on the alternating series 1 12 13 14 abstract it is shown that the best constants a and b such that inequalities 1 2n a k n 1 1 k 1 1 k 1 2n b hold for every n 1 are a 1 1 log 2 2 1 258891 and b 1

oriented measures with continuous densities and the bang bang principle abstract we introduce the notion of an oriented measure for such a measure given in l 1 a b 0 e a b whose characteristic functions have less than n discontinuity points and such that d e given a solution x to the control problem l x x n a n 1 t x n 1 a 1 t x a 0 t x 1 2 there exist two bang bang solutions y z having a contact of order n with x at a and b such that y x z reachable sets of bang bang constrained solutions are convex an application to the calculus of variations yields a density result

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

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

on leibniz series defined by convex functions abstract it is shown that for every 0 we have k n 1 1 k 1 k 1 2 n n for some strictly decreasing sequence n n 1 such that 1 2 n 1 2 1 1 2 n 1 1 hence with lim n n 1 2 this is only a particular case of more general new results on leibniz series defined by convex functions

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

half integer approximations for the partial sums of the harmonic series abstract let s n denote the n th partial sum of the harmonic series an asymptotic approximation for s n is obtained in terms of the half integer variable n 1 2 including bounds for the remainder improved estimates of the comtet function for the harmonic series are obtained as a consequence

two dimensional zonoids and chebyshev measures abstract we give an alternative proof to the well known fact that each convex compact centrally symmetric subset of r 2containing the origin is a zonoid i e the range of a two dimensional vector measure and we prove that a two dimensional zonoid whose boundary contains the origin is strictly convex if and only if it is the range of a chebyshev measure we give a condition under which a two dimensional vector measure admits a decomposition as the difference of two chebyshev measures a necessary condition on the density function for the strict convexity of the range of a measure and a characterization of two dimensional chebyshev measures

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

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

a pointwise constrained version of the liapunov convexity theorem for vectorial linear first order control systems we generalize the liapunov convexity theorem s version for vectorial control systems driven by linear odes of first order p 1p 1 in any dimension d nd n by including a pointwise state constraint more precisely given a x wp 1 a b rd solving the convexified p th order differential inclusion lpx t co u0 t u1 t um t a e consider the general problem consisting in finding bang bang solutions i e lpx t u0 t u1 t um t a e under the same boundary data x k a x k a x k b x k b k 0 1 p 1k 0 1 p 1 but restricted moreover by a pointwise state constraint of the type x t x t t a b e g 1 0 0 1 0 0 yielding x 1 t x 1 t previous results in the scalar d 1d 1 case were the pioneering amar cellina paper dealing with lpx x followed by cerf mariconda results who solved the general case of linear differential operators lplp of order p 2p 2 with c0 a b coefficients this paper is dedicated to focus on the missing case p 1p 1 i e using lpx x a x generalize the dimension of x x from the scalar case d 1d 1 to the vectorial d nd n case weaken the coefficients from continuous to integrable so that a a now becomes a d dd d integrable matrix and allow the directional vector to become a moving ac function previous vectorial results had constant no matrix i e a 0a 0 and considered constant control vertices amar mariconda and more recently integrable control vertices ourselves

integral estimates for convergent positive series abstract it is shown that for every 1 we have k n 1 1 k 1 1 n n 1 for some strictly decreasing sequence n n 1 such that 1 2 n 1 4 1 1 1 2 n 1 hence with lim n n 1 2 this is only a particular case of more general new results on series defined by convex functions

on bang bang constrained solutions of a control system given varphi 1 varphi 2 in l 1 0 t and a function x in w 2 1 0 t solving the control problem p x a 1 t x a 0 t in varphi 1 t varphi 2 t a e x 0 x 0 x t x 1 x 0 v 0 x t v 1 there exists a bang bang solution y to p satisfying y le x moreover there exists a finite union of intervals e such that y a 1 y a 0 y varphi 1 chi e varphi 2 chi 0 t e the reachable set of bang bang constrained solutions is convex an application to the calculus of variations

lattices in the cohomology of u 3 arithmetic manifolds under hypotheses required for the taylor wiles method we prove for forms of u 3 which are compact at infinity that the lattice structure on upper alcove algebraic vectors or on principal series types given by the lambda isotypic part of completed cohomology is a local invariant of the galois representation attached to lambda when this galois representation is residually irreducible locally at places dividing p as a crucial input we establish corresponding mod p multiplicity one results our main innovation is the combination of integral hecke theory and the taylor wiles method

on special representations of p adic reductive groups let f be a non archimedean locally compact field let g be a split connected reductive group over f for a parabolic subgroup q g and a ring l we consider the g representation on the l module c g q l

homological stability for completed homology we prove that the completed homology groups of mathrm gl n mathbf z in fixed degree stabilize as n goes to infinity we also prove that the action of hecke operators on stable cohomology is trivial in a precisely defined sense

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

the vector measures whose range is strictly convex let be a measure on a measure space x with values in rnandfbe the density of with respect to its total variation we show that the range r e e of is strictly convex if and only if the determinant det f x1 f xn is nonzero a e onxn we apply the result to a class of measures containing those that are generated by chebyshev systems