the measurable kesten theorem we give an explicit bound on the spectral radius in terms of the densities of short cycles in finite d regular graphs it follows that the a finite d regular ramanujan graph g contains a negligible number of cycles of size less than c log log g we prove that infinite d regular ramanujan unimodular random graphs are trees through benjamini schramm convergence this leads to the following rigidity result if most eigenvalues of a d regular finite graph g fall in the alon boppana region then the eigenvalue distribution of g is close to the spectral measure of the d regular tree in particular g contains few short cycles in contrast we show that d regular unimodular random graphs with maximal growth are not necessarily trees institute of mathematical statistics 2016

on the growth of l 2 invariants for sequences of lattices in lie groups we study the asymptotic behaviour of betti numbers twisted torsion and other spectral invariants of sequences of locally symmetric spaces our main results are uniform versions of the degeorge wallach theorem of a theorem of delorme and various other limit multiplicity theorems a basic idea is to adapt the notion of benjamini schramm convergence bs convergence originally introduced for sequences of finite graphs of bounded degree to sequences of riemannian manifolds and analyze the possible limits we show that bs convergence of locally symmetric spaces g k implies convergence in an appropriate sense of the normalized relative plancherel measures associated to l2 g this then yields convergence of normalized multiplicities of unitary representations betti numbers and other spectral invariants on the other hand when the corresponding lie group g is simple and of real rank at least two we prove that there is only one possible bs limit i e when the volume tends to infinity locally symmetric spaces always bs converge to their universal cover g k this leads to various general uniform results when restricting to arbitrary sequences of congruence covers of a fixed arithmetic manifold we prove a strong quantitative version of bs convergence which in turn implies upper estimates on the rate of convergence of normalized betti numbers in the spirit of sarnak xue an important role in our approach is played by the notion of invariant random subgroups for higher rank simple lie groups g we exploit rigidity theory and in particular the nevo stuck zimmer theorem and kazhdan s property t to obtain a complete understanding of the space of irs s of g 2017 department of mathematics princeton university

a tower of coverings of quasi projective varieties abstract the main goal of this article is to relate asymptotic geometric properties on a tower of coverings of a non compact kahler manifold of finite volume with reasonable geometric assumptions to its universal covering examples to which our findings are applicable include moduli spaces of hyperbolic punctured riemann surfaces and hermitian locally symmetric spaces of finite volume

counting and equidistribution for quaternion algebras we aim at studying automorphic forms of bounded analytic conductor in the totally definite quaternion algebra setting we prove the equidistribution of the universal family with respect to an explicit and geometrically meaningful measure it leads to answering the sato tate conjectures in this case and contains the counting law of the universal family with a power savings error term

positive representations of general commutation relations allowing wick ordering we consider the problem of representing in hilbert space commutation relations of the form graphics where the tklij are essentially arbitrary scalar coefficients examples comprise the q canonical commutation relations introduced by greenberg bozejko and speicher and the twisted canonical anti commutation relations studied by pusz and woronowicz as well as the quantum group s u 2 using these relations any polynomial in the generators ai and their adjoints can uniquely be written in wick ordered form in which all starred generators are to the left of all unstarred ones in this general framework we define the fock representation as well as coherent representations we develop criteria for the natural scalar product in the associated representation spaces to be positive definite and for the relations to have representations by bounded operators in a hilbert space we characterize the relations between the generators ai not involving a i which are compatible with the basic relations the relations may also be interpreted as defining a non commutative differential calculus for generic coefficients tklij however all differential forms of degree 2 and higher vanish we exhibit conditions for this not to be the case and relate them to the ideal structure of the wick algebra and conditions of positivity we show that the differential calculus is compatible with the involution iff the coefficients t define a representation of the braid group this condition is also shown to imply improved bounds for the positivity of the fock representation finally we study the kms states of the group of gauge transformations defined by aj exp it aj

poincare series for non riemannian locally symmetric spaces abstract we initiate the spectral analysis of pseudo riemannian locally symmetric spaces g h beyond the classical cases where h is compact automorphic forms or is trivial analysis on symmetric spaces for any non riemannian reductive symmetric space x g h on which the discrete spectrum of the laplacian is nonempty and for any discrete group of isometries whose action on x is sufficiently proper we construct l 2 eigenfunctions of the laplacian on x x for an infinite set of eigenvalues these eigenfunctions are obtained as generalized poincare series i e as projections to x of sums over the orbits of eigenfunctions of the laplacian on x we prove that the poincare series we construct still converge and define nonzero l 2 functions after any small deformation of inside g for a large class of groups thus the infinite set of eigenvalues we construct is stable under small deformations this contrasts with the classical setting where the nonzero discrete spectrum varies on the teichmuller space of a compact riemann surface we actually construct joint l 2 eigenfunctions for the whole commutative algebra of invariant differential operators on x

central limit theorems for elliptic curves and modular forms with smooth weight functions abstract in 11 the second and third named authors established a central limit theorem for the error term in the sato tate law for families of modular forms this method was adapted to families of elliptic curves in 3 by the first and second named authors in this context a central limit theorem was established only under a strong hypothesis going beyond the riemann hypothesis in the present paper we consider a smoothed version of the sato tate conjecture which allows us to overcome several limitations in particular for the smoothed version we are able to establish a central limit theorem for much smaller families of modular forms and we succeed in proving a theorem of this type for families of elliptic curves under the riemann hypothesis for l functions associated to hecke eigenforms for the full modular group

asymptotic trace formula for the hecke operators given integers m n and k we give an explicit formula with an optimal error term with square root cancelation for the petersson trace formula involving the mth and nth fourier coefficients of an orthonormal basis of s k left n right the weight k newforms with fixed square free level n provided that 4 pi sqrt mn k o left k frac 1 3 right moreover we establish an explicit formula with a power saving error term for the trace of the hecke operator mathcal t n on s k left n right averaged over k in a short interval by bounding the second moment of the trace of mathcal t n over a larger interval we show that the trace of mathcal t n is unusually large in the range 4 pi sqrt n k o left n frac 1 6 right as an application for any fixed prime p coprime to n we show that there exists a sequence k n of weights such that the error term of weyl s law for mathcal t p is unusually large and violates the prediction of arithmetic quantum chaos in particular this generalizes the result of gamburd et al j eur math soc 1 1 51 85 1999 theorem 1 4 with an improved exponent

weighted averages of modular l values using an explicit relative trace formula on gl 2 we derive a formula for averages of modular l values in the critical strip weighting by fourier coefficients hecke eigenvalues and petersson norms as an application we show that a grh holds for these averages as the weight or the level goes to we also use the formula to give explicit zero free regions of the form im s 0 for some particular modular l functions

ihara s zeta function for periodic graphs and its approximation in the amenable case abstract in this paper we give a more direct proof of the results by clair and mokhtari sharghi b clair s mokhtari sharghi zeta functions of discrete groups acting on trees j algebra 237 2001 591 620 on the zeta functions of periodic graphs in particular using appropriate operator algebraic techniques we establish a determinant formula in this context and examine its consequences for the ihara zeta function moreover we answer in the affirmative one of the questions raised in r i grigorchuk a uk the ihara zeta function of infinite graphs the kns spectral measure and integrable maps in v a kaimanovich et al eds proc workshop random walks and geometry vienna 2001 de gruyter berlin 2004 pp 141 180 by grigorchuk and uk accordingly we show that the zeta function of a periodic graph with an amenable group action is the limit of the zeta functions of a suitable sequence of finite subgraphs

on positive definite solutions to the algebraic riccati equation abstract complete necessary and sufficient conditions for the existence of a positive definite solution to the algebraic riccati equation are given it is also shown that when a positive definite solution exists it is either unique or else there are uncountably many such solutions

eigenvalues of graphs and a simple proof of a theorem of greenberg abstract in his ph d thesis greenberg proved that if x is the spectral radius of the universal cover x of a finite graph x then for each 0 a positive proportion depending only on x and of the eigenvalues of x have absolute value at least x in this paper we show that the same result holds true if we remove absolute from the previous result we also prove an analogue result for the smallest eigenvalues of x

transcendence of hecke operators in the big hecke algebra we prove a transcendence result of hecke operator t l over the ring generated by diamond operators over in a non cm component of the cyclotomic ordinary big p adic hecke algebra and we discuss its conjectural implication of a modulo p version to the vanishing of the cyclotomic invariant 2014

corps de nombres peu ramifies et formes automorphes autoduales let s be a finite set of primes p in s and q s a maximal algebraic extension of q unramified outside s and infinity assume that s 2 we show that the natural maps gal q p bar q p gal q s q are injective much of the paper is devoted to the problem of constructing selfdual automorphic cuspidal representations of gl 2n a q with prescribed properties at all places that we study via the twisted trace formula of j arthur the techniques we develop shed also some lights on the orthogonal symplectic alternative for selfdual representations of gl 2n

on the asymptotic isoperimetric constants for riemann surfaces and graphs we study the behavior of the cheeger isoperimetric constant on infinite fam ilies of graphs and riemann surfaces and its relationship to the first eigen value 1 of the laplacian we adapt probabilistic arguments of bollobas to the setting of riemann surfaces and then show that cheeger constants of the modular surfaces are uniformly bounded from above away from the maximum value we extend this result to the class of ramanujan surfaces defined below

existence condition of positive definite solutions for algebraic matrix riccati equations abstract algebraic matrix riccati equations are considered which arise in the optimal filtering as well as in control problems of continuous time invariant systems a necessary and sufficient condition is established for the existence of unique positive definite solutions and the asymptotically stable closed loop system the condition is stated in terms of the detectability and stabilizability of certain matrix pairs as was the case for the non negative definite solution simple examples are included to illustrate the new results against the existing ones

wick product for commutation relations connected with yang baxter operators and new constructions of factors we analyze a certain class of von neumann algebras generated by selfadjoint elements for satisfying the general commutation relations such algebras can be continuously embedded into some closure of the set of finite linear combinations of vectors where is an orthonormal basis of a hilbert space the operator which represents the vector is denoted by and called the wick product of the operators we describe explicitly the form of this product also we estimate the operator norm of for finally we apply these two results and prove that under the assumption all the von neumann algebras considered are ii 1 factors

hypercontractivity for a quantum ornstein uhlenbeck semigroup we prove hypercontractivity for a quantum ornstein uhlenbeck semigroup on the entire algebra mathcal b h of bounded operators on a separable hilbert space h we exploit the particular structure of the spectrum together with hypercontractivity of the corresponding birth and death process and a proper decomposition of the domain then we deduce a logarithmic sobolev inequality for the semigroup and gain an elementary estimate of the best constant

q gaussian processes non commutative and classical aspects we examine for 1 q 1 q gaussian processes i e families of operators non commutative random variables where the a t fulfill the q commutation relations for some covariance function equipped with the vacuum expectation state we show that there is a q analogue of the gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on q gaussian processes into corresponding and much simpler questions in the underlying hilbert space in particular we use this idea to show that a large class of q gaussian processes possesses a non commutative kind of markov property which ensures that there exist classical versions of these non commutative processes this answers an old question of frisch and bourret fb

noncommutative levy processes for generalized particularly anyon statistics let t mathbb r d let a function qt 2 to mathbb c satisfy q s t overline q t s and q s t 1 a generalized statistics is described by creation operators partial t dagger and annihilation operators t t in t which satisfy the q commutation relations partial s partial dagger t q s t partial dagger t partial s delta s t partial s partial t q t s partial t partial s partial dagger s partial dagger t q t s partial dagger t partial dagger s from the point of view of physics the most important case of a generalized statistics is the anyon statistics for which q s t is equal to q if s t here q in mathbb c q 1 we start the paper with a detailed discussion of a q fock space and operators partial t dagger partial t t in t in it which satisfy the q commutation relations next we consider a noncommutative stochastic process white noise omega t partial t dagger partial t lambda partial t dagger partial t t in t here lambda in mathbb r is a fixed parameter the case 0 corresponds to a q analog of brownian motion while 0 corresponds to a centered q poisson process we study q hermite q charlier respectively polynomials of infinitely many noncommutatative variables omega t t in t the main aim of the paper is to explain the notion of independence for a generalized statistics and to derive corresponding levy processes to this end we recursively define q cumulants of a field xi t t in t this allows us to define a q levy process as a field xi t t in t whose values at different points of t are q independent and which possesses a stationarity of increments in a certain sense we present an explicit construction of a q levy process and derive a nualart schoutens type chaotic decomposition for such a process

limit formulas of period integrals for a certain symmetric pair abstract let g be the unitary group of a non degenerate hermitian space and h the stabilizer of a one dimensional positive definite subspace of the hermitian space for a uniform lattice in g such that h is a uniform lattice of h we introduce the averaged h period integrals of automorphic forms on g we study their behavior as shrinks to the identity along a tower of lattices in g and prove a limit formula of the h period integrals

riesz transforms on deformed fock spaces the fock von neumann algebra equipped with its canonical trace is spanned by n hermitian operators acting on a hilbert fock space some commutation relations between and are defined by the n n hermitian matrix a we define a riesz transform where is the number operator is aninner derivation unbounded in general and let 1 p we prove that is equivalent to for every with null trace with constants which do not depend on n

periodic solutions of riccati equations applied to multirate sampling from some basic relations in estimation theory a comprehensive set of formulae are obtained for the periodic solution of riccati equations to obtain numerically stable algorithms square root formulations are introduced both the discrete time and continuous time cases are considered closed loop properties for periodic systems are also investigated the results are applied to multirate measurement sampling the control of a continuous stirred tank reactor serves as an illustrative example the temperature measurements are then complemented by concentration analysis available at a slower rate than the basic sampling and control rate the results give some guidelines about the benefits of multirate sampling

the dimensions of spaces of siegel cusp forms of general degree abstract in this paper we give a dimension formula for spaces of siegel cusp forms of general degree with respect to neat arithmetic subgroups the formula was conjectured before by several researchers cf 40 43 45 75 the dimensions are expressed by special values of shintani zeta functions for spaces of symmetric matrices at non positive integers this formula was given by shintani for only a small part of the geometric side of the trace formula see 64 to be precise it is the contribution of unipotent elements corresponding to the partitions 2 j 1 2 n 2 j where n denotes the degree and 0 j n hence our work is to show that all the other contributions vanish therefore one finds that shintani s formula means the dimension itself combining our formula and an explicit formula of the shintani zeta functions which was discovered by ibukiyama and saito cf 44 42 61 we can derive an explicit dimension formula for the principal congruence subgroups of level greater than two cf 43 45 in this explicit dimension formula the dimensions are described by degree n weight k level n and the bernoulli numbers b m

asymptotique des nombres de betti des varietes arithmetiques we study the question of the growth of betti numbers of certain arithmetic varieties in tower of congruence coverings in fact our results are about siegel varieties and varieties associated to orthogonal groups we explain how a theorem of waldspurger can be used to obtain lower and upper bound our results are in the direction of conjectures made by sarnak and xue

an explicit trace formula of jacquet zagier type for hilbert modular forms abstract we give an exact formula of the average of adjoint l functions of holomorphic hilbert cusp forms with a fixed weight and a square free level which is a generalization of zagier s formula known for the case of elliptic cusp forms on sl 2 z as an application we prove that the satake parameters of hilbert cusp forms with a fixed weight and with growing square free levels are equidistributed in an ensemble constructed by values of the adjoint l functions

fourier coefficients of automorphic forms and integrable discrete series abstract let g be the group of r points of a semisimple algebraic group g defined over q assume that g is connected and noncompact we study fourier coefficients of poincare series attached to matrix coefficients of integrable discrete series we use these results to construct explicit automorphic cuspidal realizations which have appropriate fourier coefficients 0 of integrable discrete series in families of congruence subgroups in the case of g sp 2 n r we relate our work to that of li 14 for g quasi split over q we relate our work to the result about poincare series due to khare larsen and savin 12

l2 index and the selberg trace formula abstract a method is developed for computing the l 2 index of a locally symmetric elliptic differential operator d acting on a locally symmetric manifold m g k with g semisimple of real rank one and of finite co volume based on applying the selberg trace formula to the difference of the two heat kernels associated to d the applications include an extension of the osborne warner multiplicity formula to certain non integrable discrete series derived from the l 2 spinor formula and showing the existence in some cases of non invariant l 2 cohomology classes in the middle dimension via the l 2 signature formula

on a theorem of osborne and warner multiplicities in the cuspidal spectrum osborne and warner have given a formula for the multiplicity of an integrable discrete series representation in l2 g when g is real rank one semi simple lie group and is a discrete subgroup of co finite volume we simplify and evaluate this formula to show that for most g as above the multiplicity is the formal degree of the representation times the volume of g when it is not we give a simple interpretation of the difference

random matrices nonbacktracking walks and orthogonal polynomials several well known results from the random matrix theory such as wigner s law and the marchenko pastur law can be interpreted and proved in terms of nonbacktracking walks on a certain graph orthogonal polynomials with respect to the limiting spectral measure play a role in this approach

hypercontractivity on the q araki woods algebras extending a work of carlen and lieb biane has obtained the optimal hypercontractivity of the q ornstein uhlenbeck semigroup on the q deformation of the free group algebra in this note we look for an extension of this result to the type iii situation that is for the q araki woods algebras we show that hypercontractivity from l p to l 2 can occur if and only if the generator of the deformation is bounded

elliptic curves over a finite field and the trace formula we prove formulas for power moments for point counts of elliptic curves over a finite field k such that the groups of k points of the curves contain a chosen subgroup these formulas express the moments in terms of traces of hecke operators for certain congruence subgroups of sl 2 z as our main technical input we prove an eichler selberg trace formula for a family of congruence subgroups of sl 2 z which include as special cases the groups 1 n and n our formulas generalize results of birch and ihara the case of the trivial subgroup and the full modular group and previous work of the authors the subgroups z 2z and z 2z 2 and congruence subgroups 0 2 0 4 we use these formulas to answer statistical questions about point counts for elliptic curves over a fixed finite field generalizing results of vladu gekeler howe and others

hypercontractivity in non commutative holomorphic spaces we prove an analog of janson s strong hypercontractivity inequality in a class of non commutative holomorphic algebras our setting is the q gaussian algebras q associated to the q fock spaces of bozejko kummerer and speicher for q 1 1 we construct subalgebras open image in new window a q segal bargmann transform and prove janson s strong hypercontractivity open image in new window for r an even integer

strong limit multiplicity for arithmetic hyperbolic surfaces and 3 manifolds we show that every sequence of torsion free arithmetic congruence lattices in mathrm pgl 2 mathbb r or mathrm pgl 2 mathbb c satisfies a strong quantitative version of the limit multiplicity property we deduce that for r 0 in certain range growing linearly in the degree of the invariant trace field the volume of the r thin part of any congruence arithmetic hyperbolic surface or congruence arithmetic hyperbolic 3 manifold m is of order at most mathrm vol m 11 12 as an application we prove gelander s conjecture on homotopy type of arithmetic hyperbolic 3 manifolds we show that there are constants a b such that every such manifold m is homotopy equivalent to a simplicial complex with at most a mathrm vol m vertices all of degrees bounded by b

characters l 2 betti numbers and an equivariant approximation theorem let g be a group with a finite subgroup h we define the l 2 multiplicity of an irreducible representation of h in the l 2 homology of a proper g cw complex these invariants generalize the l 2 betti numbers our main results are approximation theorems for l 2 multiplicities which extend the approximation theorems for l 2 betti numbers of luck farber and elek szabo respectively the main ingredient is the theory of characters of infinite groups and a method to induce characters from finite subgroups we discuss applications to the cohomology of arithmetic groups

smooth cuspidal automorphic forms and integrable discrete series in this paper we construct smooth cuspidal automorphic forms related to integrable discrete series of a connected semisimple lie group with finite center for classical and adelic situation as an application of the theory of schwartz spaces for automorphic forms developed by casselman in the classical situation smooth cuspidal automorphic forms are constructed via an explicit continuous map from the frechet space of smooth vectors of a banach realization inside l 1 g of an integrable discrete series into the space of smooth vectors of a strong topological dual of an appropriate schwartz space

a trace on fractal graphs and the ihara zeta function starting with ihara s work in 1968 there has been a growing interest in the study of zeta functions of finite graphs by sunada hashimoto bass stark and terras mizuno and sato to name just a few authors then clair and mokhtari sharghi studied zeta functions for infinite graphs acted upon by a discrete group of automorphisms the main formula in all these treatments establishes a connection between the zeta function originally defined as an infinite product and the laplacian of the graph in this article we consider a different class of infinite graphs they are fractal graphs i e they enjoy a self similarity property we de ne a zeta function for these graphs and using the machinery of operator algebras we prove a determinant formula which relates the zeta function with the laplacian of the graph we also prove functional equations and a formula which allows approximation of the zeta function by the zeta functions of finite subgraphs

on limit multiplicites of discrete series representations in spaces of automorphic forms of course the groups f are then arithmetic subgroups of g in this situation it seems natural to assume that as n the spectral decomposition of l2 f g will approximate that of g in particular let 3 be a discrete series representation of g if f is a subgroup of g let re f 6 be the multiplicity of 3 in the representation of g by right translations on l2 f g choose a haar measure dg on g if f is arithmetic let v f g be the finite volume of f g for the measure induced by dg let d 3 be the formal degree of 3 for the measure dg

sato tate theorem for families and low lying zeros of automorphic l functions we consider certain families of automorphic representations over number fields arising from the principle of functoriality of langlands let g be a reductive group over a number field f which admits discrete series representations at infinity let l g widehat g rtimes mathrm gal bar f f be the associated l group and r l g rightarrow mathrm gl d mathbb c a continuous homomorphism which is irreducible and does not factor through mathrm gal bar f f the families under consideration consist of discrete automorphic representations of g mathbb a f of given weight and level and we let either the weight or the level grow to infinity we establish a quantitative plancherel and a quantitative sato tate equidistribution theorem for the satake parameters of these families this generalizes earlier results in the subject notably of sarnak prog math 70 321 331 1987 and serre j am math soc 10 1 75 102 1997 as an application we study the distribution of the low lying zeros of the associated family of l functions l s pi r assuming from the principle of functoriality that these l functions are automorphic we find that the distribution of the 1 level densities coincides with the distribution of the 1 level densities of eigenvalues of one of the unitary symplectic and orthogonal ensembles in accordance with the katz sarnak heuristics we provide a criterion based on the frobenius schur indicator to determine this symmetry type if r is not isomorphic to its dual r vee then the symmetry type is unitary otherwise there is a bilinear form on mathbb c d which realizes the isomorphism between r and r vee if the bilinear form is symmetric resp alternating then r is real resp quaternionic and the symmetry type is symplectic resp orthogonal

contractivity properties of ornstein uhlenbeck semigroup for general commutation relations we study a certain class of von neumann algebras generated by selfadjoint elements i a i a i where a i a i satisfy the general commutation relations

asymptotic behavior of supercuspidal representations and sato tate equidistribution for families abstract we establish properties of families of automorphic representations as we vary prescribed supercuspidal representations at a given finite set of primes for the tame supercuspidals constructed by j k yu we prove the limit multiplicity property with error terms thereby we obtain a sato tate equidistribution for the hecke eigenvalues of these families the main new ingredient is to show that the orbital integrals of matrix coefficients of tame supercuspidal representations with increasing formal degree on a connected reductive p adic group tend to zero uniformly for every noncentral semisimple element

equidistribution theorems for holomorphic siegel modular forms for gsp_4 hecke fields and n level density this paper is a continuation of kim et al j inst math jussieu 2018 we supplement four results on a family of holomorphic siegel cusp forms for gsp 4 mathbb q first we improve the result on hecke fields namely we prove that the degree of hecke fields is unbounded on the subspace of genuine forms which do not come from functorial lift of smaller subgroups of gsp 4 second we prove simultaneous vertical sato tate theorem namely we prove simultaneous equidistribution of hecke eigenvalues at finitely many primes third we compute the n level density of degree 4 spinor l functions and thus we can distinguish the symmetry type depending on the root numbers this is conditional on certain conjecture on root numbers fourth we consider equidistribution of paramodular forms in this case we can prove the conjecture on root numbers main tools are the equidistribution theorem in our previous work and shin templier s compos math 150 12 2003 2053 2014 work

eigenvalue spacings for quantized cat maps according to one of the basic conjectures in quantum chaos the eigenvalues of a quantized chaotic hamiltonian behave like the spectrum of the typical member of the appropriate ensemble of random matrices we study one of the simplest examples of this phenomenon in the context of ergodic actions of groups generated by several linear toral automorphisms cat maps our numerical experiments indicate that for generic choices of cat maps the unfolded consecutive spacing distribution in the irreducible components of the nth quantization given by the n dimensional weil representation approaches the goe gse law of random matrix theory for certain special arithmetic transformations related to the ramanujan graphs of lubotzky phillips and sarnak the experiments indicate that the unfolded consecutive spacing distribution follows poisson statistics we provide a sharp estimate in that direction

a density theorem on automorphic l functions and some applications we establish a density theorem of automorphic l functions and give some applications on the extreme values of these l functions at s 1 and the distribution of the hecke eigenvalue of holomorphic cusp forms contents

comments on existence conditions of positive definite solutions for algebraic matrix riccati equations abstract it is pointed out that the main results in a recent paper by kano 1987 have previously been published in the literature

when kloosterman sums meet hecke eigenvalues by elaborating a two dimensional selberg sieve with asymptotics and equidistributions of kloosterman sums from ell adic cohomology as well as a bombieri vinogradov type mean value theorem for kloosterman sums in arithmetic progressions it is proved that for any given primitive hecke maass cusp form of trivial nebentypus the eigenvalue of the n th hecke operator does not coincide with the kloosterman sum mathrm kl 1 n for infinitely many squarefree n with at most 100 prime factors this provides a partial negative answer to a problem of katz on modular structures of kloosterman sums

non injectivity of the q deformed von neumann algebra in this paper we prove that the von neumann algebra generated by q gaussians is not injective as soon as the dimension of the underlying hilbert space is greater than 1 our approach is based on a suitable vector valued khintchine type inequality for wick products the same proof also works for the more general setting of a yang baxter deformation our techniques can also be extended to the so called q araki woods von neumann algebras recently introduced by hiai in this latter case we obtain the non injectivity under some asssumption on the spectral set of the positive operator associated with the deformation

optimal hypercontractivity for fermi fields and related non commutative integration inequalities optimal hypercontractivity bounds for the fermion oscillator semigroup are obtained these are the fermion analogs of the optimal hypercontractivity bounds for the boson oscillator semigroup obtained by nelson in the process several results of independent interest in the theory of non commutative integration are established

periodic solutions of matrix riccati equations in discrete time invariant systems the well known matrix riccati equations arising in optimal filtering as well as control problems for discrete time invariant systems are considered in this paper existence conditions are established both for general periodic solutions and for non negative definite periodic solutions it is shown that a pair of complex conjugate eigenvalues or a pair of real eigenvalues in a certain symplectic matrix produce periodic solutions furthermore algorithms are obtained to derive such periodic solutions using the symplectic matrix which require slight additional computations to deriving steady state solutions finally numerical examples which possess periodic solutions are presented

averages of central l values of hilbert modular forms with an application to subconvexity we use the relative trace formula to obtain exact formulas for central values of certain twisted quadratic base change l functions averaged over hilbert modular forms of a fixed weight and level we apply these formulas to the subconvexity problem for these l functions we also establish an equidistribution result for the hecke eigenvalues weighted by these l values

bounds for multiplicities of unitary representations of cohomological type in spaces of cusp forms let goo be a semisimple real lie group with unitary dual goo we produce new upper bounds for the multiplicities with which representations e of cohomological type appear in certain spaces of cusp forms on goo the main new idea is to apply noncommutative iwasawa theory to certain p adic completions of the cohomology of locally symmetric spaces

on the structure of non commutative white noises we consider the concepts of continuous bernoulli systems and non commutative white noises we address the question of isomorphism of continuous bernoulli systems and show that for large classes of quantum levy processes one can make quite precise statements about the time behaviour of their moments