quantum field theory and hopf algebra cohomology we exhibit a hopf superalgebra structure of the algebra of field operators of quantum field theory qft with the normal product based on this we construct the operator product and the time ordered product as a twist deformation in the sense of drinfeld our approach yields formulae for perturbative products and expectation values that allow for a significant enhancement in computational efficiency as compared to traditional methods employing hopf algebra cohomology sheds new light on the structure of qft and allows the extension to interacting not necessarily perturbative qft we give a reconstruction theorem for time ordered products in the spirit of streater and wightman and recover the distinction between free and interacting theory from a property of the underlying cocycle we also demonstrate how non trivial vacua are described in our approach solving a problem in quantum chemistry

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 an easy transition from operator dynamics to generating functionals by clifford algebras clifford geometric algebras of multivectors are treated in detail these algebras are built over a graded space and exhibit a grading or multivector structure the careful study of the endomorphisms of this space makes it clear that opposite clifford algebras have to be used for obtaining all endomorphisms based on these mathematics we give a fully clifford algebraic account on generating functionals which is thereby geometric the field operators are shown to be clifford and opposite clifford maps this picture relying on geometry does not need positivity in principle furthermore we propose a transition from operator dynamics to generating functionals which is based on the algebraic techniques as a calculational benefit this transition is considerably short compared to standard ones the transition is not injective unique and depends additionally on the choice of an ordering we obtain a direct and constructive connection between orderings and the explicit form of the functional hamiltonian these orderings depend on the propagator of the theory and thus on the ground state this is invisible in path integral formulations the method is demonstrated within two examples a nonlinear spinor field theory and spinor qed antisymmetrized and normal ordered functional equations are derived in both cases

the local langlands conjecture for gsp 4 we prove the local langlands conjecture for gsp4 f where f is a nonarchimedean local eld of characteristic zero

bilinear covariants and spinor fields duality in quantum clifford algebras classification of quantum spinor fields according to quantum bilinear covariants is introduced in a context of quantum clifford algebras on minkowski spacetime once the bilinear covariants are expressed in terms of algebraic spinor fields the duality between spinor and quantum spinor fields can be discussed thus by endowing the underlying spacetime with an arbitrary bilinear form with an antisymmetric part in addition to a symmetric spacetime metric quantum algebraic spinor fields and deformed bilinear covariants can be constructed they are thus compared to the classical non quantum ones classes of quantum spinor fields classes are introduced and compared with lounesto s spinor field classification a physical interpretation of the deformed parts and the underlying z grading is proposed the existence of an arbitrary bilinear form endowing the spacetime already has been explored in the literature in the context of quantum gravity s w hawking the unpredictability of quantum gravity commun

on moments of twisted l functions we study the average of the product of the central values of two l functions of modular forms f and g twisted by dirichlet characters to a large prime modulus q as our principal tools we use spectral theory to develop bounds on averages of shifted convolution sums with differences ranging over multiples of q and we use the theory of deligne and katz to prove new bounds on bilinear forms in kloosterman sums with power savings when both variables are near the square root of q when at least one of the forms f and g is non cuspidal we obtain an asymptotic formula for the mixed second moment of twisted l functions with a power saving error term in particular when both are non cuspidal this gives a significant improvement on m young s asymptotic evaluation of the fourth moment of dirichlet l functions in the general case the asymptotic formula with a power saving is proved under a conjectural estimate for certain bilinear forms in kloosterman sums

cuspidality of symmetric powers with applications the purpose of this paper is to prove that the symmetric fourth power of a cusp form on rm gl 2 whose existence was proved earlier by the first author is cuspidal unless the corresponding automorphic representation is of dihedral tetrahedral or octahedral type as a consequence we prove a number of results toward the ramanujan petersson and sato tate conjectures in particular we establish the bound q sp 1 9 sb v for unramified hecke eigenvalues of cusp forms on rm gl 2 over an arbitrary number field this is the best bound available at present

l functions for the p adic group gsp 4 in this paper we compute the local l factors for novodvorsky integrals for all generic representations of the group gsp 4 over a nonarchemidean local field 1 introduction in this paper we will study the p adic theory of novod vorsky integrals for the similitude symplectic group gsp 4 and will present the computation of the nonarchemidean l factors given by these integrals for all generic representations of the group these integrals which were introduced by m novodvorsky in the corvallis conference 17 serve as one of the few avail able integral representations for the spin l function of gsp 4 unfortunately novodvorsky s paper is somewhat sketchy and skips the proofs some of the details missing in novodvorsky s original paper have been reproduced in daniel bump s survey article 1 the reader is advised to consult bump s paper for local unramified computations the proof of the euler product decomposition of the global integral and other interesting results david soudry has generalized the integrals considered here to orthogonal groups of arbitrary odd degree the local theory of soudry s integrals appear in his memoire paper 27 our motivation for this work comes from the work of masaaki furusawa and joseph shalika on the special values of l functions of gsp 4 using the relative trace formula 8 where they need precise information about the local behavior of novodvorsky integrals we now describe our method following godement 9 the main idea is to determine the germ expansions of whittaker functions when restricted to the maximal torus in the siegel parabolic subgroup the functionals that appear as coefficients in these germ expansions are easily seen to belong to the dual of a certain twisted jacquet module we are most interested in the eigenspace decom position of this dual module under the action of the siegel torus this decomposi tion essentially determines the germ expansion and naturally the local l factor to do explicit computations one uses the fact that germ expansions must be invari ant under intertwining operators by the multiplicity one theorem of shalika 25 up to certain constants i e local coefficients these local coefficients have been extensively studied by shahidi in 22 23 15 away from the poles of inter twining operators this simple argument determines the germ expansion to extend

on the local langlands conejcture mathrm gl n the cyclic case let f be a locally compact nonarchimedean field and p its residue characteristic let f be a separable algebraic closure of f wf the weil group of f over f e a character of fx of order n and e the corresponding character of wf then there exists a canonical map from the set of equivalence classes a of irreducible degree n representations of wf such that e a a to the set of equivalence classes 7r of admissible irreducible supercuspidal representations of gl n f verifying odet x 7t this map is shown to be bijective and to preserve e factors for pairs

une preuve simple des conjectures de langlands pour gl n sur un corps p adique let f be a finite extension of p for each integer n 1 we construct a bijection from the set f 0 n of isomorphism classes of irreducible degree n representations of the absolute weil group of f onto the set f 0 n of isomorphism classes of smooth irreducible supercuspidal representations of gl n f those bijections preserve epsilon factors for pairs and hence we obtain a proof of the langlands conjectures for gl n over f which is more direct than harris and taylor s our approach is global and analogous to the derivation of local class field theory from global class field theory we start with a result of kottwitz and clozel on the good reduction of some shimura varieties and we use a trick of harris who constructs non galois automorphic induction in certain cases

higher coherent cohomology and p adic modular forms of singular weights we investigate the p adic properties of higher coherent cohomology of automorphic vector bundles of singular weights on siegel threefolds

vertex normal ordering as a consequence of nonsymmetric bilinear forms in clifford algebras we consider clifford algebras with nonsymmetric bilinear forms they parametrize the chosen ideal in the isomorphism class of the standard symmetric ones since the content of physical theories depends on the injection n nv cl v q one has to transform to the standard construction the injection is described by the antisymmetric part of the bilinear form this process results in the appropriate vertex normal ordering terms which are now obtained from the theory itself and not added ad hoc via a regularization argument

on the hopf algebraic origin of wick normal ordering a combinatorial formula of rota and stein is taken to perform wick reordering in quantum field theory wick s theorem becomes a hopf algebraic identity called cliffordization the combinatorial method relying on hopf algebras is highly efficient in computations and yields closed algebraic expressions

on the ramanujan conjecture and finiteness of poles for certain l functions in this paper we prove a number of general results about automorphic l functions which appear in the constant terms of eisenstein series for an arbitrary quasi split connected reductive algebraic group over a number field these l functions were first introduced by langlands for chevalley groups 20 and through them he was led to make some of his deep conjectures 21 one significance of these l functions is that all the automorphic l functions studied so far are among them there are three general results first we obtain a uniform line of absolute convergence for all of these l functions theorem 5 1 a uniform estimate for the hecke eigenvalues of generic cusp forms on many absolutely simple quasi split connected reductive algebraic groups over number fields corollary 5 4 and an improvement on the best available estimate for the fourier coefficients of maass wave forms corollary 5 5 also follow next we establish a meromorphic continuation and functional equation for each of these l functions theorem 6 1 finally in theorem 6 2 we prove finiteness of poles on the whole complex plane for an important class of these l functions corollaries 6 6 through 6 10 more precisely let g be a quasi split connected reductive algebraic group over a number field f set g g a f where a f is the ring of adeles of f fix a borel subgroup b of g over f and let u be its unipotent radical let p be a maximal f parabolic subgroup of g with p d b in the context of the problems studied here nothing new will be obtained if one drops the maximality condition on p write p mn a levi decomposition n c u and let b u p m and n be the corresponding groups of adelic points for every place v of f let gv g fj similarly we have bv us pv me and nv let x xv be a generic character of u f u cf section 3 then each xv is generic let r ov 7tv be a cusp form on m we shall say 7t is

a proof of langland s conjecture on plancherel measures complementary series of p adic groups analysis of p adic reductive groups our first result theorem 7 9 proves a conjecture of langlands on normalization of intertwining operators by means of local langlands root numbers and l functions at least when the group is quasi split and the inducing representation is generic assuming two natural conjectures in harmonic analysis of p adic groups we also prove the validity of the conjecture in general theorem 9 5 as our second result we obtain all the complementary series and special representations of quasi split p adic groups coming from rank one parabolic subgroups and generic supercuspidal represen

a smoothly parameterized family of stabilizable observable linear systems containing realizations of all transfer functions of mcmillan degree not exceeding n it is shown that there is a continuously parameterized family f of n dimensional single input single output siso stabilizable detectable linear system sigma p which contains at least one realization of each reduced strictly proper transfer function of mcmillan degree not exceeding n the parameterization map p to sigma p is a polynomial function in 2n indeterminates from an open convex polyhedron in r sup 2n to the linear space of all siso n dimensional linear systems

on minimal order stabilization of single loop plants abstract it is shown by example that a single loop plant of order n relative degree 1 with one right half plane pole and one right half plane zero may not be stabilizable by a compensator of order n 2 a further example shows that a minimum phase plant of order n relative degree r with one right half plane pole may not be stabilizable by a compensator of order r 2

on symmetric power l invariants of iwahori level hilbert modular forms we compute the arithmetic cal l invariants of greenberg benois of twists of symmetric powers of p adic galois representations attached to iwahori level hilbert modular forms under some technical conditions our method uses the automorphy of symmetric powers and the study of analytic galois representations on p adic families of automorphic forms over symplectic and unitary groups combining these families with some explicit plethysm in the representation theory of rm gl 2 we construct global galois cohomology classes with coefficients in the symmetric powers and provide formulae for the cal l invariants in terms of logarithmic derivatives of hecke eigenvalues

quantum clifford algebra from classical differential geometry we show the emergence of clifford algebras of nonsymmetric bilinear forms as cotangent algebras of kaluza klein kk spaces pertaining to teleparallel space times these spaces are canonically determined by the horizontal differential invariants of finsler bundles of the type b m s m where b m is the set of all the tangent frames to a differentiable manifold m and where s m is the sphere bundle if m is space time itself m4 the geometric phase space s m4 has dimension seven this reformulation of the horizontal invariants as pertaining to a kk space removes the mismatch between the dimensionality of the tangent frames to m4 and the dimensionality of s m4 in the kk space a symmetric tangent metric induces a cotangent metric which is not symmetric in general an interior covariant derivative in the sense of kaehler is defined it involves the antisymmetric part of the cotangent metric which thus enters electrodynamics and the dirac equation

a hybrid parameterization of linear single input single output systems abstract we obtain a parameterization of the set of finite dimensional linear dynamical systems of unbounded mcmillan degree in this parameterization a given system is represented in a nonunique way however a certain notion of order is available thereby providing a graded parameterization an important feature is that each graded space is diffeomorphic to a euclidean space in view of this fact we hope that many of the local results in identification valid only if the parameter space is euclidean an actually be generalized globally moreover we show that the topology induced on the space of all plants by this space is finer than the graph topology

local coefficients as artin factors for real groups introduction the purpose of this paper is to prove the equality of certain local coefficients of arithmetic significance which were attached to representations of quasi split real reductive algebraic groups in 27 with their corresponding artin factors attached by local class field theory 21 as a consequence we establish an identity satisfied by certain normalized intertwining operators it seems to be useful in applications of the trace formula 1 29 more precisely let g be the group of real points of a quasi split reductive algebraic group over r let a be the set of simple roots defined by a fixed minimal parabolic subgroup po m0a0u of g fix 6 c a and let p p9 be the corresponding standard parabolic subgroup of g and write p man for its langlands decomposition fix a nondegenerate character x f u i a h o be an irreducible admissible x gic banach in particular x generic unitary representation of m cf section 1 given v g a the complex dual of the lie algebra of a let i v o 0 be the continuously quasi unitarily if o is unitary induced representation indp ga e v 9 and let v v o 0 be its space section 0 then f a vfoo o now let w be the weyl group of ao in g choose w g w such that w 0 c a let n be the unipotent group opposite to n define n u c wn w where w is a representative of w in g for g f a 00 define

z2 gradings of clifford algebras and multivector structures let c v g be the real clifford algebra associated with the real vector space v e ndowed with a nondegenerate metric g i n this paper we study the class of z2 gradings of c v g which are somehow compatible with the multivector structure of the grassmann algebra over v a complete characterization for such z2 gradings is obtained by classifying all the even subalgebras coming from them an expression relating such subalgebras to the usual even part of c v g is also obtained finally we employ this framework to define spinor spaces and to parametrize all the possible signature changes on c v g by z2 gradings of this algebra

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

invariant distributions and time averages for horocycle flows there are infinitely many obstructions to existence of smooth solutions of the cohomological equation uu f where u is the vector field generating the horocycle flow on the unit tangent bundle sm of a riemann m surface of finite area and f is a given function on sm we study the sobolev regularity of these obstructions construct smooth solutions of the cohomological equation and derive asymptotics for the ergodic averages of horocycle flows

modularity of the rankin selberg l series and multiplicity one for sl 2 let f g be primitive cusp forms holomorphic or otherwise on the upper half plane h of levels n m respectively with unitarily normalized l functions l s f equation and l s g equation when p does not divide n resp m the inverse roots p p resp p p are nonzero with sum ap resp bp for every p prime to nm set lp s f g 1 p pp s 1 p pp s 1 p pp s 1 p pp s 1 let l s f g denote the incomplete euler product of lp s f g over all p not dividing nm this is closely related to the convolution l series sum over n 1 a sub n b sub n n s whose miraculous properties were first studied by rankin and selberg

on the arithmetic of shalika models and the critical values of l functions for gl2n let pi be a cohomological cuspidal automorphic representation of rm gl 2n bbb a over a totally real number field f suppose that pi has a shalika model we define a rational structure on the shalika model of pi f comparing it with a rational structure on a realization of pi f in cuspidal cohomology in top degree we define certain periods omega epsilon pi f we describe the behavior of such top degree periods upon twisting pi by algebraic hecke characters chi of f then we prove an algebraicity result for all the critical values of the standard l functions l s pi otimes chi here we use the recent work of b sun on the non vanishing of a certain quantity attached to pi infty as applications we obtain algebraicity results in the following cases firstly for the symmetric cube l functions attached to holomorphic hilbert modular cusp forms we also discuss the situation for higher symmetric powers secondly for certain self dual of symplectic type rankin selberg l functions for rm gl 3 times rm gl 2 assuming langlands functoriality this generalizes to certain rankin selberg l functions of rm gl n times rm gl n 1 thirdly for the degree four l functions attached to siegel modular forms of genus 2 and full level moreover we compare our top degree periods with periods defined by other authors we also show that our main theorem is compatible with conjectures of deligne and gross

grobner bases for finite temperature quantum computing and their complexity following the recent approach of using order domains to construct grobner bases from general projective varieties we examine the parity and time reversal arguments relating to the wightman axioms of quantum field theory and propose that the definition of associativity in these axioms should be introduced a posteriori to the cluster property in order to generalize the anyon conjecture for quantum computing to indefinite metrics we then show that this modification which we define via ideal quotients does not admit a faithful representation of the braid group because the generalized twisted inner automorphisms that we use to reintroduce associativity are only parity invariant for the prime spectra of the exterior algebra we then use a coordinate prescription for the quantum deformations of toric varieties to show how a faithful representation of the braid group can be reconstructed and argue that for a degree reverse lexicographic monomial ordered grobner basis the complexity class of this problem is bounded quantum polynomial

lie groups as spin groups it is shown that every lie algebra can be represented as a bivector algebra hence every lie group can be represented as a spin group thus the computational power of geometric algebra is available to simplify the analysis and applications of lie groups and lie algebras the spin version of the general linear group is thoroughly analyzed and an invariant method for constructing real spin representations of other classical groups is developed moreover it is demonstrated that every linear transformation can be represented as a monomial of vectors in geometric algebra

real zeros and size of rankin selberg l functions in the level aspect in this paper some asymptotic formulas are proved for the harmonic mollified second moment of a family of rankin selberg l functions one of the main new input is a substantial improvement of the admissible length of the mollifier which is done by solving a shifted convolution problem by a spectral method on average a first consequence is a new subconvexity bound for rankin selberg l functions in the level aspect moreover infinitely many rankin selberg l functions having at most eight non trivial real zeros are produced and some new non trivial estimates for the analytic rank of the family studied are obtained

twisted endoscopy and reducibility of induced representations for p adic groups 1 introduction this is the first in a series of papers in which we study the reducibility of representations induced from discrete series representations of the levi factors ofmaximal parabolic subgroups of p adic groups on the unitary axis the problem is equivalent to determining certain local langlands l functions which are of arithmetic significance by themselves one of the main aims of this paper is to interpret our results in the direction of the parametrization problem by means of the theory of twisted endoscopy for which our method seems to be very suitable when the representation is supercuspidal we also study the reducibility of the induced representations off the unitary axis let f be a p adic field of characteristic zero fix a positive integer n 1 and let g be either of the groups sp2n s02n or s02n l in all three cases there is a conjugacy class of maximal parabolic subgroups which has gl as its levi factor let p mn be the standard parabolic subgroup ofg in this conjugacy class then m gl let tr be a discrete series representation of m gln f and given s c let i s tr be the representation of g g f induced from tr r idet ls let i tr i 0 tr

a hopf laboratory for symmetric functions an analysis of symmetric function theory is given from the perspective of the underlying hopf and bi algebraic structures these are presented explicitly in terms of standard symmetric function operations particular attention is focused on laplace pairing sweedler cohomology for 1 and 2 cochains and twisted products rota cliffordizations induced by branching operators in the symmetric function context the latter are shown to include the algebras of symmetric functions of orthogonal and symplectic type a commentary on related issues in the combinatorial approach to quantum field theory is given