sur la presentation des representations supersingulieres de mathrm gl _2 f soit f une extension quadratique de mathbb q p nous prouvons que les representations lisses irreductibles supersingulieres a caractere central de mathrm gl 2 f ne sont pas de presentation finie

explicit description of irreducible gl2 qp representations over f p let p be an odd prime number the classi cation of irreducible representations of gl2 qp over fp is known thanks to the works of barthel livne and breuil in the present paper we illustrate an exhaustive description of such irreducible representations through the study of certain functions on the bruhat tits tree of gl2 qp in particular we are able to detect the socle ltration for the kz restriction of supersingular representations principal series and special series

moduli of finite flat group schemes and modularity we prove that under some mild conditions a two dimensional p adic galois representation which is residually modular and potentially barsotti tate at p is modular this provides a more conceptual way of establishing the shimura taniyama weil conjecture especially for elliptic curves which acquire good reduction over a wildly ramified extension of 3 the main ingredient is a new technique for analyzing flat deformation rings it involves resolving them by spaces which parametrize finite flat group scheme models of galois representations introduction

asymptotic expansions and boundary values of eigenfunctions on riemannian symmetric spaces soit x g k un espace de riemann symetrique de type non compacte a bord b k m et soit d x l algebre de tous les operateurs differentiels g invariants sur x on obtient des developpements asymptotiques pour une grande classe de fonctions jointives f sur x

serre weights for rank two unitary groups we study the weight part of a generalisation of serre s conjecture for mod l galois representations associated to automorphic representations on rank two unitary groups for odd primes l we propose a conjectural set of serre weights agreeing with all conjectures in the literature and under a mild assumption on the image of the mod l galois representation we are able to show that any modular representation is modular of each conjectured weight we make no assumptions on the ramification or inertial degrees of l our main innovations are the use of outer forms of text gl 2 to avoid the parity difficulties with inner forms and the use of the lifting techniques and change of weight results introduced in barnet lamb et al j am math soc 24 2 411 469 2011 duke math j 161 8 1521 1580 2012 potential automorphy and change of weight preprint 2010

filtration de certains espaces de fonctions sur un espace symetrique reductif we introduce a filtration of a g k module of some space of functions on a reductive symmetric space g h and compute the associated grading as a direct sum of induced representations as an application of this result to the reductive groups viewed as symmetric spaces we are able to realize any harish chandra module as a subquotient of a direct sum of induced representations from parabolic subgroups the inducing representations being trivial on the unipotent radical

the fontaine mazur conjecture for gl _2 introduction 641 1 breuil mezard conjecture and the p adic local langlands 644 1 1 the breuil mezard conjecture 644 1 2 review of colmez s functor 647 1 3 hilbert samuel multiplicities 650 1 4 representations and pseudo representations 653 1 5 gl2 qp representations mod p 657 1 6 local patching and multiplicities 661 1 7 from pseudo representations to representations 669 2 modularity via the breuil mezard conjecture 675 2 1 quaternionic forms 675 2 2 global patching and multiplicities 677 2 3 the breuil mezard conjecture 686 addendum to ki 1 687 references 689

modular representations of pgl2 and automorphic forms for shimura curves in this paper we study certain infinite dimensional representations of the general linear group gl2 over a local field k in vector spaces over k these representations are closely related to the theory of analytic functions on the p adic upper half plane over k they were originally introduced by morita mor who studied them intensively in the case when the characteristic of k is zero morita s work has subsequently been greatly extended by schneider and stuhler ss and schneider s to representations of gl k and higher p adic symmetric spaces in this paper we re consider some of morita s representations of gl2 from the point of view of integral and modular representation theory it seems quite natural to study p adic representations by reducing them modp and we supply a technique for doing this in particular suppose that ydenotes the bruhat tits tree of pglz k we consider the representations of pgl2 k on the harmonic functions on w of e v e n weight that is the harmonic functions on the edges of j taking values in even symmetric powers of the standard representation of gl2 in this paper we find an invariant integral lattice in these representations and we describe the modular representations of pgl2 k obtained by reducing this lattice modp the modular representations obtained from morita s representations are quite simple and natural objects although they seem a bit peculiar at first suppose that f is the residue field of k that v is a vector space over f and that p gl2 f end v is a finite dimensional modular representation notice that p may be viewed in a natural way as a modular representation of gl r where r is the valuation ring of k we show that a typical infinite dimensional representation of morita s when reduced mod p has a filtration whose factors are isomorphic to i cl2 r wj t n l2 t note that there are only finitely many irreducible finite dimensional modular representations of glz f the appearance of these finitely many

sur quelques representations supersingulieres de gl2 qpf abstract fix a prime number p 5 an integer f 1 and let q p f be the unramified extension of q p of degree f by c breuil and v pask nas 2007 5 to a generic semi simple continuous representation gal q p q p f gl 2 f p we can associate a parameterized family of smooth admissible representations of gl 2 q p f with coefficients in f p in this article we prove that there are more parameters than those known

a paley wiener theorem for real reductive groups le groupe sl 3 r prequantification des orbites de dimension minimale parmi celles non triviales dans la representation co adjointe de sl 3 r noyaux de blattner kostant steinberg irreductibilite et unicite du prolongement de certaines representations de p 3 a sl 3 r etude du prolongement des representations e a sl 3 r la representation type s s 2 minimal parametres de langlands et operateur d entrelacement avec la serie principale pour les representations e et

the spherical fourier transform of rapidly decreasing functions a simple proof of a characterization due to harish chandra helgason trombi and varadarajan abstract harish chandra s helgason p c trombi and v s varadarajan have described the image of the l p schwartz spaces 0 p 2 under the spherical fourier transform we give a short proof of their results based on the paley wiener theorem

some applications of the g g expansion method to non linear partial differential equations in the present paper we construct the traveling wave solutions involving parameters of the 2 1 dimensional higher order broer kaup equations the 2 1 dimensional breaking soliton equations the 2 1 dimensional asymmetric nizhnik novikov vesselov equations and the 2 1 dimensional bkp equations in terms of the hyperbolic functions trigonometric functions and the rational functions by using a new approach namely the g g expansion method where g g x satisfies a second order linear ordinary differential equation when the parameters are taken special values the solitary waves are derived from the traveling waves the traveling wave solutions are expressed by the hyperbolic functions the trigonometric functions and the rational functions

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

on the universal module of p adic spherical hecke algebras let widetilde g be a split connected reductive group with connected center z over a local non archimedean field f of residue characteristic p let widetilde k be a hyperspecial maximal compact open subgroup in widetilde g let r be a commutative ring let v be a finitely generated r free r widetilde k module for an r algebra b and a character chi frak h v widetilde g widetilde k rightarrow b of the spherical hecke algebra frak h v widetilde g widetilde k rm end r widetilde g rm ind widetilde k widetilde g v we consider the specialization m chi v rm ind widetilde k widetilde g v otimes frak h v widetilde g widetilde k chi b of the universal frak h v widetilde g widetilde k module rm ind widetilde k widetilde g v for large classes of r including cal o f and overline bbb f p v b and chi arguing geometrically on the bruhat tits building we give a sufficient criterion for m chi v to be b free and to admit a widetilde g equivariant resolution by a koszul complex built from finitely many copies of rm ind widetilde k z widetilde g v this criterion is the exactness of certain fairly small and explicit frak n equivariant r module complexes where frak n is the group of cal o f valued points of the unipotent radical of a borel subgroup in widetilde g we verify it if f bbb q p and if v is an irreducible overline bbb f p widetilde k representation with highest weight in the closed bottom p alcove or a lift of it to cal o f we use this to construct p adic integral structures in certain locally algebraic representations of widetilde g

sur le theoreme de paley wiener d arthur the fourier transform of a c function f with compact support on a real reductive lie group g is given by a collection of operators o p e ip e f for a suitable family of representations of g which depends on a family indexed by p in a finite set of parabolic subgroups of g of pairs of parameters e varying in a set of discrete series e lying in a complex finite dimensional vector space the ip e are generalized principal series induced from p it is easy to verify the holomorphy of the fourier transform in the complex parameters also it satisfies some growth properties moreover an intertwining operator between two representations ip e ip e of the family implies an intertwining property for o p e and o p e there is also a way to introduce hsuccessive partial derivatives h of the family of representations ip e along the parameter e and intertwining operators between subquotients of these successive derivatives imply the intertwining property for the successive derivatives of the fourier transform o we show that these properties characterize the collections of operators p e o p e which are fourier transforms of a c function with compact support for g linear the proof which uses harish chandra fs plancherel formula rests on a similar result for left and right k finite functions which is due to j arthur we give also a proof of arthur fs result purely in term of representations involving the work of a knapp and e stein on intertwining integrals and langlands and vogan fs classifications of irreducible representations of g

on the modularity of 2 adic potentially semi stable deformation rings using p adic local langlands correspondence for text gl 2 mathbb q 2 and an ordinary r mathbb t theorem we prove that the support of patched modules for quaternionic forms meet every irreducible component of the potentially semi stable deformation ring this gives a new proof of the breuil mezard conjecture for 2 dimensional representations of the absolute galois group of mathbb q 2 which is new in the case overline r a twist of an extension of the trivial character by itself as a consequence a local restriction in the proof of fontaine mazur conjecture in pask nas algebra number theory 10 6 1301 1358 2016 is removed

the breuil mezard conjecture when l neq p let l and p be primes let f qp be a finite extension with absolute galois group gf let f be a finite field of characteristic l and let w gf gln f be a continuous representation let r be the universal framed deformation ring for if l d p then the breuil mezard conjecture as recently formulated by emerton and gee relates the mod l reduction of certain cycles in r to the mod l reduction of certain representations of gln of we state an analogue of the breuil mezard conjecture when l p and we prove it whenever l 2 using automorphy lifting theorems we give a local proof when l is quasibanal for f and is tamely ramified we also analyze the reduction modulo l of the types defined by schneider and zink

multiplicity theorems modulo p for mathbf gl _ 2 mathbf q _p let f be a non archimedean local field pi an admissible irreducible mathbf gl 2 f representation with complex coefficients for a quadratic extension l f and an l times character chi a classical result of tunnell and saito establish a precise connection between the dimension of the hom space hbox hom l times big pi vert l times chi big and the normalized local factor of the pair pi chi the study of analogous hom spaces for complex valued representations has recently been generalized to mathbf gl n in aizenbud et al ann math 172 1407 1434 2010 and their connections with local factors have been established by work of moeglin and waldspurger la conjecture locale de gross prasad pour les groupes speciaux orthogonaux le cas general preprint available at http www math jussieu fr moeglin gp pdf 2009 in this paper we approach the analogous problem in the context of the p modular langlands correspondence for mathbf gl 2 mathbf q p we describe the restriction to cartan subgroups of an irreducible p modular representation pi of mathbf gl 2 mathbf q p and deduce generalized multiplicity results on the dimension of the ext spaces mathrm ext fancyscript o l times i big pi vert fancyscript o l times chi big where fancyscript o l times is the ring of integers of a quadratic extension of mathbf q p and chi a smooth character of fancyscript o l times

on serre s conjecture for mod l galois representations over totally real fields in 1987 serre conjectured that any mod two dimensional irre ducible odd representation of the absolute galois group of the rationals came from a modular form in a precise way we present a generalisation of this conjecture to 2 dimensional representations of the absolute galois group of a totally real eld where is unramied the hard work is in formulating an analogue of the weight part of serre s conjecture serre furthermore asked whether his conjecture could be rephrased in terms of a mod langlands philosophy using ideas of emerton and vign eras we formulate a mod local global principle for the group d where d is a quaternion algebra over a totally real eld split above and at 0 or 1 innite places and show how it implies the conjecture

serre s modularity conjecture ii we provide proofs of theorems 4 1 and 5 1 of khare and wintenberger invent math doi 10 1007 s00222 009 0205 7 2009

fourier inversion on a reductive symmetric space let x be a semisimple symmetric space in previous papers 8 and 9 we have dened an explicit fourier transform for x and shown that this transform is injective on the space c 1 c x ofcompactly supported smooth functions on x in the present paper which is a continuation of these papers we establish an inversion formula for this transform

the classification of irreducible admissible mod p representations of a p adic gln let f be a finite extension of p using the mod p satake transform we define what it means for an irreducible admissible smooth representation of an f split p adic reductive group over overline mathbb f p to be supersingular we then give the classification of irreducible admissible smooth gln f representations over overline mathbb f p in terms of supersingular representations as a consequence we deduce that supersingular is the same as supercuspidal these results generalise the work of barthel livne for n 2 for general split reductive groups we obtain similar results under stronger hypotheses

weights of galois representations associated to hilbert modular forms let f be a totally real field p 3 a rational prime unramified in f and p a place of f over p let gal f f gl2 fp be a two dimensional mod p galois representation which is assumed to be modular of some weight and whose restriction to a decomposition subgroup at p is irreducible we specify a set of weights determined by the restriction of to inertia at p which contains all the modular weights for this proves part of a conjecture of diamond buzzard and jarvis which provides an analogue of serre s epsilon conjecture for hilbert modular forms mod p

iwahori hecke model for supersingular representations of gl2 qp abstract in this paper we realize a regular supersingular representation of gl 2 q p as a quotient of a representation induced from the iwahori subgroup of gl 2 q p we also show that this realization provides a uniform way of looking at all the self extensions of which have a four dimensional space of i 1 invariants

theoreme de type paley wiener pour les groupes de lie semi simples reels avec une seule classe de conjugaison de sous groupes de cartan abstract in this article we characterize the space of fourier transforms of the smooth compactly supported functions on a semisimple lie group with one conjugacy class of cartan subgroups cf th 6 we get also results on the space of fourier transforms of the elements of the enveloping algebra of the lie algebra of such a group cf th 1 2 and 3

multiplicites modulaires et representations de rm gl sb 2 mathbf z sb p et de rm gal overline mathbf q sb p mathbf q sb p en ell p appendice par guy henniart sur l unicite des types pour rm gl sb 2

on the breuil mezard conjecture we give a new local proof of the breuil mezard conjecture for 2 dimensional representations of the absolute galois group of qp when p 5 and the representation has scalar endomorphisms

exact and explicit solutions to some nonlinear evolution equations by utilizing the g g expansion method in this paper we demonstrate the effectiveness of the so called g g expansion method by examining some nonlinear evolution equations with physical interest our work is motivated by the fact that the g g expansion method provides not only more general forms of solutions but also periodic and solitary waves if we set the parameters in the obtained wider set of solutions as special values then some previously known solutions can be recovered the method appears to be easier and faster by means of a symbolic computation system

a canonical torsion theory for pro p iwahori hecke modules abstract let f be a locally compact nonarchimedean field with residue characteristic p and g the group of f rational points of a connected split reductive group over f we define a torsion pair in the category mod h of modules over the pro p iwahori hecke k algebra h of g where k is an arbitrary field we prove that under a certain hypothesis the torsionfree class embeds fully faithfully into the category mod i g of smooth k representations of g generated by their pro p iwahori fixed vectors if the characteristic of k is different from p then this hypothesis is always satisfied and the torsionfree class is the whole category mod h if k contains the residue field of f then we study the case g s l 2 f we show that our hypothesis is satisfied and we describe explicitly the torsionfree and the torsion classes if f q p and p 2 then an h module is in the torsion class if and only if it is a union of supersingular finite length submodules it lies in the torsionfree class if and only if it does not contain any nonzero supersingular finite length module if f q p the torsionfree class is the whole category mod h and we give a new proof of the fact that mod h is equivalent to mod i g these results are based on the computation of the h module structure of certain natural cohomology spaces for the pro p iwahori subgroup i of g

un calcul d anneaux de deformations potentiellement barsotti tate let f be an unramified extension of qp the first aim of this work is to develop a purely local method to compute the potentially barsotti tate deformations rings with tame galois type of irreducible two dimensional representations of the absolute galois group of f we then apply our method in the particular case where f has degree 2 over q p and determine this way almost all these deformations rings in this particular case we observe a close relationship between the structure of these deformations rings and the geometry of the associated kisin variety as a corollary and still assuming that f has degree 2 over q p we prove except in two very particular cases a conjecture of kisin which predicts that intrinsic galois multiplicities are all equal to 0 or 1

extended auxiliary equation method and its applications for finding the exact solutions for a class of nonlinear schrodinger type equations in this paper we extended the auxiliary equation method proposed by sirendaoreji and kudryashov to construct new types of jacobi elliptic function solutions of nonlinear partial differential equations pdes in mathematical physics the effectiveness of the extended method is demonstrated by applications to three nonlinear pdes namely the 2 1 dimensional nonlinear cubic quintic ginzburg landau equation the 1 1 dimensional resonant nonlinear schrodinger s equation with dual power law nonlinearity and the generalized zakharov system of equations the solitary wave solutions or trigonometric functions solutions are obtained from the jacobi elliptic function solutions when the modulus of the jacobi elliptic functions approaches to one or zero respectively comparison between our new results and the well known results is given

smooth duality in natural characteristic abstract we develop a duality theory for admissible smooth representations of p adic lie groups on vector spaces over fields of characteristic p to this end we introduce certain higher smooth duality functors and relate our construction to the auslander duality of completed group rings we study the behavior of smooth duality under tensor products inflation and induction and discuss the dimension theory of smooth mod p representations of a p adic reductive group finally we compute the higher smooth duals of the irreducible smooth representations of gl 2 q p in characteristic p and relate our results to the contragredient operation of colmez

automorphic lifts of prescribed types we prove a variety of results on the existence of automorphic galois representations lifting a residual automorphic galois representation we prove a result on the structure of deformation rings of local galois representations and deduce from this and the method of khare and wintenberger a result on the existence of modular lifts of specified type for galois representations corresponding to hilbert modular forms of parallel weight 2 we discuss some conjectures on the weights of n dimensional mod p galois representations finally we use recent work of taylor to prove level raising and lowering results for n dimensional automorphic galois representations

potentially semi stable deformation rings let k qp be a finite extension and gk gal k k the galois group of an alge braic closure k let f be a finite field of characteristic p and vf a finite dimensional f vector space equipped with a continuous action of gkthe study of the defor mation theory of galois representations was initiated by mazur ma who showed that if vf has no non trivial endomorphisms then it admits a universal deformation ring ryw after the work of wiles wi and the conjectures of fontaine mazur fm it became clear that for arithmetic applications it was important to understand cer tain quotients of ry corresponding to deformations satisfying certain conditions for example wiles uses deformations which arise from finite flat group schemes and the corresponding quotient of ryw was constructed by ramakrishna ra suppose that l k is a finite extension and let a b be integers it seems to be a kind of folklore conjecture that there should be a quotient of ry whose points in finite extensions of w f l p correspond to deformations of vf which become semi stable over l and have hodge tate weights in the interval a b this is closely related to the final conjecture of fo 3 special cases of this when vf is 2 dimensional are conjectured in the papers of fontaine mazur fm p 191 breuil conrad diamond taylor bcdt conj 1 1 1 and breuil m zard bm 2 2 2 4 the purpose of this paper is to prove such a result

the plancherel decomposition for a reductive symmetric space i spherical functions we prove the plancherel formula for spherical schwartz functions on a reductive symmetric space our starting point is an inversion formula for spherical smooth compactly supported functions the latter formula was earlier obtained from the most continuous part of the plancherel formula by means of a residue calculus in the course of the present paper we also obtain new proofs of the uniform tempered estimates for normalized eisenstein integrals and of the maass selberg relations satisfied by the associated c functions

new traveling wave solutions for higher dimensional nonlinear evolution equations using a generalized big frac g prime g big expansion method in this paper we construct new traveling wave solutions of some nonlinear evolution equations in mathematical physics via the 3 1 dimensional potential ytsf equation the 3 1 dimensional modified kdv zakharov kuznetsev equation the 3 1 dimensional kadomtsev petviashvili equation and the 1 1 dimensional kdv equation by using a generalized expansion method where g g satisfies the jacobi elliptic equation g 2 p g here we assume that p g is a polynomial of fourth order many new exact solutions in terms of the jacobi elliptic functions are obtained

on 2 adic deformations we compute the versal deformation ring of a split generic 2 dimensional representation chi 1 oplus chi 2 of the absolute galois group of mathbb q p as an application we show that the breuil mezard conjecture for both non split extensions of chi 1 by chi 2 and chi 2 by chi 1 implies the breuil mezard conjecture for chi 1 oplus chi 2 the result is new for p 2 the proof works for all primes

lattices in the cohomology of shimura curves we prove the main conjectures of breuil j reine angew math 2012 including a generalisation from the principal series to the cuspidal case and dembele j reine angew math 2012 subject to a mild global hypothesis that we make in order to apply certain r mathbb t theorems more precisely we prove a multiplicity one result for the mod p cohomology of a shimura curve at iwahori level and we show that certain apparently globally defined lattices in the cohomology of shimura curves are determined by the corresponding local p adic galois representations we also indicate a new proof of the buzzard diamond jarvis conjecture in generic cases our main tools are the geometric breuil mezard philosophy developed in emerton and gee j inst math jussieu 2012 and a new and more functorial perspective on the taylor wiles kisin patching method along the way we determine the tamely potentially barsotti tate deformation rings of generic two dimensional mod p representations generalising a result of breuil and mezard bull soc math de france 2012 in the principal series case

modules simples en caracteristique p de l algebre de hecke du pro p iwahori de gl3 f resume pour f un corps p adique et n n n 2 on ne connait pas de facon generale les representations lisses irreductibles modulo p de gl n f le foncteur des invariants par le pro p iwahori i 1 invite a les apprehender par l etude des modules simples a droite sur l algebre de hecke h f p gl n f i 1 dans le cas n 2 on connait les h f p gl n f i 1 modules simples ayant un caractere central parmi eux on distingue les modules que l on appelle supersinguliers ils ne correspondent pas via le foncteur des invariants par le pro p iwahori a des sous quotients d induites paraboliques de gl 2 f m f vigneras representations modulo p of the p adic group gl 2 f compos math 140 2004 333 358 nous considerons le cas n 3 il s agit de donner la classification des h f p gl 3 f i 1 modules simples ayant un caractere central et de decrire conjecturalement les modules supersinguliers l ensemble de ces modules supersinguliers est en bijection non unique avec l ensemble des f p representations irreductibles de dimension 3 du groupe de weil de f

a satake isomorphism for representations modulo p of reductive groups over local fields let f be a local field with finite residue field of characteristic p let g be a connected reductive group over f and b a minimal parabolic subgroup of g with levi decomposition b zu let k be a special parabolic subgroup of g in good position relative to z u fix an absolutely irreducible smooth representation of k on a vector space v over some field c of characteristic p writing h g k v for the intertwining hecke algebra of v in g we define a natural algebra homomorphism from h g k v to h z z k v u k we show it is injective and identify its image we thus generalize work of f herzig who assumed f of characteristic 0 g unramified and k hyperspecial and took for c an algebraic closure of the prime field fp we show that in the general case h g k v need not be commutative that is in contrast with the cases herzig considers and with the more classical situation where v is trivial and the field of coefficients is the field of complex numbers

new application of the g g expansion method to higher order nonlinear equations in this paper we extend the g g expansion method to higher order nonlinear equations as a result a series of new travelling wave solutions of the modified kawahara equation and the fifth order kdv equation have been obtained the travelling wave solutions are expressed by the hyperbolic functions the trigonometric functions and the rational functions it is shown that the g g expansion method presents a wider applicability for handling nonlinear evolution equations

on the validity and reliability of the g g expansion method by using higher order nonlinear equations in this study we demonstrate the validity and reliability of the so called g g expansion method via symbolic computation for illustrative examples we choose the sixth order boussinesq equation and the ninth order korteweg de vries equation as a result the power of the employed method is confirmed

on crystabelline deformation rings of mathrm gal overline mathbb q _p mathbb q _p with an appendix by jack shotton we prove that certain crystabelline deformation rings of two dimensional residual representations of mathrm gal overline mathbb q p mathbb q p are cohen macaulay as a consequence this allows to improve kisin s r 1 p mathbb t 1 p theorem to an r mathbb t theorem

functoriality smith theory and the brauer homomorphism if is an automorphism of order p of the semisimple group g there is a natural correspondence between mod p cohomological automorphic forms on g and g we describe this correspondence in the global and local settings

the g g expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics i the present paper we construct the traveling wave solutions involving parameters of the combined korteweg de vries modified korteweg de vries equation the reaction diffusion equation the compound kdv burgers equation and the generalized shallow water wave equation by using a new approach namely the g g expansion method where g g satisfies a second order linear ordinary differential equation when the parameters take special values the solitary waves are derived from the traveling waves the traveling wave solutions are expressed by the hyperbolic functions the trigonometric functions and the rational functions

the oshima sekiguchi and liouville theorems on heintze groups abstract let l be an elliptic operator on a riemannian manifold m a function f annihilated by l is said to be l harmonic f is said to have moderate growth if and only if f grows at most exponentially in the riemannian distance if m is a rank one symmetric space and l is the laplace beltrami operator for m the oshima sekiguchi theorem t oshima j sekiguchi eigenspaces of invariant differential operators on an affine symmetric space invent math 57 1980 1 81 states that a l harmonic function f has moderate growth if and only if f is the poisson integral of a distribution on the furstenberg boundary in this work we prove that this result generalizes to a very large class of homogeneous riemannian manifolds of negative curvature we also i prove a liouville type theorem that characterizes the polynomial like harmonic functions which vanish on the boundary in terms of their growth properties ii describe all polynomial like harmonic functions and iii give asymptotic expansions for the poisson kernel one consequence of this work is that every schwartz distribution on the boundary is the boundary value for a l harmonic function f which is uniquely determined modulo polynomial like harmonic functions

using g g expansion method to seek the traveling wave solution of kolmogorov petrovskii piskunov equation abstract in this paper we use the g g expansion method to seek the traveling wave solutions of the kolmogorov petrovskii piskunov equation the solutions obtained in this paper are more general than the solutions given in refs 23 24 25 and the computation procedure is much simpler it is shown that the g g expansion method provides a very effective and powerful tool for solving nonlinear equations in mathematical physics

reductions of galois representations of slope 1 abstract we compute the reductions of irreducible crystalline two dimensional representations of g q p of slope 1 for primes p 5 and all weights we describe the semisimplification of the reductions completely in particular we show that the reduction is often reducible we also investigate whether the extension obtained is peu or tres ramifiee in the relevant reducible non semisimple cases the proof uses the compatibility between the p adic and mod p local langlands correspondences and involves a detailed study of the reductions of both the standard and non standard lattices in certain p adic banach spaces

a twisted invariant paley wiener theorem for real reductive groups let g be the group of real points of a possibly disconnected linear reductive algebraic group defined over r which is generated by the real points of a connected component g let k be a maximal compact subgroup of the group of real points of the identity component of this algebraic group we characterize the space of maps 7 tr f where is an irreducible tempered representation of g and f varies over the space of smooth compactly supported functions on g which are leftand right k finite this work is motivated by applications to the twisted arthur selberg trace formula

p adic l functions and unitary completions of representations of p adic reductive groups in the first half of the article we introduce the notion of the universal unitary completion of a continuous representation of a p adic reductive group on a locally convex p adic vector space and we prove that such a completion exists under appropriate hypotheses the problem of studying unitary completions has been raised by breuil 4 in connection with his work on a possible p adic local langlands correspondence for gl2 and we relate our construction to certain conjectures of breuil in 4 sec 1 3 for the group gl2 p in particular we show that the universal unitary completion of the locally analytic parabolic induction of a locally algebraic character coincides with the universal unitary completion of the corresponding locally algebraic induction provided that the character being induced satisfies a noncritical slope condition see prop 2 5 in the second half of the article we consider a certain unitary banach space representation of gl2 p obtained by p adically completing the cohomology of classical modular curves the mere existence of this representation implies that those locally algebraic parabolically induced representations of gl2 p which arise from classical finite slope newforms have a nontrivial universal unitary completion verifying a conjecture of breuil in 4 sec 1 3 for these representations while applying proposition 2 5 in this context enables us to give a new construction of p adic l functions attached to p stabilized newforms of noncritical slope combining our construction with the representation theoretic definition of l invariants implicit in 5 cor 1 1 7 we are able to give a simple proof of the mazur tate teitelbaum exceptional zero conjecture in 18 p 46 in terms of breuil s definition of the l invariant

on the universal mod p supersingular quotients for gl2 f over f p for a general f qp abstract let f q p be a finite extension we explore the universal supersingular mod p representations of gl 2 f by computing a basis for their spaces of invariants under the pro p iwahori subgroup this generalizes works of breuil and schein from q p and the totally ramified cases to an arbitrary extension f q p using these results we then construct for an unramified f q p quotients of the universal supersingular modules which have as quotients all the supersingular representations of gl 2 f with a gl 2 o f socle that is expected to appear in the mod p local langlands correspondence a construction in the case of an extension of q p with inertia degree 2 and suitable ramification index is also presented

scattering theory for the laplacian on symmetric spaces of noncompact type and its application to a conjecture of strichartz abstract in this paper we develop the scattering theory for the laplacian on symmetric spaces of noncompact type we study the asymptotic properties of the resolvent in the framework of the agmon hormander space our approach is based on a detailed analysis of the helgason fourier transform and generalized spherical functions on symmetric spaces of noncompact type as an application of our scattering theory we prove a conjecture by strichartz concerning a characterization of a family of generalized eigenfunctions of the laplacian

critere d irreductibilite pour les series principales de gln f en caracteristique p resume soient p un nombre premier et f un corps p adique parmi les f p representations lisses irreductibles de gl 2 f ayant un caractere central l barthel et r livne ont classifie les sous quotients d induites paraboliques et montre qu il en existe d autres qu ils n ont pas classifiees et qu ils ont appelees supersingulieres l barthel r livne irreducible modular representations of gl 2 of a local field duke math j 75 2 1994 261 292 l barthel r livne modular representations of gl 2 of a local field the ordinary unramified case j number theory 55 1995 en 2001 c breuil a obtenu la classification de ces dernieres dans le cas particulier ou f q p c breuil sur quelques representations modulaires et p adiques de gl 2 q p i compos math 138 2 2003 165 188 pour n 3 on ne connait pas meme la classification des f p representations lisses irreductibles de gl n f qui sont des sous quotients d induites paraboliques nous nous interessons ici au cas des series principales le cadre de la caracteristique p fait jouer un role particulier a l unique pro p sylow du sous groupe d iwahori de gl n f toute f p representation lisse non nulle de gl n f possede un vecteur non trivial fixe par ce pro p iwahori il est ainsi naturel d approcher les representations lisses de gl n f via l etude des modules sur la f p algebre de hecke du pro p iwahori de gl n f les travaux de m f vigneras sur la structure de cette algebre ont permis de definir ses modules standards m f vigneras pro p iwahori hecke ring and supersingular f p representations math ann 331 3 2005 523 556 erratum in math ann 333 3 2005 699 701 a l aide des modules standards dits reguliers nous etudions l espace des invariants sous l action du pro p iwahori des series principales de gl n f en caracteristique p nous en deduisons un critere d irreductibilite pour ces series principales

pro p iwahori hecke ring and supersingular representations the motivation of this paper is the search for a langlands correspondence modulo p we show that the pro p iwahori hecke ring open image in new window of a split reductive p adic group g over a local field f of finite residue field fq with q elements admits an iwahori matsumoto presentation and a bernstein z basis and we determine its centre we prove that the ring open image in new window is finitely generated as a module over its centre these results are proved in 11 only for the iwahori hecke ring let p be the prime number dividing q and let k be an algebraically closed field of characteristic p a character from the centre of open image in new window to k which is as null as possible will be called null the simple open image in new window modules with a null central character are called supersingular when g gl n we show that each simple open image in new window module of dimension n containing a character of the affine subring open image in new window is supersingular using the minimal expressions of haines generalized to open image in new window and that the number of such modules is equal to the number of irreducible k representations of the weil group wf of dimension n when the action of an uniformizer pf in the hecke algebra side and of the determinant of a frobenius frf in the galois side are fixed i e the number nn q of unitary irreducible polynomials in fq x of degree n one knows that the converse is true by explicit computations when n 2 10 and when n 3 rachel ollivier

analytic study on two nonlinear evolution equations by using the g g expansion method abstract the validity and reliability of the so called g g expansion method is tested by applying it to two nonlinear evolutionary equations solutions in more general forms are obtained when the parameters are taken as special values it is observed that the previously known solutions can be recovered new rational function solutions are also presented being concise and less restrictive the method can be applied to many nonlinear partial differential equations

eigenspaces of the laplacian on hyperbolic spaces composition series and integral transforms abstract let x be a projective real complex or quaternion hyperbolic space realized as the pseudo riemannian symmetric space x g h with g o p q u p q or sp p q these are the classical isotropic symmetric spaces let be the g invariant laplace beltrami operator on x a complete description by k types for each c of all closed g invariant subspaces of the eigenspace f c x f f is given the eigenspace representations are compared with principal series representations using poisson transformations similar results are obtained also for the exceptional isotropic symmetric space the langlands parameters of the spherical discrete series representations are determined