on a decomposition lemma for positive semi definite block matrices abstract this short note in part of expository nature points out several new or recent consequences of a quite nice decomposition for positive semi definite matrices

the griffiths bundle is generated by groups first the griffiths line bundle of a mathbf q vhs mathscr v is generalized to a griffiths character mathrm grif mathbf g mu r associated to any triple mathbf g mu r where mathbf g is a connected reductive group over an arbitrary field f mu in x mathbf g is a cocharacter over overline f and r mathbf g rightarrow gl v is an f representation the classical bundle studied by griffiths is recovered by taking f mathbf q mathbf g the mumford tate group of mathscr v r mathbf g rightarrow gl v the tautological representation afforded by a very general fiber and pulling back along the period map the line bundle associated to mathrm grif mathbf g mu r the more general setting also gives rise to the griffiths bundle in the analogous situation in characteristic p given by a scheme mapping to a stack of mathbf g zips when mathbf g is f simple we show that up to positive multiples the griffiths character mathrm grif mathbf g mu r and thus also the griffiths line bundle is essentially independent of r with central kernel and up to some identifications is given explicitly by mu as an application we show that the griffiths line bundle of a projective mathbf g text mathtt zip mu scheme is nef

more extensions of a determinant inequality of hartfiel abstract hartfiel s determinant inequality first proved for two positive definite matrices has been recently extended to the case of sector matrices by lin hou and dong this paper contributes an improvement of lin s result with a simple argument we also present a complement of hou and dong s result

algebraic cycles and higher k theory here g x is the grothendieck group of coherent sheaves on x 13 gr refers to the graded group defined by the y filtration on g x cf kratzer 14 soult 20 and ch x is the chow group of codimension i algebraic cycles defined by fulton 9 the left hand isomorphism is a formal consequence of the existence of a i structure on g x while the existence of r is the central theme of the b f m rr theorem the main purpose of this paper is to define a theory of higher chow groups ch x n n 2 0 so as to obtain isomorphisms

galois representations arising from some compact shimura varieties our aim is to establish some new cases of the global langlands correspondence for glm along the way we obtain a new result on the description of the cohomology of some compact shimura varieties let f be a cm eld with complex conjugation c and

extensions of fischer s inequality abstract denote by b n 1 n k the set of block matrices whose i j blocks are n i n j complex matrices let a i b n 1 n k be positive semidefinite and d i b n 1 n k be block diagonal matrices for 1 i m we obtain the following extension of fischer s inequality det i 1 m d i a i p i d i j 1 k det i 1 m d i j a i j p i d i j 0 p i 1 where a i j is the j th main diagonal block of a i in addition if a i and d i are nonsingular the reverse inequality holds when 1 p i 0 we also extend these two results to a larger class of matrices namely matrices whose numerical ranges are contained in a sector

generalized twisted group rings abstract let r be a domain and g a group let g g r 0 be a generalized 2 cocycle i e not necessarily taking its values in the units of r and consider the generalized twisted group ring r g first we investigate the graded structure of r g in particular we give conditions under which r g is gr hereditary respectively a gr maximal order next we derive criteria for r g to be a tame order respectively a maximal order over some central subring we also derive conditions under which r g is an azumaya algebra

from log determinant inequalities to gaussian entanglement via recoverability theory many determinantal inequalities for positive definite block matrices are consequences of general entropy inequalities specialized to gaussian distributed vectors with prescribed covariances in particular strong subadditivity ssa yields ln det v ac ln det v bc ln det v abc ln det v c geq 0 for all 3 times 3 block matrices v abc where subscripts identify principal submatrices we shall refer to the above mentioned inequality as ssa of log det entropy in this paper we develop further insights on the properties of the above mentioned inequality and its applications to classical and quantum information theory in the first part of this paper we show how to find known and new necessary and sufficient conditions under which saturation with equality occurs subsequently we discuss the role of the classical transpose channel also known as petz recovery map in this problem and find its action explicitly we then prove some extensions of the saturation theorem by finding faithful lower bounds on a log det conditional mutual information in the second part we focus on quantum gaussian states whose covariance matrices are not only positive but obey additional constraints due to the uncertainty relation for gaussian states the log det entropy is equivalent to the renyi entropy of order 2 we provide a strengthening of log det ssa for quantum covariance matrices that involves the so called gaussian renyi 2 entanglement of formation a well behaved entanglement measure defined via a gaussian convex roof construction we then employ this result to define a log det entropy equivalent of the squashed entanglement measure which is remarkably shown to coincide with the gaussian renyi 2 entanglement of formation this allows us to establish useful properties of such measure s such as monogamy faithfulness and additivity on gaussian states

galois representations and torsion in the coherent cohomology of hilbert modular varieties let f be a totally real number eld and let p be a prime unramied in f we prove the existence of galois pseudo representations attached to modp m hecke eigenclasses of paritious weight occurring in the coherent cohomology of hilbert modular varieties forf of level prime to p

motivic periods and grothendieck arithmetic invariants abstract we construct a period regulator for motivic cohomology of an algebraic scheme over a subfield of the complex numbers for the field of algebraic numbers we formulate a period conjecture for motivic cohomology by saying that this period regulator is surjective showing that a suitable betti de rham realization of 1 motives is fully faithful we can verify this period conjecture in several cases the divisibility properties of motivic cohomology imply that our conjecture is a neat generalization of the classical grothendieck period conjecture for algebraic cycles on smooth and proper schemes these divisibility properties are treated in an appendix by b kahn extending previous work of bloch and colliot thelene raskind

vanishing theorems for torsion automorphic sheaves on general pel type shimura varieties abstract we consider the interior cohomology and the hodge graded pieces in the case of the de rham realization of general not necessarily compact pel type shimura varieties with coefficients in the local systems corresponding to sufficiently regular algebraic representations of the associated reductive group for primes p bigger than an effective bound we prove that the f p and z p cohomology groups are concentrated in the middle degree that the z p cohomology groups are free of p torsion and that every f p cohomology class lifts to a z p cohomology class

maximal graded orders over crystalline graded rings abstract crystalline graded rings are generalizations of certain classes of rings like generalized twisted group rings generalized weyl algebras and generalized skew crossed products under certain conditions in particular the group is finite it is proven that the global dimension of a crystalline graded ring coincides with the global dimension of its base ring when in addition the base ring is a commutative dedekind domain two constructions are given for producing maximal graded orders on the way a new concept appears so called spectrally twisted group some general properties of it are studied at the end of the paper several examples are considered

mixed hodge complexes on algebraic varieties using the theory of mixed hodge modules we introduce the notion of mixed hodge complex on an algebraic variety and establish the relation between the filtered complex of du bois and the corresponding complex of mixed hodge modules some application to the du bois singularity is given

on torsion in the cohomology of locally symmetric varieties the main result of this paper is the existence of galois representations associated with the mod p or mod p m cohomology of the locally symmetric spaces for gln over a totally real or cm eld proving conjectures of ash and others following an old suggestion of clozel recently realized by harris lan taylor thorne for characteristic 0 cohomology classes one realizes the cohomology of the locally symmetric spaces for gln as a boundary contribution of the cohomology of symplectic or unitary shimura varieties so that the key problem is to understand torsion in the cohomology of shimura varieties thus we prove new results on the p adic geometry of shimura varieties of hodge type namely the shimura varieties become perfectoid when passing to the inverse limit over all levels at p and a new period map towards the ag variety exists on them called the hodge tate period map it is roughly analogous to the embedding of the hermitian symmetric domain which is roughly the inverse limit over all levels of the complex points of the shimura variety into its compact dual the hodge tate period map has several favorable properties the most important being that it commutes with the hecke operators away from p for the trivial action of these hecke operators on the ag variety and that automorphic vector bundles come via pullback from the ag variety

majorization criterion for distillability of a bipartite quantum state bipartite quantum states are classified into three categories separable states bound entangled states and free entangled states it is of great importance to characterize these families of states for the development of quantum information science in this letter i show that the separable states and the bound entangled states have a common spectral property more precisely i prove that for undis tillable separable and bound entangled states the eigenvalue vector of the global system is majorized by that of the local system this result constitutes a new sufficient condition for distillability of bipartite quantum states this is achieved by proving that if a bipartite quantum state satisfies the reduction criterion for distillability then it satisfies the majorization criterion for separability

extending a refinement of koteljanskiĭ s inequality abstract the goal of this paper is two fold first of all a recent refinement of koteljanski s inequality by dong and hou 9 is further extended to sector matrices moreover by making use of the crabtree haynsworth quotient formula we present two analogues of yan s result 16 on 3 3 block positive definite matrices

the filtered ogus realisation of motives abstract we construct the filtered ogus realisation of voevodsky motives over a number field k this realisation extends the functor defined on 1 motives by andreatta barbieri viale and bertapelle as an illustration we note that the analogue of the tate conjecture holds for k3 surfaces

generalized μ ordinary hasse invariants abstract we give a short proof for the existence of ordinary hasse invariants for the good reduction special fiber of shimura varieties of hodge type using the stack of g zips introduced by moonen wedhorn and pink wedhorn ziegler we give an explicit formula for the power of the hodge bundle that admits a hasse invariant when g is a weil restriction of g 1 we relate the ekedahl oort strata of g and those of g 1

generalized 2 cocycles of finite groups and maximal orders abstract letrbe a dedekind domain with quotient fieldkand letgbe a finite group let g g r 0 be a generalized 2 cocycle i e not necessarily taking its values in the units ofr and consider the generalized twisted group ringr g in this paper we derive criteria forr gto be a graded maximalr order resp a hereditaryr order or a maximalr order ink g

a completely ppt map abstract a map is completely ppt if it is completely positive and the resulting matrix is positive partial transpose ppt we show that the map x x tr x i is completely ppt some interesting consequences are included

deligne s conjecture on 1 motives we reformulate a conjecture of deligne on 1 motives by using the integral weight filtration of gillet and soule on cohomology and prove it this implies the original conjecture up to isogeny if the degree of cohomology is at most two we can prove the conjecture for the hodge realization without isogeny and even for 1 motives with torsion

the neron severi group of a proper seminormal complex variety we prove a lefschetz 1 1 theorem for proper seminormal varieties over the complex numbers

inequalities related to partial transpose and partial trace abstract in this paper we present inequalities related to partial transpose and partial trace for positive semidefinite matrices some interesting results involving traces and eigenvalues are also included

some new results for hua type operator matrices abstract let a i i 1 2 n be strict contractions on a hilbert space h the n n operator matrix h n a 1 a 2 a n i a j a i 1 i j 1 n is called a hua type operator matrix in this note we mainly investigate some results which are related to the hua type operator matrix we firstly give some equivalent conditions for the positivity of n n operator matrices i a j a i i j 1 n then the equation min h 2 a 1 a 2 a 1 1 a 2 1 2 is shown in particular some equivalent conditions for strict contractions a 1 and a 2 such that h 2 a 1 a 2 2 are obtained

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

quasi constant characters motivation classification and applications abstract in 13 initially motivated by questions about the hodge line bundle of a hodge type shimura variety we singled out a generalization of the notion of minuscule character which we termed quasi constant here we prove that the character of the hodge line bundle is always quasi constant furthermore we classify the quasi constant characters of an arbitrary connected reductive group over an arbitrary field as an application we observe that if is a quasi constant cocharacter of an f p group g then our construction of group theoretical hasse invariants in loc cit applies to the stack g zip without any restrictions on p even if the pair g is not of hodge type and even if is not minuscule we conclude with a more speculative discussion of some further motivation for considering quasi constant cocharacters in the setting of our program outlined in loc cit

quasi constant fundamental weights in terms of levi weyl groups abstract in previous joint work with j s koskivirta we introduced the notion of quasi constant character of a maximal torus of a connected reductive group over a field we showed that over an algebraically closed field it naturally unifies the notions minuscule and co minuscule in this note we characterize quasi constant fundamental weights in terms of the weyl group of the corresponding maximal levi subgroup equivalently purely in the language of root systems the result characterizes special and co special vertices of dynkin diagrams in terms of the weyl group of the corresponding maximal sub root system

ekedahl oort and newton strata for shimura varieties of pel type we study the ekedahl oort stratification for good reductions of shimura varieties of pel type these generalize the ekedahl oort strata defined and studied by oort for the moduli space of principally polarized abelian varieties the siegel case they are parameterized by certain elements w in the weyl group of the reductive group of the shimura datum we show that for every such w the corresponding ekedahl oort stratum is smooth quasi affine and of dimension ell w and in particular non empty some of these results have previously been obtained by moonen vasiu and the second author using different methods we determine the closure relations of the strata we give a group theoretical definition of minimal ekedahl oort strata generalizing oort s definition in the siegel case and study the question whether each newton stratum contains a minimal ekedahl oort stratum as an interesting application we determine which newton strata are non empty this criterion proves conjectures by fargues and by rapoport generalizing a conjecture by manin for the siegel case we give a necessary criterion when a given ekedahl oort stratum and a given newton stratum meet

the n cohomology of limits of discrete series abstract let g be a semisimple lie group which has a compact cartan subgroup h let k be a maximal compact subgroup of g containing h and let n be the sum of the negative root spaces of g corresponding to an arbitrary choice of a positive root system of g h we compute the n cohomology of the k finite vectors in a limit of a discrete series representation of g in the special case when is a discrete series representation our result reduces to the result of w schmid in the special case when is a holomorphic limit we interpret its multiplicity in l 2 g cohomologically where is a discrete subgroup of g in the general case we present a conjecture for this multiplicity

on the shafarevich and tate conjectures for hyperkahler varieties 1 problems and results 2 polarized hyperk ihler varieties and cubic fourfoids 3 the period mapping 4 the kuga satake construction 5 the kuga satake construction in the relative setting 6 proof of theorem 1 5 1 7 proof of theorem 1 6 1 8 descent 9 proof of theorem 1 3 1 appendix 1 spinorial shimura varieties appendix 2 a short proof of the hodge conjecture for cubic fourfolds 205 210 212 216 220 224 229 233 240 244 245

bogomolov s inequality for higgs sheaves in positive characteristic we prove bogomolov s inequality for higgs sheaves on varieties in positive characteristic p that can be lifted modulo p 2 this implies the miyaoka yau inequality on surfaces of non negative kodaira dimension liftable modulo p 2 this result is a strong version of shepherd barron s conjecture our inequality also gives the first algebraic proof of bogomolov s inequality for higgs sheaves in characteristic zero solving the problem posed by narasimhan

contragredient representations and characterizing the local langlands correspondence we consider the question what is the contragredient in terms of l homomorphisms we conjecture that it corresponds to the chevalley automorphism of the l group and prove this in the case of real groups the proof uses a characterization of the local langlands correspondence over r we also consider the related notion of hermitian dual in the case of gl n bbb r

a quotient of the lubin tate tower ii in this article we construct the quotient mathcal m mathbf 1 p k of the infinite level lubin tate space mathcal m mathbf 1 by the parabolic subgroup p k subset mathrm gl n k of block form n 1 1 as a perfectoid space generalizing the results of ludwig forum math sigma 5 e17 41 2017 to arbitrary n and k mathbb q p finite for this we prove some perfectoidness results for certain harris taylor shimura varieties at infinite level as an application of the quotient construction we show a vanishing theorem for scholze s candidate for the mod p jacquet langlands and mod p local langlands correspondence an appendix by david hansen gives a local proof of perfectoidness of mathcal m mathbf 1 p k when n 2 and shows that mathcal m mathbf 1 q k is not perfectoid for maximal parabolics q not conjugate to p

variation of mixed hodge structure i around 1970 griffiths introduced the notion of a variation of hodge structure on a complex manifold s see 17 w 2 it constitutes the axiomatization of the features possessed by the local systems of cohomology associated to a family of compact k ihler manifolds esp smooth projective varieties parametrized by s these are for a variation of hodge structure u of weight m 0 a locally constant sheaf u of q vector spaces or u of real vector spaces 1 hodge filtration horizontality a decreasing filtration o r of gs by holomorphic subbundles such that the evident connection v on takes o p into q 2 hodge decompositions if gs denotes the sheaf of functions on s and if

some applications of two completely copositive maps abstract a linear map m n m k is called completely copositive if the resulting matrix a j i i j 1 m is positive semidefinite for any integer m and positive semidefinite matrix a i j i j 1 m in this paper we present some applications of the completely copositive maps x tr x i x and x tr x i x some new extensions about trace inequalities of positive semidefinite 3 3 block matrices are included

krull dimension of generalized weyl algebras and iterated skew polynomial rings commutative coefficients abstract many rings that have enjoyed growing interest in recent years e g quantum enveloping algebras quantum matrices certain witten algebras can be presented as generalized weyl algebras in the paper we develop techniques for calculating dimensions here mainly the krull dimension in the sense of rentschler gabriel of such generalized weyl algebras and specify the results for some popular algebras including those mentioned above

the simple modules of certain generalized crossed products abstract we consider a dedekind domaindand az graded ring r i z r i withr0 dand eachri dvibeing a freed module of rank 1 the structure ofris described by an automorphism ofdand a generalized 2 cocyclec z z dnot necessarily taking its values in the units ofd the aim of this paper is to classify the simpler modules say in the case wherec i j 0 for alli j z we also deal with this problem in then graded case as a consequence we obtain a description of the simple modules of some classical algebras and of generalized weyl algebras

simple holonomic modules over the second weyl algebra a2 abstract for simple generalized weyl algebras of gelfand kirillov dimension 4 a class including the second weyl algebra a 2 some simple factor algebras of the universal enveloping algebra of the lie algebra sl 2 sl 2 and of u q sl 2 u q sl 2 etc the simple holonomic modules are classified up to pairs of irreducible elements of certain noncommutative euclidean ring

limits of discrete series with infinitesimal character zero for a connected linear semisimple lie group this paper considers those nonzero limits of discrete series representations having infinitesimal character 0 calling them totally degenerate such representations exist if and only if has a compact cartan subgroup is quasisplit and is acceptable in the sense of harish chandra totally degenerate limits of discrete series are natural objects of study in the theory of automorphic forms in fact those automorphic representations of adelic groups that have totally degenerate limits of discrete series as archimedean components correspond conjecturally to complex continuous representations of galois groups of number fields the automorphic representations in question have important arithmetic significance but very little has been proved up to now toward establishing this part of the langlands conjectures there is some hope of making progress in this area and for that one needs to know in detail the representations of under consideration the aim of this paper is to determine the classification parameters of all totally degenerate limits of discrete series in the knapp zuckerman classification of irreducible tempered representations i e to express these representations as induced representations with nondegenerate data the paper uses a general argument based on the finite abelian reducibility group attached to a specific unitary principal series representation of first an easy result gives the aggregate of the classification parameters then a harder result uses the easy result to match the classification parameters with the representations of under consideration in representation by representation fashion the paper includes tables of the classification parameters for all such groups

nori 1 motives let mathsf ehm be nori s category of effective homological mixed motives in this paper we consider the thick abelian subcategory mathsf ehm 1 subset mathsf ehm generated by the i th relative homology of pairs of varieties for i in 0 1 we show that mathsf ehm 1 is naturally equivalent to the abelian category t mathcal m 1 of 1 motives with torsion this is our main theorem along the way we obtain several interesting results firstly we realize t mathcal m 1 as the universal abelian category obtained using nori s formalism from the betti representation of an explicit diagram of curves secondly we obtain a conceptual proof of a theorem of vologodsky on realizations of 1 motives thirdly we verify a conjecture of deligne on extensions of 1 motives in the category of mixed realizations for those extensions that are effective in nori s sense

on some generalizations of the brunn minkowski inequality abstract recently liu 2016 9 obtained some determinantal inequalities for sector matrices in this paper we make considerable improvement on liu s results

generalized 1 motivic sheaves abstract let k be a field of characteristic 0 we extend the definition of the category of 1 motivic sheaves introduced by barbieri viale and kahn allowing quotients of connected algebraic k groups by formal k groups we show that its bounded derived category is equivalent to the bounded derived category of the category of generalized 1 motives with torsion

strata hasse invariants hecke algebras and galois representations we construct group theoretical generalizations of the hasse invariant on strata closures of the stacks mathop g hbox mathtt zip nolimits mu restricting to zip data of hodge type we obtain a group theoretical hasse invariant on every ekedahl oort stratum closure of a general hodge type shimura variety a key tool is the construction of a stack of zip flags mathop g hbox mathtt zipflag nolimits mu fibered in flag varieties over mathop g hbox mathtt zip nolimits mu it provides a simultaneous generalization of the classical case homogeneous complex manifolds studied by griffiths schmid and the flag space for siegel varieties studied by ekedahl van der geer four applications are obtained 1 pseudo representations are attached to the coherent cohomology of hodge type shimura varieties modulo a prime power 2 galois representations are associated to many automorphic representations with non degenerate limit of discrete series archimedean component 3 it is shown that all ekedahl oort strata in the minimal compactification of a hodge type shimura variety are affine thereby proving a conjecture of oort 4 part of serre s letter to tate on mod p modular forms is generalized to general hodge type shimura varieties

ogus realization of 1 motives abstract after introducing the ogus realization of 1 motives we prove that it is a fully faithful functor more precisely following a framework introduced by ogus considering an enriched structure on the de rham realization of 1 motives over a number field we show that it yields a full functor by making use of an algebraicity theorem of bost

canonical sections of the hodge bundle over ekedahl oort strata of shimura varieties of hodge type abstract we construct canonical non vanishing global sections of powers of the hodge bundle on each ekedahl oort stratum of a hodge type shimura variety in particular we recover the quasi affineness of the ekedahl oort strata in the projective case this gives a very short proof of non emptiness of ekedahl oort strata it follows that the newton strata are also nonempty by a result of s nie from the canonicity of our construction we deduce the fact that the ordinary locus is determined by the ekedahl oort strata of its image under any embedding of shimura varieties

modularity lifting beyond the taylor wiles method we prove new modularity lifting theorems for p adic galois representations in situations where the methods of wiles and taylor wiles do not apply previous generalizations of these methods have been restricted to situations where the automorphic forms in question contribute to a single degree of cohomology in practice this imposes several restrictions one must be in a shimura variety setting and the automorphic forms must be of regular weight at infinity in this paper we essentially show how to remove these restrictions our most general result is a modularity lifting theorem which on the automorphic side applies to automorphic forms on the group mathrm gl n over a general number field it is contingent on a conjecture which in particular predicts the existence of galois representations associated to torsion classes in the cohomology of the associated locally symmetric space we show that if this conjecture holds then our main theorem implies the following if e is an elliptic curve over an arbitrary number field then e is potentially automorphic and satisfies the sato tate conjecture in addition we also prove some unconditional results for example in the setting of mathrm gl 2 over mathbf q we identify certain minimal global deformation rings with the hecke algebras acting on spaces of p adic katz modular forms of weight 1 such algebras may well contain p torsion moreover we also completely solve the problem for p odd of determining the multiplicity of an irreducible modular representation overline rho in the jacobian j 1 n where n is the minimal level such that overline rho arises in weight two

on some inequalities related to positive block matrices abstract let a i j i j 1 m m m m n be a positive semidefinite matrix we will show that min m n tr a i j i n a i j i j 1 m is a block ppt matrix according to this result we present some inequalities of positive block matrices which complement some existing results

positive semi definite 2 2 block matrices and norm inequalities abstract let m a x x b c 2 n 2 n be positive semi definite 2 2 block matrix where a b x c n n a characterization for the matrix m with a b k i to be positive partial transpose is given in terms of its spectral norm using this result a counter example is constructed for the conjecture that m a b when x is normal for all unitarily invariant norms

crystalline realizations of 1 motives we consider the crystalline realization of deligne s 1 motives in positive characteristics and prove a comparison theorem with the de rham realization of formal liftings to zero characteristic we then show that one dimensional crystalline cohomology of an algebraic variety defined by forcing universal cohomological descent via de jong s alterations coincides with the crystalline realization of the picard 1 motive over perfect fields of characteristic 2

trace inequalities for positive semidefinite block matrices abstract if a 2 2 block matrix a b b c is positive semidefinite where each block is square then the following trace inequality holds tr a c tr b b tr a tr c tr b 2 this improves a result of besenyei 2 moreover we show that tr b m tr a m 2 c m 2 m 2 3

sharp de rham realization we introduce the sharp universal extension of a 1 motive with additive factors and torsion over a field of characteristic zero we define the sharp de rham realization t by passing to the lie algebra over the complex numbers we then show a sharp de rham comparison theorem in the category of formal hodge structures for a free 1 motive along with its cartier dual we get a canonical connection on their sharp extensions yielding a perfect pairing on sharp realizations thus we show how to provide one dimensional sharp de rham cohomology h dr1 of algebraic varieties

on galois representations associated to hilbert modular forms in this paper we prove that to any hilbert cuspidal eigenform one may attach a compatible system of galois representations this result extends the analogous results of deligne and deligne serre for elliptic modular forms the principal work on this conjecture was carried out by carayol and taylor but their results left one case remaining which we complete in this paper we also investigate the compatibility of our results with the local langlands correspondence and prove that whenever the local component of the automorphic representation is not special then the results coincide

on a norm inequality for a positive block matrix abstract for a positive semidefinite matrix h a x x b we consider the norm inequality h a b we show that this inequality holds under certain conditions some related topics are also investigated

new properties for certain positive semidefinite matrices abstract we bring in some new notions associated with 2 2 block positive semidefinite matrices these notions concern the inequalities between the singular values of the off diagonal blocks and the eigenvalues of the arithmetic mean or geometric mean of the diagonal blocks we investigate some relations between them many examples are included to illustrate these relations