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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., Comment: 53 p

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., Comment: 17 pages

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., Comment: Minor edits

We prove that the non-backtracking random walk on Ramanujan graphs with large girth exhibits the fastest possible cutoff with a bounded window., Comment: 16 pages, the second version fixes some minor typos

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.

Explicit examples of {\bf positive} crystalline measures and Fourier quasicrystals are constructed using pairs of stable of polynomials, answering several open questions in the area., Comment: 15 pages, 1 figure

For integers $k$, we consider the affine cubic surface $V_{k}$ given by $M({\bf x})=x_{1}^2 + x_{2}^2 +x_{3}^2 -x_{1}x_{2}x_{3}=k$. We show that for almost all $k$ the Hasse Principle holds, namely that $V_{k}(\mathbb{Z})$ is non-empty if $V_{k}(\mathbb{Z}_p)$ is non-empty for all primes $p$, and that there are infinitely many $k$'s for which it fails. The Markoff morphisms act on $V_{k}(\mathbb{Z})$ with finitely many orbits and a numerical study points to some basic conjectures about these "class numbers" and Hasse failures. Some of the analysis may be extended to less special affine cubic surfaces., Comment: This is the final version of this paper. The published version of this paper contains an abridged Sec. 10 on computations. We make the full version accessible here. There are various updates, in particular references to the papers [CTWX20], [LM20] and [GMS22] on failures to the Hasse Principle. The published version has some additional differences

To each of the symmetry groups of the Platonic solids we adjoin a carefully designed involution yielding topological generators of PU(2) which have optimal covering properties as well as efficient navigation. These are a consequence of optimal strong approximation for integral quadratic forms associated with certain special quaternion algebras and their arithmetic groups. The generators give super efficient 1-qubit quantum gates and are natural building blocks for the design of universal quantum gates.

We prove results pertaining to strong approximation for Markoff triples in the case of prime moduli., Comment: This is the first of three papers giving detailed proofs of the results announced in [BGS16]

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., Comment: Final version. To appear in the special issue of the Bulletin of the Iranian Mathematical Society (BIMS) in honor of Freydoon Shahidi's 70th birthday

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\"of 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., Comment: 43 pages, 11 figures

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

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., Comment: 9 pages. arXiv admin note: substantial text overlap with arXiv:1303.4957

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.

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., Comment: 49 pages, 16 figures

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.

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., Comment: 42 pages

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.

This paper deals with some questions that have received a lot of attention since they were raised by Hejhal and Rackner in their 1992 numerical computations of Maass forms. We establish sharp upper and lower bounds for the $L^2$-restrictions of these forms to certain curves on the modular surface. These results, together with the Lindelof Hypothesis and known subconvex $L^\infty$-bounds are applied to prove that locally the number of nodal domains of such a form goes to infinity with its eigenvalue., Comment: To appear in GAFA

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, Comment: 2 figures. Included reviewer comments. Accepted for the Contemporary Mathematics proceedings of Constructive Functions 2014

We formulate and prove a finite version of Vinogradov's bilinear sum inequality. We use it together with Ratner's joinings theorems to prove that the Mobius function is disjoint from discrete horocycle flows on $\Gamma \backslash SL_2(\mathbb{R}).

We establish the main saturation conjecture in [BGS10] 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.

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.

We prove various orbital counting statements for Fuchsian groups of the second kind. These are of independent interest, and also are used in the companion paper [BourgainKontorovich2009] to produce primes in the Affine Linear Sieve., Comment: 32 pages

A celebrated theorem of Selberg states that for congruence subgroups of SL(2,Z) there are no exceptional eigenvalues below 3/16. We prove a generalization of Selberg's theorem for infinite index "congruence" subgroups of SL(2,Z). Consequently we obtain sharp upper bounds in the affine linear sieve, where in contrast to \cite{BGS} we use an archimedean norm to order the elements.

We develop the affine sieve in the context of orbits of congruence subgroups of semi-simple groups acting linearly on affine space. In particular we give effective bounds for the saturation numbers for points on such orbits at which the value of a given polynomial has few prime factors. In many cases these bounds are of the same quality as what is known in the classical case of a polynomial in one variable and the orbit is the integers. When the orbit is the set of integral matrices of a fixed determinant we obtain a sharp result for the saturation number, and thus establish the Zariski density of matrices all of whose entries are prime numbers. Among the key tools used are explicit approximations to the generalized Ramanujan conjectures for such groups and sharp and uniform counting of points on such orbits when ordered by various norms.

We determine the variance for the fluctuations of the arithmetic measures obtained by collecting all closed geodesics on the modular surface with the same discriminant and ordering them by the latter. This arithmetic variance differs by subtle factors from the variance that one gets when considering individual closed geodesics when ordered by their length. The arithmetic variance is the same one that appears in the fluctuations of measures associated with quantum states on the modular surface., Comment: Added a brief comment concerning Duke's theorem on the unit tangent bundle

The existence of a strong spectral gap for quotients $\Gamma\bs G$ of noncompact connected semisimple Lie groups is crucial in many applications. For congruence lattices there are uniform and very good bounds for the spectral gap coming from the known bounds towards the Ramanujan-Selberg Conjectures. If $G$ has no compact factors then for general lattices a strong spectral gap can still be established, however, there is no uniformity and no effective bounds are known. This note is concerned with the strong spectral gap for an irreducible co-compact lattice $\Gamma$ in $G=\PSL(2,\bbR)^d$ for $d\geq 2$ which is the simplest and most basic case where the congruence subgroup property is not known. The method used here gives effective bounds for the spectral gap in this setting., Comment: 35 pages. Final version (to appear in JEMS)

Motivated by problems of mathematical physics (quantum chaos) questions of equidistribution of eigenfunctions of the Laplace operator on a Riemannian manifold have been studied by several authors. We consider here, in analogy with arithmetic hyperbolic surfaces, orthonormal bases of eigenfunctions of the Laplace operator on the two dimensional unit sphere which are also eigenfunctions of an algebra of Hecke operators which act on these spherical harmonics. We formulate an analogue of the equidistribution of mass conjecture for these eigenfunctions as well as of the conjecture that their moments tend to moments of the Gaussian as the eigenvalue increases. For such orthonormal bases we show that these conjectures are related to the analytic properties of degree eight arithmetic L-functions associated to triples of eigenfunctions. Moreover we establish the conjecture for the third moments and give a conditional (on standard analytic conjectures about these arithmetic L-functions) proof of the equdistribution of mass conjecture., Comment: 18 pages, an appendix gives corrections to the article "On the central critical value of the triple product L-function" (In: Number Theory 1993-94, 1-46. Cambridge University Press 1996) by Siegfried Boecherer and Rainer Schulze-Pillot. Revised version (minor revisions, new abstract), paper to appear in Communications in Mathematical Physics

We study the distribution of normalized spacings between the fractional parts of an^2, n=1,2,.... We conjecture that if a is "badly approximable" by rationals, then the sequence of fractional parts has Poisson spacings, and give a number of results towards this conjecture. We also present an example of a Diophantine number a for which the higher correlation functions of the sequence blow up., Comment: Substantial revisions in the exposition. Accepted for publication in Inventiones mathematicae

In Iwaniec-Sarnak [IS] the percentages of nonvanishing of central values of families of GL_2 automorphic L-functions was investigated. In this paper we examine the distribution of zeros which are at or neat s=1/2 (that is the central point) for such families of L-functions. Unlike [IS], most of the results in this paper are conditional, depending on the Generalized Riemann Hypothesis (GRH). It is by no means obvious, but on the other hand not surprising, that this allows us to obtain sharper results on nonvanishing., Comment: Abstract added in migration (from introduction)

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 functorial 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., Comment: 5 pages