent development of the theory of modular forms and associated zeta functions, together with all its arithmetic significance, is quite pleasing, and our knowledge in this field is evergrowing, … The recent development of the theory of modular forms and associated zeta functions, together with all its arithmetic significance, is quite pleasing, and our knowledge in this field is evergrowing, but the forms of half integral weight have attracted only casual attention, in spite of their importance and ancientness. Indeed, the connection of such forms with zeta functions was never clarified. When Hecke developed his theory of Euler product forthe forms of integral weight, he pointed out the impossibility of a similar theory for the forms of half integral weight, and that only partial information could be obtained for the Fourier coefficients of such forms (Werke, p. 639). He explained this in more detail in his last paper [3], which might have given a rather negative and somewhat misleading impression that one would not be able to do much except in some special cases. A treatment of a more general type of modular form was given by Wohlfahrt [12]. In fact, he defined Hecke operators whose degree is the square of a prime, and showed a certain multiplicative relation, as predicted by Hecke, for the Fourier coefficients, but discussed neither Euler product, nor connection with zeta functions. In the present paper, we try to reveal a more affirmative aspect of the subject. To be specific, put, for each positive integer N,

Affine linear sieve, expanders, and sum-product

2009-11-25

Jean Bourgain , Alex Gamburd , Peter Sarnak

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Table of Integrals, Series, and Products

1980-01-01

Analytic Number Theory

2004-06-08

Henryk Iwaniec , Emmanuel Kowalski

Symmetric random walks on groups

1959-01-01

Harry Kesten

Small eigenvalues of Laplacian for $Γ_{0}(N)$

1990-01-01

Henryk Iwaniec

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Functorial Products for GL 2 × GL 3 and the Symmetric Cube for GL 2

2002-05-01

Henry H. Kim , Freydoon Shahidi

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Topics in Classical Automorphic Forms

1997-10-10

Henryk Iwaniec

Introduction The classical modular forms Automorphic forms in general The Eisenstein and the Poincare series Kloosterman sums Bounds for the Fourier coefficients of cusp forms Hecke operators Automorphic $L$-functions Cusp … Introduction The classical modular forms Automorphic forms in general The Eisenstein and the Poincare series Kloosterman sums Bounds for the Fourier coefficients of cusp forms Hecke operators Automorphic $L$-functions Cusp forms associated with elliptic curves Spherical functions Theta functions Representations by quadratic forms Automorphic forms associated with number fields Convolution $L$-functions Bibliography Index.

Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups

2000-01-01

A. Borel , Nolan R. Wallach

The Description for this book, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups. (AM-94), will be forthcoming. The Description for this book, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups. (AM-94), will be forthcoming.

Zeta Functions of Simple Algebras

1972-01-01

Roger Godement , Hervé Jacquet

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Pseudo-laplaciens. I

1982-01-01

Yves Colin de Verdìère

We construct, on a 2 or 3-dimensional Riemannian manifold, the self-adjoint extensions Δ α,x 0 (α∈R/πZ) of the Laplace operator restricted to the functions vanishing in some neigbhourhood of some … We construct, on a 2 or 3-dimensional Riemannian manifold, the self-adjoint extensions Δ α,x 0 (α∈R/πZ) of the Laplace operator restricted to the functions vanishing in some neigbhourhood of some point x 0 of X. We compute explicitely the eigenvalues of Δ α,x 0 .

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On Euler Products and the Classification of Automorphic Representations I

1981-06-01

H. Jacquet , J. A. Shalika

Hyperbolic distribution problems and half-integral weight Maass forms

1988-02-01

William Duke

Number of Points of Varieties in Finite Fields

2000-01-01

Serge Lang , André Weil

Ramanujan graphs

1988-09-01

Alexander Lubotzky , Ralph S. Phillips , Peter Sarnak

Values ofL-series of modular forms at the center of the critical strip

1981-06-01

Winfried Kohnen , Don Zagier

Fourier coefficients of modular forms of half-integral weight

1987-06-01

Henryk Iwaniec

The limit set of a Fuchsian group

1976-01-01

S. J. Patterson

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Uniform distribution of eigenfunctions on compact hyperbolic surfaces

1987-12-01

Steven Zelditch

On demontre que les fonctions propres {φ k } du laplacien sur une surface hyperbolique compacte X devient uniformement distribuee sur X quand k→∞ On demontre que les fonctions propres {φ k } du laplacien sur une surface hyperbolique compacte X devient uniformement distribuee sur X quand k→∞

On the number of nodal domains of random spherical harmonics

2009-10-01

Fëdor Nazarov , Mikhail Sodin

Let $N(f)$ be a number of nodal domains of a random Gaussian spherical harmonic $f$ of degree $n$. We prove that as $n$ grows to infinity, the mean of $N(f)/n^2$ … Let $N(f)$ be a number of nodal domains of a random Gaussian spherical harmonic $f$ of degree $n$. We prove that as $n$ grows to infinity, the mean of $N(f)/n^2$ tends to a positive constant $a$, and that $N(f)/n^2$ exponentially concentrates around $a$. This result is consistent with predictions made by Bogomolny and Schmit using a percolation-like model for nodal domains of random Gaussian plane waves.

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Zeroes of zeta functions and symmetry

1999-01-01

Nicholas Katz , Peter Sarnak

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On the Holomorphy of Certain Dirichlet Series

1975-07-01

Goro Shimura