maass operators and eisenstein series let g k be a hermitian symmetric space of the non compact type and let f c g be an arithmetic subgroup it is obvious that the enveloping algebra u g 0 of the complexified lie algebra of g gives rise to operators which transform automorphic forms on n with respect to f to f automorphic forms of a different type however it is in general not easy to write these differential operators explicitly in the coordinates of in the case g sp n ir when n is the siegel upper half plane of genus n maass constructed certain explicit operators of this type as a consequence of his construction he was able to show that his operators transform certain simple eisenstein series for f to other specific simple eisenstein series for f for a summary of his results and for references see 6 in this paper we generalize these results of maass to arbitrary hermitian symmetric spaces of the non compact type namely we show that the enveloping algebra u g 0 gives rise to differential operators a such that for certain pairs of canonical automorphy factors j j on g we have aj j cf theorem 2 4 and corollary 3 4 we do this abstractly using the transformation properties of the automorphy factors in particular we do not give explicit formulas for our differential operators which we call maass operators it would be interesting to do so in general we have also restricted our attention to cases in which one of the automorphy factors j j is scalar valued the same methods probably work more generally our interest in these operators arose from their use by shimura in his study of rationality properties for special values of zeta functions see 8 9 and subsequent papers katz 4 has also introduced these operators in another form the author has extended shimura s method to the case of siegel modular forms in 3 and intends to use the results of the present paper in future work of the same kind for this reason a brief and somewhat obtuse discussion of the rationality of maass operators has been included in sect 4 in any specific i e coordinatized example the results of sect 4 can be improved upon

random walks on combs we develop techniques to obtain rigorous bounds on the behaviour of random walks on combs using these bounds we calculate exactly the spectral dimension of random combs with infinite teeth at random positions or teeth with random but finite length we also calculate exactly the spectral dimension of some fixed non translationally invariant combs we relate the spectral dimension to the critical exponent of the mass of the two point function for random walks on random combs and compute mean displacements as a function of walk duration we prove that the mean first passage time is generally infinite for combs with anomalous spectral dimension

boundedness of automorphic l functions in vertical strips the purpose of this paper is to prove the boundedness in vertical strips of finite width for all the l functions that appear in constant terms of eisenstein series under a certain natural assumption on local normalized intertwining operators as a corollary we prove the boundedness for a number of important l functions among them the symmetic cube and triple product l functions attached to cusp forms on gl 2 as well as several rankin selberg product l functions where the local assumption was alredy proved to be valid in each case under the same assumption the finiteness of poles on all of c for all of these l functions is also proved a necessary step in the proof of boundedness in vertical strips

on certain zeta functions attached to two hilbert modular forms ii the case of automorphic forms on a quaternion algebra as to the meaning of the symbols the reader is referred to the introduction of part i we recall here only that f is a totally real algebraic number field of degree n and w denotes the fourier coefficients of an elliptic modular form 52 z e wq a e2q iaz in part i we investigated d assuming that ex a n q s is an l function of a cm field with an algebraic valued hecke character in the present part ii we treat the case where x a can be obtained as eigenvalues of hecke operators t a on a quaternion algebra b over f thus we begin our study by recalling the theory of automorphic forms on the product o r of r copies of the upper half plane with respect to congruence subgroups of b x here r is the number of archimedean primes of f unramified in b then we introduce in section 2 the notion of arithmeticity or q rationality of such automorphic forms and show that the space of automorphic forms can be spanned by the arithmetic ones our main theorems will be stated in section 3 and proved in section 6 sections 4 and 5 are preliminaries to the proof the central result specialized to the case r 1 asserts that if g is a q rational eigenform such that gj t a x a g for all integral ideals a of f then d s0 for certain integers so are algebraic numbers times 7tdg g where d is an integer determined by the weight of g and is the petersson inner product

on mod p properties of siegel modular forms we prove two results on mod p properties of siegel modular forms first we use theta series in order to construct of a siegel modular form of weight p 1 which is congruent to 1 mod p second we define a theta operator vartheta on q expansions and show that the algebra of siegel modular forms mod p is stable under vartheta by exploiting the relation between vartheta and generalized rankin cohen brackets

diluted banded random matrices scaling behavior of eigenfunction and spectral properties we demonstrate that the normalized localization length of the eigenfunctions of diluted sparse banded random matrices follows the scaling law the scaling parameter of the model is defined as where is the average number of non zero elements per matrix row n is the matrix size and additionally we show that also scales the spectral properties of the model up to certain sparsity characterized by the spacing distribution of eigenvalues

special cycles on unitary shimura varieties i unramified local theory the supersingular locus in the fiber at p of a shimura variety attached to a unitary similitude group gu 1 n 1 over is uniformized by a formal scheme mathcal n in the case when p is an inert prime we define special cycles mathcal z bold x in mathcal n associated to collections bold x of m special homomorphisms with fundamental matrix t hermm ok when m n and t is nonsingular we show that the cycle mathcal z bold x is either empty or is a union of components of the ekedahl oort stratification and we give a necessary and sufficient condition in terms of t for mathcal z bold x to be irreducible when mathcal z bold x is zero dimensional in which case it reduces to a single point we determine the length of the corresponding local ring by using a variant of the theory of quasi canonical liftings we show that this length coincides with the derivative of a representation density for hermitian forms

uniform infinite planar triangulations the existence of the weak limit as n of the uniform measure on rooted triangulations of the sphere with n vertices is proved some properties of the limit are studied in particular the limit is a probability measure on random triangulations of the plane

determination of holomorphic modular forms by primitive fourier coefficients we prove that siegel modular forms of degree greater than one integral weight and level n with respect to a dirichlet character chi of conductor mathfrak f chi are uniquely determined by their fourier coefficients indexed by matrices whose contents run over all divisors of n mathfrak f chi the cases of other major types of holomorphic modular forms are included

arithmetic properties of fourier jacobi expansions of automorphic forms in several variables in this paper we establish some basic arithmetic results concerning automorphic forms in several variables our results are in part analogues of the classical facts for elliptic modular forms and results of shimura on hilbert and siegel modular forms 9 and 10 in essence this involves relating rationality properties of fourier jacobi coefficients to rationality properties in algebraic geometry our automorphic forms are c valued holomorphic functions on certain cartesian powers hnd of siegel upper half spaces hn we consider appropriate arithmetic groups r acting on hnd such that r h d is noncompact in particular we reduce our study to that of congruence subgroups r of reductive linear algebraic groups g such that the derived group g is almost simple over q further let k be a maximal compact subgroup of g r the connected component of the identity in the real

fonctions holomorphes a croissance moderee et vecteurs distributions the space of distribution vectors of a holomorphic discrete series representation of scalar type can be identified to a space of holomorphic functions with moderate growth this is the main result of the paper we give an application of it to the hankel transform and to multivariate laguerre series expansions

eisenstein congruence on unitary groups and iwasawa main conjectures for cm fields the purpose of this article is to prove iwasawa main conjecture for cm fields in certain cases through an extensive study on the divisibility relation between p adic l functions for cm fields and eisenstein ideals of unitary groups of degree three

generic ising trees the ising model on a class of infinite random trees is defined as a thermodynamic limit of finite systems a detailed description of the corresponding distribution of infinite spin configurations is given as an application we study the magnetization properties of such systems and prove that they exhibit no spontaneous magnetization furthermore the values of the hausdorff and spectral dimensions of the underlying trees are calculated and found to be respectively and

the adelic zeta function associated with the space of binary cubic forms with coefficients in a function field in this paper we study the adelic zeta function associated with the prehomogeneous vector space of binary cubic forms defined over a function field we establish its rationality find its poles and residues and a simple functional equation that this zeta function satisfies introduction the subject of integral equivalence of binary cubic forms although not as popular or well developed as the theory of integral equivalence of binary quadratic forms has received a fair amount of attention the first paper on this topic 9 written by eisenstein appeared in 1844 later on several other first rate mathematicians tried their hand at the subject among others the list includes arndt cayley hermite and davenport the classical method of attacking this problem is to draw upon a one to one discriminant preserving relation between binary cubic and binary quadratic forms which assigns to a cubic form its hessian and then use the reduction theory for binary quadratic forms to obtain results on class numbers of binary cubic forms the best of the classical results are due to davenport 5 who showed that e h ir n 2 36x o x15 16 nax e h ir n v2 12x o x 5116 nax where h ir n is the class number of irreducible binary cubic forms of discriminant n the credit for a new surge of interest in the study of integral equivalence of binary cubic forms must indisputably go to shintani for it was his 1972 paper 25 that put this study on modern footing in that paper shintani introduced a certain zeta function which as it turns out decomposes into a linear combination of products of dirichlet series whose coefficients are weighted class numbers of binary cubic forms and certain mellin transforms by studying analytic properties of the zeta function shintani was able to improve the error term in davenport s result to o x576 it is of interest to note that the zeta function and the dirichlet series it received by the editors september 3 1985 1980 mathematics subject classification 1985 revision primary 12a70 secondary iocio 12a90 1987 american mathematical society 0002 9947 87 1 00 25 per page

multiply refined enumeration of alternating sign matrices four natural boundary statistics and two natural bulk statistics are considered for alternating sign matrices asms specifically these statistics are the positions of the 1 s in the first and last row and column of an asm and the numbers of generalized inversions and 1 s in an asm previously known results for the exact enumeration of asms with prescribed values of some of these statistics are reviewed in detail a quadratic relation which fully determines the generating function associated with all six statistics is then obtained the derivation of the relation involves combining the desnanot jacobi determinant identity with the izergin korepin formula for the partition function of the six vertex model with domain wall boundary conditions

an explicit formula for siegel series combining induction formulas for local densities with a functional equation for the siegel series we give an explicit formula for the siegel series by this formula we also give an explicit formula for the fourier coefficients of the siegel eisenstein series

correction to poles and residues of standard l functions attached to siegel modular forms correction to my paper on the poles of standard l functions attached to siegel modular forms

localization in band random matrix models with and without increasing diagonal elements it is shown that localization of eigenfunctions in the wigner band random matrix model with increasing diagonal elements can be related to localization in a band random matrix model with random diagonal elements the relation is obtained by making use of a result of a generalization of brillouin wigner perturbation theory which shows that reduced hamiltonian matrices with relatively small dimensions can be introduced for nonperturbative parts of eigenfunctions and by employing intermediate basis states which can improve the method of the reduced hamiltonian matrix the latter model deviates from the standard band random matrix model mainly in two aspects i the root mean square of diagonal elements is larger than that of off diagonal elements within the band and ii statistical distributions of the matrix elements are close to the levy distribution in their central parts except in the high top regions

on the arithmetic of siegel hilbert cuspforms petersson inner products and fourier coefficients the latter is a sort of l indistinguishability result concerning the presumably transcendental petersson norms squared of hecke eigenfunctions the general assertion is essential in discussion of special values of l functions obtained as inner products incidental to the proof of this fact we obtain a very short proof of the important known result that the space of holomorphic cuspforms of such weight and with respect to a principal congruence subyroup is spanned by those with rational fourier coefficients for weights as above rather than starting from the theory of canonical models we begin with consideration of the arithmetic of the fourier coefficients of siegel s eisenstein series this is a relatively elementary issue amenable to direct calculation although ironically the best reference currently available seems to be h3 wherein canonical models results are invoked by contrast previous proofs of results concerning rationality properties of fourier coefficients have relied essentially upon the theory of canonical models the paradigms are the two papers sh2 and sh3 of shimura which depend upon the sequence of his papers culminating in shl which developed the necessary theory of canonical models

theta dichotomy for unitary groups some recent work of gross and prasad 14 suggests that the root numbers attached to certain symplectic representations of the weil deligne group of a local field f control certain branching rules for representations of orthogonal groups over f on a global level this local phenomenon should have implications for the structure of the value at the center of symmetry for certain l functions of arithmetic interest 12 this conjectural picture is based to some extent on the now classic work of tunnell 47 and waldspurger 49 as well as on the case of the triple product l function 13 34 17 in all of these examples the local root number detects the existence of a certain type of invariant linear functional it is possible to set up analogous conjectures for the isometry groups of hermitian or quaternion hermitian forms it turns out that root numbers also play a role in the local and global theta correspondence roughly speaking certain local root numbers should control the occurrence of representations in the local theta correspondence between groups of the same size e g for dual pairs of the form sp n o 2n 1 and u n u n as will be seen the theta correspondence for such pairs is connected with a certain induced representation in s at the point s 0 on the unitary axis by contrast the correspondence for the pairs sp n o 2n and sp n o 2n 2 discussed by prasad 35 is connected to the behavior of a similar induced representation at the points 12 in which case no epsilon factor arises in this paper we consider the local theta correspondence for unitary groups in the non archimedean case let f be a non archimedean local field of characteristic not equal to 2 and let e be a quadratic extension of f for further notation see the notation section at the end of this introduction let v and w be e vector spaces of dimensions m and n equipped respectively with a hermitian form v v e and a skew hermitian form w w e then the isometry groups g v and g w of the spaces v and w form a dual reductive pair in the symplectic group sp w where w v e w viewed as an f vector space of dimension 2mn and equipped with the symplectic form

the fourier expansion of eisenstein series for 3 the fourier expansions of eisenstein series for gl 3 z are obtained by two methods one analogous to the classical method used by many number theorists including weber in his derivation of the kronecker limit formula the other method is analogous to that used by siegel to obtain fourier expansions of eisenstein series for the siegel modular group the expansions involve matrix argument k bessel functions recently studied by tom bengtson these k bessel functions are natural generalizations of the ordinary k bessel function which arise when considering harmonic analysis on the symmetric space of the general linear group using a certain system of coordinates

values of symmetric cube l functions and fourier coefficients of siegel eisenstein series of degree 3 we obtain formulas for certain weighted sums of values of the symmetric square and triple product l functions as a consequence we get exact values at the right critical point for the symmetric square and symmetric cube l functions attached to certain cuspforms we also give applications to fourier coefficients of modular forms

the spectral dimension of random brushes we consider a class of random graphs called random brushes which are constructed by adding linear graphs of random lengths to the vertices of viewed as a graph we prove that for d 2 all random brushes have spectral dimension ds 2 for d 3 we have and for d 4 we have 3 ds d

zeta functions and eisenstein series on classical groups we construct an euler product from the hecke eigenvalues of an automorphic form on a classical group and prove its analytic continuation to the whole complex plane when the group is a unitary group over a cm field and the eigenform is holomorphic we also prove analytic continuation of an eisenstein series on another unitary group containing the group just mentioned defined with such an eigenform as an application of our methods we prove an explicit class number formula for a totally definite hermitian form over a cm field

the selberg trace formula for sl 3 ℤ sl 3 ℝ so 3 ℝ in this paper we compute the trace formula for sl 3 z in detail and refine it to a greater extent than has previously been done we show that massive cancellation occurs in the parabolic terms leading to a far simpler formula than had been thought possible

derivative of the standard p adic l function associated with a siegel form this work has greatly bene ted from an excellent long stay at columbia university founded by a fwo travel grant v4 260 14n while the author was a fwo phd fellow at ku leuven

zeta functions and eisenstein series on metaplectic groups the main objectives of this paper are 1 to construct an euler product from a hecke eigenform of half integral weight 2 to prove its meromorphic continuation to the whole plane 3 to examine its relationship with the fourier coefficients of the form and 4 to apply these results to the investigations of eisenstein series of half integral weight the automorphic forms are holomorphic ones considered on the metaplectic cover m of g where g n sp n f with a totally real algebraic number field f the first three are generalizations of the corresponding results in the elliptic and hilbert modular cases obtained in 2 and 6 at the same time all four closely parallel similar results in the case of integral weight in 9 and s10 to describe our results let us take for simplicity f to be q in which case the weight k is half an odd positive integer we consider a holomorphic form f of weight k with respect to a principal congruence subgroup of sp n z of level n which is an eigenform in the sense that f it m 2 m f with a complex number 2 m where n is a multiple of 4 and t m is a hecke operator defined for each positive integer m in a natural way as our first main result we shall prove theorem 4 4 that

appearance of vertices of infinite order in a model of random trees we study an equilibrium statistical mechanical model of tree graphs which are made up of a linear subgraph the spine to which leaves are attached we prove that the model has two phases a generic phase where the spine becomes infinitely long in the thermodynamic limit and all vertices have finite order and a condensed phase where the spine is finite with probability 1 and a single vertex of infinite order appears in the thermodynamic limit we calculate the spectral dimension of the graphs in both phases and prove the existence of a gibbs measure we discuss generalizations of this model and the relationship with models of nongeneric random trees

on the bloch kato conjecture for elliptic modular forms of square free level let kappa ge 6 be an even integer m an odd square free integer and f in s 2 kappa 2 gamma 0 m a newform we prove that under some reasonable assumptions that half of the lambda part of the bloch kato conjecture for the near central critical value l kappa f is true we do this by bounding the ell valuation of the order of the appropriate bloch kato selmer group below by the ell valuation of algebraic part of l kappa f we prove this by constructing a congruence between the saito kurokawa lift of f and a cuspidal siegel modular form

on certain zeta functions attached to two hilbert modular forms i the case of hecke characters to make our exposition smooth let us first introduce some notational conventions for an algebraic number field k of finite degree we denote by jk the set of all embeddings of k into c and by ik the free z module generated by the elements of jk we then put rik ik0 r and cik ik0 c if p eapau e ik with a e jk and po e z we put xp fl xa pfor 0 x e k this is meaningful for p e cik if xa are all real and positive throughout the paper we denote by d the complex upper half plane and by f a totally real algebraic number field of degree n now the zeta function to be studied in this paper when suitably specialized has the form

on differential operators attached to certain representations of classical groups classification et construction de sous espaces irreductibles de sr integrales contenant les applications polynomiales l operateur differentiel d operateurs differentiels de type arithmetique applications aux series de eisenstein

condensation in nongeneric trees we study nongeneric planar trees and prove the existence of a gibbs measure on infinite trees obtained as a weak limit of the finite volume measures it is shown that in the infinite volume limit there arises exactly one vertex of infinite degree and the rest of the tree is distributed like a subcritical galton watson tree with mean offspring probability m 1 we calculate the rate of divergence of the degree of the highest order vertex of finite trees in the thermodynamic limit and show it goes like 1 m n where n is the size of the tree these trees have infinite spectral dimension with probability one but the spectral dimension calculated from the ensemble average of the generating function for return probabilities is given by 2 2 if the weight wn of a vertex of degree n is asymptotic to n

statistical ensemble of scale free random graphs a thorough discussion of the statistical ensemble of scale free connected random tree graphs is presented methods borrowed from field theory are used to define the ensemble and to study analytically its properties the ensemble is characterized by two global parameters the fractal and the spectral dimensions which are explicitly calculated it is discussed in detail how the geometry of the graphs varies when the weights of the nodes are modified the stability of the scale free regime is also considered when it breaks down either a scale is spontaneously generated or else a singular node appears and the graphs become crumpled a new computer algorithm to generate these random graphs is proposed possible generalizations are also discussed in particular more general ensembles are defined along the same lines and the computer algorithm is extended to arbitrary degenerate scale free random graphs

confluent hypergeometric functions on tube domains our problem specialized to the case of the siegel upper half space h of degree m concerns the fourier expansion of a series s z fl det z a det a a l here z is a variable on h l is a lattice in the space v of all real symmetric matrices of size m and fl ec it can be shown that this is convergent if re f l m and also that if re m 2 and r e f l m 2 it has a fourier expansion of the form

scaling properties of multilayer random networks multilayer networks are widespread in natural and manmade systems key properties of these networks are their spectral and eigenfunction characteristics as they determine the critical properties of many dynamics occurring on top of them here we numerically demonstrate that the normalized localization length ensuremath beta of the eigenfunctions of multilayer random networks follows a simple scaling law given by ensuremath beta x 1 x with x ensuremath gamma b mathrm eff 2 l ensuremath delta ensuremath delta ensuremath sim 1 and b mathrm eff being the effective bandwidth of the adjacency matrix of the network whose size is l the scaling law for ensuremath beta that we validate on real world networks might help to better understand criticality in multilayer networks and to predict the eigenfunction localization properties of them

terms in the selberg trace formula for 3 ℒ 3 ℛ 3 ℛ associated to eisenstein series coming from a minimal parabolic subgroup in this paper we compute the contribution to the trace formula for sl 3 z of the integrals associated to inner products of eisenstein series we show these reduce to corresponding integrals for a lower rank trace formula plus a few residual terms

critical values of l functions on gsp 2 gl 2 this paper deals with deligne s conjecture on the critical values of l functions let z g h s denote the tensor product l function attached to a siegel modular form g of weight k and an elliptic cusp form h of weight l we assume that the first fourier jacobi coefficient of g is not identically zero then deligne s conjecture is fully proven for z g h s when l 2k 2 and partly for the remaining case

correspondence among eisenstein series of weight 2 abstract there is a known correspondence among modular forms jacobi forms and siegel modular forms of genus 2 in this paper we show this correspondence can be extended to non holomorphic eisenstein series in particular among e 2 2 z 0 e 2 1 z 0 and f 3 2 0

the spectral dimension of generic trees we define generic ensembles of infinite trees these are limits as n of ensembles of finite trees of fixed size n defined in terms of a set of branching weights among these ensembles are those supported on trees with vertices of a uniformly bounded order the associated probability measures are supported on trees with a single spine and hausdorff dimension dh 2 our main result is that the spectral dimension of the ensemble average is ds 4 3 and that the critical exponent of the mass defined as the exponential decay rate of the two point function along the spine is 1 3