on the genericity of cuspidal automorphic forms of so2n 1 this is a sequel of our work js06 we extend moeglin s results m97a and m97b from the even orthogonal groups to old orthogonal groups and complete our proof of the cap conjecture for irreducible cuspidal automoprhic representations of so2n 1 a with special bessel models we also give characterization of the vanishing of the central value of the standard l function of so2n 1 a in terms of theta correspondence as result we obtain the weak langlands functorial transfer from so2n 1 a to gl2n a for irreducible cuspidal automoprhic representations of so2n 1 a with special bessel models

potential symmetries of the nonlinear wave equation utt uux x and related exact and approximate solutions this paper explores potential symmetries of the nonlinear wave equation utt uux x as well as related new similarity reductions and exact solutions of this equation new approximate solutions of the perturbed nonlinear equations stemming from the exact solutions of the equation are obtained by applying a new approach to the use of the lie group technique for differential equations dependent on a small parameter in addition some nonlinear wave equations exactly reducible to the equation utt uux x are constructed using this approach

non classical symmetry reduction example of the boussinesq equation a symmetry of an equation will leave the set of all solutions invariant a conditional symmetry will leave only a subset of solutions defined by some differential condition invariant the authors show how a specific class of conditional symmetries can be used to reduce a partial differential equation to an ordinary one in particular for the boussinesq equation these conditional symmetries together with the ordinary ones provide all possible reductions to ordinary differential equations a group theoretical explanation of the recently obtained new reductions is provided

remarks on classical invariant theory a uniform formulation applying to all classical groups simultaneously of the first fundamental theory of classical invariant theory is given in terms of the weyl algebra the formulation also allows skew symmetric as well as symmetric variables examples illustrate the scope of this formulation

on the μ invariant of anticyclotomic p adic l functions for cm fields following hida s ideas we determine the invariant of the anticyclotomic katz p adic l functions for self dual characters when the branch conductor is not divisible by inert or ramified primes above 2 thus we generalize finis result to general cm fields in this case

automorphic forms with degenerate fourier coefficients the main theme of this paper is that singular automorphic forms on classical groups are given by theta series liftings we establish several inequalities relating the automorphic multiplicities of a given representation and that of its abstract theta lift our methods allow one to lift noncuspidal square integrable automorphic forms when the dual pair involved is in the stable range in this way we construct new families of singular automorphic forms many of which are clearly unipotent in fact starting from one dimensional representations and repeating the procedure of lifting in the stable range one may obtain all automorphic forms which are quadratic unipotent in the sense of moeglin

new similarity reductions of the boussinesq equation some new similarity reductions of the boussinesq equation which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering are presented these new similarity reductions including some new reductions to the first second and fourth painleve equations cannot be obtained using the standard lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method that involves no group theoretical techniques

conditional lie backlund symmetry and reduction of evolution equations we suggest a generalization of the notion of invariance of a given partial differential equation with respect to a lie backlund vector field such a generalization proves to be effective and enables us to construct principally new ansatz reducing evolution type equations to several ordinary differential equations in the framework of the said generalization we obtain principally new reductions of a number of nonlinear heat conductivity equations ut uxx f u ux with poor lie symmetry and obtain their exact solutions it is shown that these solutions cannot be constructed by means of the symmetry reduction procedure

solution free sets for sums of binary forms in this paper we obtain quantitative estimates for the asymptotic density of subsets of the integer lattice z2 that contain only trivial solutions to an additive equation involving binary forms in the process we develop an analogue of vinogradov s mean value theorem applicable to binary forms

the regularized siegel weil formula the second term identity and the rallis inner product formula in this paper we establish the second term identity of the siegel weil formula in full generality and derive the rallis inner product formula for global theta lifts for any dual pair as a corollary we resolve the non vanishing problem of global theta lifts initiated by steve rallis

when nonlinear differential equations are equivalent to linear differential equations a necessary and sufficient condition is established for the existence of a 1 1 transformation of a system of nonlinear differential equations to a system of linear equations the obtained theorems enable one to construct such transformations from the invariance groups of differential equations the hodograph transformation the legendre transformation and lie s transformation of the monge ampsre equation are shown to be special cases noninvertible transformations are also considered examples include burgers equation a nonlinear diffusion equation and the liouville equation introduction in this work we study transformations mapping nonlinear differen tial equations to linear differential equations in a 1 1 manner based upon the group analysis of differential equations we obtain necessary and sufficient conditions for the existence of such transformations the established theorems not only allow us to determine the existence of the transformations but also enable us to actually construct these transformations from invariance groups of the nonlinear equations in the following analysis two types of transformations are considered 1 the invariance groups of differential equations and 2 the mappings which transform nonlinear differential equations to linear differential equations theorems will be proved based upon the following observations clearly if there exists a 1 1 mapping between any two differential equations it must inject properties of one equation into the other including their invariance properties for this reason the often ignored fact that any linear differential equation admits an invariance group related to the superpo sition principle becomes very significant the generator of this particular group depends upon an arbitrary solution of the linear equation it follows then that any nonlinear equation transformable to a linear equation by a 1 1 mappingmust admit an invariance group whose generator depends upon an arbitrary solution of some linear differential equation the idea of comparing invariance groups of differential equations in the search of mappings connecting the equations was first used by bluman in his study of burgers equation 1 and it was applied to the study of the mappings of one dimensional linear parabolic equations to the heat equation 2 in the first section of this paper some basic properties of various transformations are summarized the theorems are presented in the second section the examples in the third section include the hodograph transformation the legendre transformation lie s transformation of themonge ampre equation and the equation ux uxx uyy 0 in the last section we examine properties of noninvertible mappings of burgers equation the diffusion equation u 2ux uy 0 and the liouville equation u y e u respectively into linear equations

new exact solutions of burgers s equation an extension to the direct method of clarkson and kruskal new exact solutions of burgers s equation which is the simplest evolution equation to embody nonlinearity and dissipation are obtained these solutions are obtained by extending the method for finding symmetry reductions due to clarkson and kruskal new similarity solutions of the boussinesq equation j math phys 30 2201 2213 1989 the novel feature of this extended method is that one can seek reductions to a system of ordinary differential equations rather than the usual single equation and this leads to a wider class of solutions one is able to complete the calculations necessary for burgers s equation in all but a single case where it is necessary to integrate an abel equation of the second kind this indicates that the method is practicable in contrast it is often the case that new methods are of limited use in practice in particular solutions in terms of a class of confluent hypergeometric functions are computed by comparison solutions found by the original method due to clarkson and kruskal and other reduction methods are in terms of parabolic cylinder functions or airy functions which are special cases of this class

regularized theta liftings and periods of modular functions in this paper we use regularized theta liftings to construct weak maass forms of weight 1 2 as lifts of weak maass forms of weight 0 as a special case we give a new proof of some of recent results of duke toth and the third author on cycle integrals of the modular j invariant and extend these to any congruence subgroup moreover our methods allow us to settle the open question of a geometric interpretation for periods of j along infinite geodesics in the upper half plane in particular we give the central value of the non existent l function for j the key to the proofs is the construction of a kind of simultaneous green function for both the cm points and the geodesic cycles which is of independent interest

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

improved kinetic renormalisation group approach to diffusion limited aggregation an improved kinetic renormalisation group approach to diffusion limited aggregation is presented this approach is based on the growth process itself and accounts for the dispersity in the growth probabilities it yields the multifractal spectrum d q with better values at smaller q on the 2d square lattice the authors find df 1 694 for the fractal dimensions of the cluster and that of its interface and d 1 1 01 for the information dimension the former agrees with the simulation results df approximately 1 70 and the latter compares very well with the exact value d 1 1

on approximate symmetry and approximate solutions of the nonlinear wave equation with a small parameter the concept of approximate symmetry is introduced the authors describe all nonlinearities f u with which the nonlinear wave equation square operator u lambda u3 epsilon f u 0 with a small parameter epsilon is approximately scale and conformally invariant some approximate solutions of wave equations in question are obtained using the approximate symmetry

on the siegel weil formula for unitary groups following s s kudla and s rallis we extend the siegel weil formula for unitary groups which relates a value of a siegel eisenstein series to the convergent integral of a theta function

new similarity reductions of the steady state boundary layer equations new similarity reductions and exact solutions of the steady state boundary layer equations which are of considerable importance in physics are presented besides similarity reductions obtained by applying the non classical group method due to bluman and cole some similarity solutions which figure neither among classical nor among non classical reductions are determined using an extension of the non classical method

endoscopy theta liftings and period integrals for the unitary group in three variables l packets for the quasi split unitary group in three variables u 3 we shall give an explicit parametrization of these l packets using theta liftings and describe some relations between the structure of l packets and certain period integrals we also prove that an endoscopic l packet contains a unique generic representation in local and global cases suppose that g is a reductive algebraic group over a local or global field

the mathrm pgsp _ 1 1 mathbb a pgsp1 1 a distinguished representations in this paper we use the regularized quaternionic siegel weil formula to deal with the mathrm pgsp 1 1 mathbb a period problem for a cuspidal representation tau of mathrm pgsp 4 mathbb a e where e f is a quadratic extension of number fields and e is contained in the 4 dimensional quaternion division algebra d of f with s d 2

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

transcending classical invariant theory let sp2 r sp2 sp be the real symplectic group of rank n let sp denote the two fold cover of sp there is a unitary representation constructed by shale sh and weil wa of sp that is of considerable interest in the theory of automorphic forms we denote this representation by co and call it the oscillator representation the purpose of this paper is to establish about c a fact that should be useful for clarifying the structure of spaces of automorphic forms constructed using co we begin by formulating the result let e be a reductive subgroup of sp and let e denote the inverse image of e in sp denote by r e the set of infinitesimal equivalence classes of continuous irreducible admissible representations of e on locally convex spaces let co be the smooth representation of sp associated to co let co be realized on a hilbert space and let the subspace of smooth vectors on which space co is defined be written denote by r9 e co the set of elements of e which are realized as quotients by co e invariant closed subspaces of t g consider a reductive dual pair g g c sp h2 it is not hard to show that g and g commute with one another hence we may regard co i g g as a representation of gx g it is well known f that r gx g s9 g xr9 g the identification associates to p e r g and p e i g the tensor product p x p we note that p x p is not defined as a topological vector space nevertheless the infinitesimal equivalence class of p op is well defined select p x p e g x g suppose that in fact p p e 9 g g c then clearly p e r g c and p e 9 g c hence g co defines the graph of a correspondence between certain subsets of r9 g cv and j g co in fact the situation is quite precise

a weak second term identity of the regularized siegel weil formula for unitary groups following w t gan and s takeda we obtain a weak second term identity of the regularized siegel weil formula for the unitary dual pair u n n u v where v is a split hermitian space of dimension 2r with r 1 le n le 2r 1 as an application we obtain a rallis inner product formula for theta lifts from u w to u v for a skew hermitian space w of dimension n

solutions of the nonlinear wave equation invariant under conditional symmetries in this paper we discuss the application of conditional symmetries for obtaining exact and approximate solutions of the equation we propose to use the invariance properties of the infinitesimal conditional symmetries with respect to the classical lie symmetries in order to explicitly solve the determining equations for the conditional symmetries exact solutions invariant under the conditional symmetries are given by explicit formulae approximate solutions are calculated and presented in the graphical form by using mathematica

l functions and theta correspondence for classical groups using the doubling method of piatetski shapiro and rallis we develop a theory of local factors of representations of classical groups and apply it to give a necessary and sufficient condition for nonvanishing of global theta liftings in terms of analytic properties of the l functions and local theta correspondence

applications of nonclassical symmetries in this paper we discuss new applications of the nonclassical method of symmetry reduction the equations treated are well known in mathematical physics and are used to describe multisoliton solutions of the burger equation they are also useful in the study of kac moody algebras and in the examination of nonlinear waves in the theory of elasticity with the help of the nonclassical symmetry method the partial differential equations are reduced to ordinary differential equations which are solvable explicitly

faltings heights of abelian varieties with complex multiplication let m be the shimura variety associated with the group of spinor similitudes of a quadratic space over q of signature n 2 we prove a conjecture of bruinier kudla yang relating the arithmetic intersection multiplicities of special divisors and big cm points on m to the central derivatives of certain l functions as an application of this result we prove an averaged version of colmez s conjecture on the faltings heights of cm abelian varieties

some local global non vanishing results for theta lifts from orthogonal groups we firstly improve a theorem of b roberts which characterizes non vanishing of a global theta lift from o x to sp n in terms of non vanishing of local theta lifts in particular we will remove all the archimedean conditions imposed upon his theorem sec ondly following roberts we will apply our theorem to theta lifting of low rank similitude groups namely we characterize the non vanishing condition of a global theta lift from go 4 to gsp 2 in our improved setting also we consider non vanishing conditions of a global theta lift from go 4 to gsp 1 and explicitly compute the lift when it exists

trilinear forms and the central values of triple product l functions we give an explicit formula for certain global trilinear forms that appear in jacquet s conjecture in terms of local trilinear forms and the central values of triple product l functions

doubling constructions and tensor product l functions the linear case we present an integral representation for the tensor product l function of a pair of automorphic cuspidal representations one of a classical group the other of a general linear group our construction is uniform over all classical groups and is applicable to all cuspidal representations it does not require genericity the main new ideas of the construction are the use of generalized speh representations as inducing data for the eisenstein series and the introduction of a new global and local model which generalizes the whittaker model here we consider linear groups but our construction also extends to arbitrary degree metaplectic coverings this will be the topic of an upcoming work

green forms and the arithmetic siegel weil formula we construct natural green forms for special cycles in orthogonal and unitary shimura varieties in all codimensions and for compact shimura varieties of type mathrm o p 2 and mathrm u p 1 we show that the resulting local archimedean height pairings are related to special values of derivatives of siegel eisentein series a conjecture put forward by kudla relates these derivatives to arithmetic intersections of special cycles and our results settle the part of his conjecture involving local archimedean heights

new classes of symmetries for partial differential equations new classes of symmetries for partial differential equations are introduced by writing a given partial differential equation s in a conserved form a related system t with potentials as additional dependent variables is obtained the lie group of point transformations admitted by t induces a symmetry group of s new symmetries may be obtained for s that are neither point nor lie backlund symmetries they are determined by a completely algorithmic procedure significant new symmetries are found for the wave equation with a variable wave speed and the nonlinear diffusion equation

generalised form of a conjecture of jacquet and a local consequence following the work of harris and kudla we prove a general form of a conjecture of jacquet relating the non vanishing of a certain period integral to non vanishing of the central critical value of a certain l function as a consequence we deduce a theorem relating the existence of gl 2 k invariant linear forms on irreducible admissible representations of gl 2 k for a commutative semi simple cubic algebra k over a non archimedean local field k in terms of local epsilon factors which was proved only in some cases by the first author in his earlier work in d prasad invariant forms for representations of gl 2 over a local field amer j math 114 1992 no 6 1317 1363 this has been achieved by globalising a locally distinguished supercuspidal representation to a globally distinguished representation a result of independent interest which is proved by an application of the relative trace formula

a regularized siegel weil formula for unitary groups for the quasi split unitary group u n n we prove a refined analogue of the result of s s kudla and s rallis on an identity a regularized siegel weil formula between a residue of an eisenstein series and the integral of a certain theta function

the hodge conjecture and arithmetic quotients of complex balls let s be a closed shimura variety uniformized by the complex n ball associated with a standard unitary group the hodge conjecture predicts that every hodge class in h 2k s mathbb q k 0 dots n is algebraic we show that this holds for all degrees k away from the neighborhood bigl tfrac13n tfrac23n bigr of the middle degree we also prove the tate conjecture for the same degrees as the hodge conjecture and the generalized form of the hodge conjecture in degrees away from an interval depending on the codimension c of the subvariety centered at the middle dimension of s these results are derived from a general theorem that applies to all shimura varieties associated with standard unitary groups of any signature the proofs make use of arthur s endoscopic classification of automorphic representations of classical groups as such our results rely on the stabilization of the trace formula for the disconnected groups gl n rtimes langle theta rangle associated with base change

on the non vanishing of theta liftings of tempered representations of u p q abstract in this paper we give an explicit determination of the non vanishing of the theta liftings of tempered representations for unitary dual pairs u p q u r s for arbitrary non negative integers p q r s for discrete series representations in terms of harish chandra parameters we give a complete criterion when the theta lifts are nonzero for tempered representations we determine the non vanishing in terms of the local langlands correspondence assuming the local gan gross prasad conjecture

e factor of representations of classical groups abstract we let g be the isometry group of a nondegenerate symmetric or skew symmetric form h on a space v over some local or global field k i e g u v h we let lg be the l group of go the identity component of g and r be the natural representation of this group we develop here a theory of l functions and e factors attached to this r in particular we develop a theory of local and global zeta integrals for such g we use very special intertwining operators therein we require an analysis of the proper normalization of such operators with such techniques we determine the correct e factors associated to local representations the issue of the correct local l factor is discussed

group invariant solutions of differential equations we introduce the concept of a weak symmetry group of a system of partial differential equations that generalizes the nonclassical method introduced by bluman and cole for finding group invariant