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

pullbacks of saito kurokawa lifts pullbacks of siegel eisenstein series have been studied by garrett 17 18 bocherer 6 7 heim 26 and play a key role in the proof of the algebraicity of critical values of certain automorphic l functions more generally one might consider pullbacks of siegel cusp forms for example ikeda 30 gave a conjectural formula for pullbacks of ikeda lifts 29 in terms of critical values of l functions for spn gl2 also the gross prasad conjecture 22 23 8 27 8 would relate pullbacks of siegel cusp forms of degree 2 to central critical values of l functions for gsp2 gl2 gl2 indeed bocherer furusawa and schulze pillot 8 gave an explicit formula for pullbacks of yoshida lifts 57 in this paper we give an explicit formula for pullbacks of saito kurokawa lifts and prove the algebraicity of central critical values of certain l functions for sp1 gl2 to be precise let be an odd positive integer let f s2 sl2 z be a normalized hecke eigenform and h s 1 2 0 4 a hecke eigenform associated to f by the shimura correspondence let f s 1 sp2 z be the saito kurokawa lift of h for each normalized hecke eigenform g s 1 sl2 z we consider the period integral f h h g g given by f h h g g

some local global non vanishing results of theta lifts for symplectic orthogonal dual pairs following the approach of b roberts we characterize the non vanishing of the global theta lift for symplectic orthogonal dual pairs in terms of its local counterpart in particular we replace the temperedness assumption present in robert s work by a certain weaker assumption and apply our results to small rank similitude groups among our applications is a certain instance of langlands functorial transfer of a non generic cuspidal automorphic representation of gsp 4 to gl 4

theta dichotomy and doubling epsilon factors for spn f in this work we aim to prove the theta dichotomy conjecture for dual pairs of type sp w o v with dim v odd the crux of the proof relies on the analysis of the reducibility of a certain degenerate principal series representation of the group sp w having double the rank of sp w this analysis is carried out by way of a normalized standard intertwining operator which is also used to define an epsilon factor coming from the doubling integral construction of piatetski shapiro and rallis in this paper we also illustrate the connection between the doubling epsilon factor and the theta dichotomy conjecture

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

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

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

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

triple product formula and the subconvexity bound of triple product l function in level aspect in this paper we derive a nice general formula for the local integrals of triple product formula whenever one of the representations has sufficiently higher level than the other two as an application we generalize venkatesh and woodbury s work on the subconvexity bound of triple product l function in level aspect allowing joint ramifications higher ramifications general unitary central characters and general special values of local epsilon factors

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

on the siegel weil formula for quaternionic unitary groups we extend the siegel weil formula to all quaternion dual pairs applications include the classification problem of skew hermitian forms over a quaternion algebra over a number field and a product formula for the weighted average of the representation numbers of a skew hermitian form by another skew hermitian form

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

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

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

construction of discrete series for classical p adic groups in this paper the authors complete the classification of irreducible square integrable representations of classical p adic groups assuming a natural assumption which is expected to hold in general this classification implies a parameterization of irreducible tempered representations of these groups and it implies a classification of the non unitary duals of these groups modulo cuspidal data

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

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

one point second order curved boundary condition for lattice boltzmann simulation of suspended particles abstract the lattice boltzmann method lbm has been widely considered as a distinctive and reliable approach for simulating the complex particulate flows as an intrinsic kinetic scheme it is quite convenient for lbm to apply the bounce back bb type methods to handle the moving boundaries with complicated geometry which is a tricky task for general interface resolved methods however the two major schemes in lbm i e the simple bb rule and the curved boundary condition cbc are presently encountered by the problems of low precision and loss of local computation respectively to overcome those two deficiencies in the boundary treatment simultaneously a one point second order cbc is proposed in this paper information of only a single node is required in the present scheme and the second order accuracy is validated in the channel flow and cylindrical couette flow applications to the particulate flows are further implemented to verify the present scheme numerical results are in good agreement with those in the literature

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

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

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

a note on operator inequalities of tsallis relative operator entropy tsallis relative operator entropy was defined as a parametric extension of relative operator entropy and the generalized shannon inequalities were shown in the previous paper after the review of some fundamental properties of tsallis relative operator entropy some operator inequalities related to tsallis relative operator entropy are shown in the present paper our inequalities give the upper and lower bounds of tsallis relative operator entropy the operator equality on tsallis relative operator entropy is also shown by considering the tensor product this relation generalizes the pseudoadditivity for tsallis entropy as a corollary of our operator equality derived from the tensor product manipulation we show several operator inequalities including the superadditivity and the subadditivity for tsallis relative operator entropy our results are generalizations of the superadditivity and the subadditivity for tsallis entropy