rigorous uniform approximation of d finite functions using chebyshev expansions a wide range of numerical methods exists for computing polynomial approximations of solutions of ordinary differential equations based on chebyshev series expansions or chebyshev interpolation polynomials we consider the application of such methods in the context of rigorous computing where we need guarantees on the accuracy of the result and from the complexity point of view it is well known that the order n truncation of the chebyshev expansion of a function over a given interval is a near best uniform polynomial approximation of the function on that interval in the case of solutions of linear differential equations with polynomial coefficients the coefficients of the expansions obey linear recurrence relations with polynomial coefficients unfortunately these recurrences do not lend themselves to a direct recursive computation of the coefficients owing among other things to a lack of initial conditions we show how they can nevertheless be used as part of a validated process to compute good uniform approximations of d finite functions together with rigorous error bounds and we study the complexity of the resulting algorithms our approach is based on a new view of a classical numerical method going back to clenshaw combined with a functional enclosure method

first steps towards p adic langlands functoriality by the theory of colmez and fontaine a de rham representation of the galois group of a local field roughly corresponds to a representation of the weil deligne group equipped with an admissible filtration on the underlying vector space using a modification of the classical local langlands correspondence we associate with any pair consisting of a weil deligne group representation and a type of a filtration admissible or not a specific locally algebraic representation of a general linear group we advertise the conjecture that this pair comes from a de rham representation if and only if the corresponding locally algebraic representation carries an invariant norm in the crystalline case the weil deligne group representation is unramified and the associated locally algebraic representation can be studied using the classical satake isomorphism by extending the latter to a specific norm completion of the hecke algebra we show that the existence of an invariant norm implies that our pair indeed comes from a crystalline representation we also show by using the formalism of tannakian categories that this latter fact is compatible with classical unramified langlands functoriality and therefore generalizes to arbitrary split reductive groups resume par la theorie de colmez et fontaine une representation de de rham du groupe de galois d un corps local correspond essentiellement a une representation du groupe de weil deligne dont l espace sous jacent est muni d une filtration admissible en modifiant la correspondance locale de langlands on associe a chaque couple forme d une representation du groupe de weil deligne et des poids d une filtration admissible ou pas une representation localement algebrique particuliere d un groupe lineaire general on conjecture qu un couple provient d une representation de de rham si et seulement si la representation localement algebrique correspondante possede une norme invariante dans le cas cristallin la representation du groupe de weil deligne est non ramifiee et la representation localement algebrique associee peut s etudier gr ce a l isomorphisme de satake classique en prolongeant ce dernier a une completion de l algebre de hecke on montre que l existence d une norme invariante comme ci dessus implique que le couple provient effectivement d une representation cristalline on montre aussi en utilisant le formalisme des categories tannakiennes que ce dernier fait est compatible avec la fonctorialite de langlands non ramifiee classique et donc qu il se generalise a tout groupe reductif deploye 2 c breuil p schneider

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

the image of the thom map for eilenberg maclane spaces fundamental classes in bp cohomology of eilenberg mac lane spaces are defined the image of the thom map from bp cohomology to mod p cohomology is determined for arbitrary eilenberg mac lane spaces this image is a polynomial subalgebra generated by infinitely many elements obtained by applying a maximum number of milnor primitives to the fundamental class in mod p cohomology this subalgebra in mod p cohomology is invariant under the action of the steenrod algebra and it is annihilated by all milnor primitives we also show that bp cohomology determines morava k cohomology for eilenberg mac lane spaces

cohomology of infinitesimal and discrete groups an infinitesimal algebraic group over a field k is a connected algebraic group scheme whose coordinate ring is a finite dimension al k algebra if one begins with a reductive algebraic group g defined over fp then one has a close relationship between the representation theory of g that of its infinitesimal subgroups g defined as the kernels of the frobenius endomorphisms rr g g and that of its finite chevalley subgroups g fq of fq rational points q pd a basic theorem of cline et al i 9 theorem 6 6 relates the cohomology of g in a rational representation to that of the finite groups g fq related work of cline et al 6 and of the authors 1 16 relates the cohomology of g to that of its infinitesimal subgroups a determination of the cohomology of the first infinitesimal subgroup g 1 of g with trivial coefficients has enabled the authors in 14 to make certain explicit cohomology calculations for the finite groups g fq with certain nontrivial coefficients in this paper we continue our study of infinitesimal subgroups of simple algebraic groups by proving various qualitative properties of their cohomology and by extending our computational knowledge in particular we provide a computation which applies in a range of degrees to chevalley groups over the prime field we also investigate the cohomology of infinitesimal subgroups of nilpotent groups enabling us for example to determine the cohomology of their lie algebras in certain cases we apply this lie algebra cohomology computation to determine the integral cohomology of various torsion free nilpotent groups modulo small prime torsion thereby extending a theorem of dwyer 12 in more detail the paper begins in sect 1 with a spectral sequence converging to the cohomology of the first infinitesimal subgroup g which was previously studied in 14 from this spectral sequence it easily follows that h g1 m is a noetherian h g1 k module for each finite dimensional rational gl module m we then apply the work of andersen and jantzen 1 on the gl cohomology of induced modules to obtain the analogous result for the second infinitesimal subgroup gz our analysis also permits a determination of the generic

representations lisses p temperees des groupes p adiques we consider certain asymptotic properties of smooth p adically valued functions and representations of a p adic reductive group g first we continue the study of the so called p tempered and p discrete representations as defined in a former paper and apply this to get a classification of locally integral representations i e those representations such that for any open compact subgroup h the h invariant subspace admits hecke invariant lattices then we show that the space of square integrable smooth functions as defined in the text is an algebra under convolution to which the action of the hecke algebra on any p tempered representation extends naturally we formulate a plancherel like formula but prove it only for sl 2

a holonomic systems approach to special functions identities abstract we observe that many special functions are solutions of so called holonomic systems bernstein s deep theory of holonomic systems is then invoked to show that any identity involving sums and integrals of products of these special functions can be verified in a finite number of steps this is partially substantiated by an algorithm that proves terminating hypergeometric series identities and that is given both in english and in maple

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

chebyshev series expansion of inverse polynomials the chebyshev series expansion n 0 antn x of the inverse of a polynomial j 0kbjtj x is well defined if the polynomial has no roots in 1 1 if the inverse polynomial is decomposed into partial fractions the an are linear combinations of simple functions of the polynomial roots also if the first k of the coefficients an are known the others become linear combinations of these derived recursively from the bj s on a closely related theme finding a polynomial with minimum relative error towards a given f x is approximately equivalent to finding the bj in f x 0k bj tj x 1 k 1 an tn x a newton algorithm produces these if the chebyshev expansion of f x is known

efficient and accurate computation of upper bounds of approximation errors for purposes of actual evaluation mathematical functions f are commonly replaced by approximation polynomials p examples include floating point implementations of elementary functions quadrature or more theoretical proof work involving transcendental functions replacing f by p induces a relative error p f 1 in order to ensure the validity of the use of p instead of f the maximum error i e the supremum norm i must be safely bounded above over an interval i whose width is typically of order 1 numerical algorithms for supremum norms are efficient but they cannot offer the required safety previous validated approaches often require tedious manual intervention if they are automated they have several drawbacks such as the lack of quality guarantees in this article a novel automated supremum norm algorithm on univariate approximation errors is proposed achieving an a priori quality on the result it focuses on the validation step and paves the way for formally certified supremum norms key elements are the use of intermediate approximation polynomials with bounded approximation error and a non negativity test based on a sum of squares expression of polynomials the new algorithm was implemented in the sollya tool the article includes experimental results on real life examples

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

optimal and nearly optimal algorithms for approximating polynomial zeros abstract we substantially improve the known algorithms for approximating all the complex zeros of an n th degree polynomial p x our new algorithms save both boolean and arithmetic sequential time versus the previous best algorithms of schonhage 1 pan 2 and neff and reif 3 in parallel nc implementation we dramatically decrease the number of processors versus the parallel algorithm of neff 4 which was the only nc algorithm known for this problem so far specifically under the simple normalization assumption that the variable x has been scaled so as to confine the zeros of p x to the unit disc x x 1 our algorithms which promise to be practically effective approximate all the zeros of p x within the absolute error bound 2 b by using order of n arithmetic operations and order of b n n 2 boolean bitwise operations in both cases up to within polylogarithmic factors the algorithms allow their optimal work preserving nc parallelization so that they can be implemented by using polylogarithmic time and the orders of n arithmetic processors or b n n 2 boolean processors all the cited bounds on the computational complexity are within polylogarithmic factors from the optimum in terms of n and b under both arithmetic and boolean models of computation in the boolean case under the additional realistic assumption that n o b

complexity for modules over finite chevalley groups and classical lie algebras 1 1 let g be a connected reductive algebraic group defined over a finite prime field fp of p elements for a given rational g module m one can consider the restrictions of m to the restricted lie algebra lie g and to the finite group g fp of fp rational points even though there is no direct functorial connection between the categories of restricted lie g modules and g fp modules the ambient algebraic group g can serve as a bridge linking these two important module categories early results of curtis in the 1960s provided a one to one correspondence between the simple restricted modules over k fp of lie g and those of g fp this correspondence is given by simply restricting certain classes of simple g modules more importantly these results along with work of steinberg allow one to easily transfer the questions pertaining to the computation of irreducible characters among these three categories for over thirty years this approach has been one of the main ideas used to relate the representation and cohomology theory of algebraic groups lie algebras and finite groups of lie type in the defining characteristic see e g cpsvan jan2 hum3 and jan1 given the previous results in this area a natural course of investigation should be to find relationships between canonical modules like projective modules for lie g and g fp more precisely

new fast algorithms for polynomial interpolation and evaluation on the chebyshev node set abstract for a polynomial p x of a degree n we study its interpolation and evaluation on a set of chebyshev nodes x cos 2 1 2n 2 0 1 n this is easily reduced to applying discrete fourier transforms dfts to the auxiliary polynomial q n p x where 2 x 1 exp 1 2n we show the back and forth transition between p x and q based on the respective back and forth transformations of the variable 1 z 1 z y x 1 x 1 y z 2 all these transformations like the dfts are performed by using o n log n arithmetic operations which thus suffice in order to support both interpolation and evaluation of p x on the chebychev set as well as on some related sets of nodes this improves by factor log n the known arithmetic time bound for chebyshev interpolation and gives an alternative supporting algorithm for the record estimate of o n log n for chebyshev evaluation obtained by gerasoulis in 1987 and based on a distinct algorithm

cohomology of sheaves on the building and r representations this article contains four parts the subject of the first one is to compare open compact subgroups of a reductive connected group g over a local non archimedean field associated to certain natural concave functions by the theory of bruhat tits bt1 we compare the groups which appear in the theory of unrefined minimal types by moy and prasad mp1 mp2 with those which appear in the theory of complex sheaves on the building x by schneider and stuhler ss schneider and stuhler give to special vertices a predominant role hiding in this way some symetry that we restaure we need all vertices in particular we describe how the groups vary along a geodesic starting from any point in a discussion with kutzko and with schneider it appeared that there are in the literature two different definitions for an iwahori subgroup the connected and the non connected as i was not aware of this and as there are may be other definitions i include here the unique decomposition property for the possible iwahori subgroups this part is entirely geometric and is certainly trivial for someone who is familiar with the theory of bruhat and tits

a unified approach to the tau method and chebyshev series expansion techniques abstract in this paper we show the full equivalence between the recursive 1 and operational 2 formulations of the tau method we then use such methods as analytic tools in the simulation of other numerical techniques for the approximate solution of differential equations as tau methods with special perturbation terms in this paper we consider as examples of this approach two classical numerical techniques based on chebyshev series expansions we introduce a numerical parameter the length of a method to compare different numerical techniques with reference to a given basis our results make possible the recursive formulation of a variety of series expansion methods

a fast and well conditioned spectral method a spectral method is developed for the direct solution of linear ordinary differential equations with variable coefficients and general boundary conditions the method leads to matrices that are al