sufficiency and duality in nonsmooth multiobjective optimization involving generalized f α π d type i functions in this paper new classes of generalized f a p d type i functions are introduced for a nonsmooth multiobjective programming problem based upon these generalized functions sufficient optimality conditions are established weak strong and strict converse duality theorems are also derived for wolfe and mond weir type multiobjective dual programs in order to relate the efficient and weak efficient solutions of primal and dual problems

on elliptic curves of prime power conductor over imaginary quadratic fields with class number 1 the main result of this paper is to extend from q to each of the nine imaginary quadratic fields of class number one a result of serre 1987 and mestre oesterle 1989 namely that if e is an elliptic curve of prime conductor then either e or a 2 3 or 5 isogenous curve has prime discriminant for four of the nine fields the theorem holds with no change while for the remaining five fields the discriminant of a curve with prime conductor is either prime or the square of a prime the proof is conditional in two ways first that the curves are modular so are associated to suitable bianchi newforms and second that a certain level lowering conjecture holds for bianchi newforms we also classify all elliptic curves of prime power conductor and non trivial torsion over each of the nine fields in the case of 2 torsion we find that such curves either have cm or with a small finite number of exceptions arise from a family analogous to the setzer neumann family over q

a variational like inequality for multifunctions with applications abstract we study a variational like inequality problem for a multifunction a certain relevant complementarity problem is defined we establish some existence theorems for the above two problems an application to convex mathematical programming is also shown

wave equation based image warping in seismic processing and reservoir characterization we often need to measure relative displacements between different realizations of data over the years many methods have been developed using different measures of similarity such alignment or warping methods are often effective signal or image processing tools however none of the available methods are directly driven by the physics of seismic imaging we have found that a seismic image can be considered as a field governed by the wave equation we visualized different image realizations as snapshots of the wavefield at different times and these conveyed the required displacements or time shifts by formulating the problem in a physical context we obtained displacements that honored the directionality of the wave propagation for example 4d time shifts on migrated stacks were obtained in a direction normal to the reflectors we have computed these shifts in an inverted finite difference scheme to overcome limitations of the two way wave equation in this application we factorized it to its one way counterparts the method was demonstrated on synthetic and real data sets

optimality and duality in multiobjective fractional programming involving ρ semilocally type i preinvex and related functions abstract optimality conditions are obtained for a nonlinear fractional multiobjective programming problem involving semidifferentiable functions also a general dual is formulated and a duality result is proved using concepts of generalized semilocally type i preinvex functions

stable explicit depth extrapolation of seismic wavefields stable explicit filters for depth extrapolation of seismic wavefields may be derived through a modification of the conventional taylor series method the modified taylor series method described here yields extrapolators with maximally flat amplitude spectra in their passband while ensuring that no spectral components in the wavefield are amplified by any factor greater than one the price for stability is increased phase error for low normalized frequencies implicit extrapolators are more accurate than the 39 coefficient stable explicit extrapolator described here however the small spatial sampling intervals required to obtain high phase accuracy in implicit extrapolation imply that this accuracy is rarely achieved in practice over the wide range of normalized frequencies likely to be encountered in practice stable explicit extrapolators outperform implicit ones the method presented here for deriving stable explicit extrapolators is in no formal sense optimal it is only guaranteed to yield stable extrapolators in my limited experience with alternative methods for designing stable extrapolators the method presented here produced the least phase error while ensuring stability nevertheless a simple method for designing optimal in some sense stable explicit extrapolators would be preferred over the method presented here 7 refs 15 figs

cohomology with twisted one dimensional coefficients for congruence subgroups of sl4 z and galois representations abstract we extend the computations in 2 3 4 to find the cohomology in degree five of a congruence subgroup of sl 4 z with coefficients in a field twisted by a nebentype character along with the action of the hecke algebra on the cohomology this is the top cuspidal degree for each hecke eigenclass we find we produce the unique galois representation that appears to be attached to it the computations require serious modifications to our previous algorithms nontrivial coefficients add a layer of complication to our data structures new possibilities must be taken into account in the galois finder the code that finds the galois representations we have improved the galois finder to report when the attached galois representation is uniquely determined by our data

efficiency through variational like inequalities with lipschitz functions in this work first we introduce several notions of invexity and pseudoinvexity for a locally lipschitz function by means of the generalized jacobian we study relationships between these concepts in particular the implications between preinvexity and invexity next we obtain necessary and sufficient optimality conditions for efficient and weak efficient solutions of finite dimensional non necessarily pareto vector optimization problems with locally lipschitz objective functions through solutions of vector variational like inequality problems these conditions are stated via the generalized jacobian and under pseudoinvexity hypotheses and they show that a vector optimization problem can be reformulated as a vector variational like inequality problem this work extends and improves several previous papers where the objective function of the vector optimization problem is assumed to be differentiable or being locally lipschitz the authors consider the componentwise subdifferential based on the clarke s generalized gradients of the components of the objective function throughout the paper some simple examples are given in order to illustrate the main concepts and results

generalized mixed quasi variational like inequalities in this paper we use the auxiliary principle technique to suggest and analyze some iterative methods for solving generalized mixed quasi variational like inequalities we show that the convergence of the proposed method requires either pseudomonotonicity or partially relaxed strongly monotonicity under some mild condition as special cases one can obtain a number of new and known algorithms for solving variational inequalities and complementarity problems our results represent significant and important refinements of the previously known results

seismic depth migration with the dirac equation we demonstrate the applicability of the dirac equation in seismic wavefield extrapolation by presenting a new explicit one way prestack depth migration scheme the method is in principle accurate up to 90 from the vertical and it tolerates lateral velocity variations this is achieved by performing the extrapolation step of migration with the dirac equation implemented in the space frequency domain the dirac equation is an exact linearization of the square root wave equation and is equivalent to keeping infinitely many terms in a taylor series or continued fraction expansion of the square root operator an important property of the new method is that the local velocity and the spatial derivatives decouple in separate terms within the extrapolation operator therefore we do not need to precompute and store large tables of convolutional extrapolator coefficients depending on velocity the main drawback of the explicit scheme is that evanescent energy must be removed at each depth step to obtain numeric

equivalence of saddle points and optima and duality for a class of non smooth non convex problems abstract the equivalence between saddle points and optima and duality theorems are established for a much larger class of non smooth non convex problems in which functions are locally lipschitz and are satisfying invex type conditions of hanson and craven

comparison of semi simplifications of galois representations abstract let be the absolute galois group of a global field let 1 and 2 be two p adic finite dimensional representations of then there exists a finite number of primes q such that if the characteristic polynomials of 1 frob q and 2 frob q are equal then 1 and 2 have isomorphic semi simplifications and so the same l functions we give a method to compute a sufficient list of primes based on the ramification and the dimension of the representations we use then the result to prove a conjecture by b van geemen and j top

a class of b p r invex functions and mathematical programming abstract invexity of a function is generalized the new class of nonconvex functions called b p r invex functions with respect to and b being introduced includes many well known classes of generalized invex functions as its subclasses some properties of the introduced class of b p r invex functions with respect to and b are studied further mathematical programming problems involving b p r invex functions with respect to and b are considered the equivalence between saddle points and optima and different type duality theorems are established for this type of optimization problems

binary hermitian forms over a cyclotomic field abstract let be a primitive fifth root of unity and let f be the cyclotomic field f q let o f be the ring of integers we compute the voronoi polyhedron of binary hermitian forms over f and classify gl 2 o conjugacy classes of perfect forms the combinatorial data of this polyhedron can be used to compute the cohomology of the arithmetic group gl 2 o and hecke eigenforms

hemiequilibrium like problems abstract in this paper we introduce and consider a new class of equilibrium problems known as hemiequilibrium like problems this new class includes hemiequilibrium equilibrium like problems and several classes of variational inequalities as special cases a number of iterative methods for solving hemiequilibrium like problems are suggested and analyzed by using the auxiliary principle technique we also study the convergence analysis of these iterative methods under some mild conditions the results obtained in this paper can be considered as a novel application of the auxiliary principle technique

relationships between pre invex concepts abstract it is well known that some properties of convex optimization problems especially sufficient conditions for optimality and duality conditions are preserved when convexity is weakened to pre invexity however there are several distinct pre invex concepts b vex b pre invex pre univex p r pre invex and these are not equivalent the pre invexity concepts mentioned above have also some properties which are characteristic and invariant for functions belonging to these classes of functions

spectral methods in time for hyperbolic equations a pseudospectral numerical scheme for solving linear periodic hyperbolic problems is described it has infinite accuracy both in time and in space the high accuracy in time is achieved without increasing the computational work and memory space which is needed for a regular one step explicit scheme the algorithm is shown to be optimal in the sense that among all the explicit algorithms of a certain class it requires the least amount of work to achieve a certain given resolution the class of algorithms referred to consists of all explicit schemes which may be represented as a polynomial in the spatial operator

on minimax fractional optimality conditions with invexity abstract under different forms of invexity conditions sufficient kuhn tucker conditions and three dual models are presented for the minimax fractional programming

composite nonsmooth multiobjective programs with v ρ invexity abstract in this paper some problems consisting of nonsmooth composite multiobjective programs have been treated with v invexity type of conditions in particular we prove the generalized karush kuhn tucker sufficient optimality theorem and duality theorems for nonsmooth composite multiobjective programs also weak vector saddle point theorems are obtained for the composite programs under v invexity conditions the results obtained generalize the results of hun kuk et al to some extent

minkowski s conjecture well rounded lattices and topological dimension let a c sln r be the diagonal subgroup and identify sln r sln z with the space of unimodular lattices in rn in this paper we show that the closure of any bounded orbit a lcsln r sln z meets the set of well rounded lattices this assertion implies minkowski s conjecture for n 6 and yields bounds for the density of algebraic integers in totally real sextic fields the proof is based on the theory of topological dimension as reflected in the combinatorics of open covers of rn and tn

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

on p r invexity type nonlinear programming problems abstract mathematical programming problems involving p r invex functions with respect to are considered we introduce new classes of nonlinear programming problems called kt p r invex wd p r invex and hc p r invex problems where p r are some real numbers it is shown that for these types of problems kuhn tucker conditions are both necessary and sufficient for optimality furthermore these p r invexity type problems with r 0 are not sufficient for wolfe weak duality in this way it was shown that the optimization problems possessing some kind of invexity need not be equivalent to the class of optimization problems for which kuhn tucker necessary conditions for optimality are also sufficient and wolfe weak duality holds

a nonreflecting boundary condition for discrete acoustic and elastic wave equations one of the nagging problems which arises in application of discrete solution methods for wave propagation calculations is the presence of reflections or wraparound from the boundaries of the numerical mesh these undesired events eventually override the actual seismic signals which propagate in the modeled region the solution to avoiding boundary effects used to be to enlarge the numerical mesh thus delaying the side reflections and wraparound longer than the range of times involved in the modeling obviously this solution considerably increases the expense of computation more recently nonreflecting boundary conditions were introduced for the finite difference method clayton and enquist 1977 reynolds 1978 these boundary conditions are based on replacing the wave equation in the boundary region by one way wave equations which do not permit energy to propagate from the boundaries into the numerical mesh this approach has been relatively successful except that its effectiveness degrades for event

a note on invex problems with nonnegative variable we provide sufficient optimality conditions involving invexity for mathematical programming problems with nonnegative variable we also show that upon certain assumptions on the scaling function these problems are convex transformable hence certain associated convex problems should be solved some applications are shown

optimality and duality for nonsmooth multiobjective fractional programming with generalized invexity abstract in this paper we consider a class of nonsmooth multiobjective fractional programming problems in which functions are locally lipschitz we establish generalized karush kuhn tucker necessary and sufficient optimality conditions and derive duality theorems for nonsmooth multiobjective fractional programming problems containing v invex functions

modular representations of gl 3 fp symmetric squares and mod p cohomology of gl 3 z abstract we prove certain cases of nonabelian reciprocity between mod p galois representations and hecke eigenclasses in the mod p cohomology of gl 3 z using the symmetric square liftings from gl 2 and congruences on hecke eigenvalues we need to develop some modular representation theory for gl 3 z p in some cases we also need to invoke certain computer calculations to finish the proof

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

existence and iterative approximations of solutions for mixed quasi variational like inequalities in banach spaces abstract in this paper we introduce and investigate a new class of mixed quasi variational like inequalities in reflexive banach spaces by applying a minimax inequality due to ding tan and a lemma due to chang we establish some existence and uniqueness results of solution for the mixed quasi variational like inequality next by using a kkm theorem due to fan and an auxiliary principle technique due to cohen we suggest two iterative algorithms and study the convergence criteria of iterative sequences generated by the iterative algorithms our results extend improve and unify several known results in the literature

necessary and sufficient conditions in constrained optimization additional conditions are attached to the kuhn tucker conditions giving a set of conditions which are both necessary and sufficient for optimality in constrained optimization under appropriate constraint qualifications necessary and sufficient conditions are also given for optimality of the dual problem duality and converse duality are treated accordingly

an analogue of serre s conjecture for galois representations and hecke eigenclasses in the mod p cohomology of gl n ℤ

sufficiency and duality in nondifferentiable multiobjective programming involving generalized type i functions abstract recently hachimi and aghezzaf defined generalized f d type i functions a new class of functions that unifies several concepts of generalized type i functions in this paper the generalized f d type i functions are extended to nondifferentiable functions by utilizing the new concepts we obtain several sufficient optimality conditions and prove mixed type and mond weir type duality results for the nondifferentiable multiobjective programming problem

on rigid analytic uniformizations of jacobians of shimura curves the main goal of this article is to give an explicit rigid analytic uniformization of the maximal toric quotient of the jacobian of a shimura curve over bbb q at a prime dividing exactly the level this result can be viewed as complementary to the classical theorem of v c erednik and drinfeld which provides rigid analytic uniformizations at primes dividing the discriminant as a corollary we offer a proof of a conjecture formulated by m greenberg in his paper on stark heegner points and quaternionic shimura curves thus making greenberg s construction of local points on elliptic curves over bbb q unconditional

galois representations with conjectural connections to arithmetic cohomology in this paper we extend a conjecture of a ash and w sinnott relating niveau 1 galois representations to the mod p cohomology of congruence subgroups ofsln z to include galois representations of higher niveau we then present computational evidence for our conjecture in the case n 3 in the form of three dimensional galois representations which appear to correspond to cohomology eigenclasses as predicted by the conjecture our examples include galois representations with nontrivial weight and level as well as irreducible three dimensional representations that are in no obvious way related to lower dimensional representations in addition we prove that certain symmetric square representations are actually attached to cohomology eigenclasses predicted by the conjecture

nonsmooth invex functions and sufficient optimality conditions in this paper the various definitions of nonsmooth invex functions are gathered in a general scheme by means of the concept of k directional derivative characterizations of nonsmooth k invexity are derived as well as results concerning constrained optimization without any assumption of convexity of the k directional derivatives

on sufficiency of the kuhn tucker conditions if all the functions fo x f x f x are convex on c then these conditions are also sufficient various classes of functions have been defined for the purpose of weakening this limitation of convexity in mathematical programming mangasarian 11 has speculated that pseudo convexity of x and quasiconvexity off x are the weakest conditions that can be imposed so that the above conditions are sufficient for optimality it will be shown that there are other wide classes of functions for which the conditions are sufficient 545

optimality and duality for generalized fractional programming involving nonsmooth f ρ convex functions using a parametric approach we establish necessary and sufficient conditions and derive duality theorems for a class of nonsmooth generalized minimax fractional programming problems containing f r convex functions

optimality and duality in fractional multiple objective programming involving semilocally preinvex and related functions necessary and sufficient optimality conditions are obtained for a nonlinear fractional multiple objective programming problem involving semidifferentiable functions also a general dual is formulated and duality results are proved using concepts of generalized semilocally preinvex functions

nondifferentiable multiobjective programming under generalized d univexity abstract in this paper we are concerned with a nondifferentiable multiobjective programming problem with inequality constraints we introduce four new classes of generalized convex functions by combining the concepts of weak strictly pseudoinvex strong pseudoinvex weak quasi invex weak pseudoinvex and strong quasi invex functions in aghezzaf and hachimi numer funct anal optim 22 2001 775 d invex functions in antczak europ j oper res 137 2002 28 and univex functions in bector et al univex functions and univex nonlinear programming proc admin sci assoc canada 1992 p 115 by utilizing the new concepts we derive a karush kuhn tucker sufficient optimality condition and establish mond weir type and general mond weir type duality results for the nondifferentiable multiobjective programming problem

optimality and duality for the multiobjective fractional programming with the generalized f ρ convexity abstract a class of multiobjective fractional programmings mfp are first formulated where the involved functions are local lipschitz and clarke subdifferentiable in order to deduce our main results we give the definitions of the generalized f convex class about the clarke subgradient under the above generalized convexity assumption the alternative theorem is obtained and some sufficient and necessary conditions for optimality are also given related to the properly efficient solution for the problems finally we formulate the two dual problems md and md1 corresponding to mfp and discuss the week strong and reverse duality

multi objective infinite programming using the theory of abstract optimization problems in infinite dimensional spaces we provide necessary optimality conditions of first and second order for weakly efficient solutions of the multi objective infinite programming problem sufficient conditions are given under invexity assumptions we generalize the notion of kkt invexity for the multi objective infinite problem and show that this notion is a necessary and sufficient condition for every vector kkt solution to be a weakly efficient solution moreover we develop a theorem of the alternative useful for proving some of our results

nonsmooth minimax programming under locally lipschitz φ ρ invexity abstract minimax programming problems involving locally lipschitz invex functions are considered the parametric and non parametric necessary and sufficient optimality conditions for a class of nonsmooth minimax programming problems are obtained under nondifferentiable invexity assumption imposed on objective and constraint functions when the sufficient conditions are utilized parametric and non parametric dual problems in the sense of mond weir and wolfe may be formulated and duality results are derived for the considered nonsmooth minimax programming problem with the reference to the said functions we extend some results of optimality and duality for a larger class of nonsmooth minimax programming problems

generalized proper efficiency and duality for a class of nondifferentiable multiobjective variational problems withv invexity abstract a mond weir type dual for a class of nondifferentiable multiobjective variational problems in which every component of the objective function contains a term involving the square root of a certain positive semidefinite quadratic form is considered and various duality results viz weak strong and converse duality theorems are developed for conditionally properly efficient solutions these results are obtained under v invexity assumptions and its generalizations on objective and constraint functions this work extends many results on variational problems established earlier

multiobjective programming under d invexity abstract in this paper a generalization of convexity is considered in the case of nonlinear multiobjective programming problem where the functions involved are nondifferentiable by considering the concept of pareto optimal solution and substituting d invexity for convexity the fritz john type and karush kuhn tucker type necessary optimality conditions and duality in the sense of mond weir and wolfe for nondifferentiable multiobjective programming are given

geometric families of 4 dimensional galois representations with generically large images we study compatible families of four dimensional galois representations constructed in the etale cohomology of a smooth projective variety we prove a theorem asserting that the images will be generically large if certain conditions are satisfied we only consider representations with coefficients in an imaginary quadratic field we apply our result to an example constructed by jasper scholten a non selfdual 4 dimensional galois representation www math uiuc edu algebraic number theory 0183 1999 obtaining a family of linear groups and one of unitary groups as galois groups over mathbb q

on the images of modular and geometric three dimensional galois representations we study compatible families of three dimensional galois representations constructed in the etale cohomology of a smooth projective variety we prove a theorem asserting that the residual images will be generically large if certain easy to check conditions are satisfied we only consider representations with coefficients in an imaginary quadratic field for primes inert in this field the residual representations when irreducible are unitary we apply our result to an example constructed by van geemen and top obtaining a family of special linear groups and one of special unitary groups as galois groups over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i we also consider the case of cohomological modular forms for a congruence subgroup of sl inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i assuming clozel s conjecture stating that a geometric family of three dimensional galois representations can be attached to them we verify for three examples the conditions guaranteeing generically large images and explicitly bound the finite set of primes with non maximal image we also discuss what intrinsic conditions a modular form should verify to guarantee that the images of the attached galois representations will be generically large

the generalized quasi variational inequality problem with applications a new problem called the generalized quasi variational inequality problem is introduced this new problem extends all the existing variational inequality problems in finite dimensional spaces and enlarges the class of problems that can be approached by the variational inequality theory some existence results without convexity assumptions are presented applications to mathematical and equilibrium programming are given

uniformization of modular elliptic curves via p adic periods abstract the langlands programme predicts that a weight 2 newform f over a number field k with integer hecke eigenvalues generally should have an associated elliptic curve e f over k in 19 we associated building on works of darmon 8 and greenberg 20 a p adic lattice to f under certain hypothesis and implicitly conjectured that is commensurable with the p adic tate lattice of e f in this paper we present this conjecture in detail and discuss how it can be used to compute directly from f a weierstrass equation for the conjectural e f we develop algorithms to this end and implement them in order to carry out extensive systematic computations in which we compute weierstrass equations of hundreds of elliptic curves some with huge heights over dozens of number fields the data we obtain give extensive support for the conjecture and furthermore demonstrate that the conjecture provides an efficient tool to building databases of elliptic curves over number fields

smith theory and hecke operators abstract of the connections between the cohomology of arithmetic groups and representations of galois groups some are known and more are conjectured this paper proves one of these conjectures for certain monomial galois representations let p be a prime we show that the representation of the absolute galois group of q induced from an f p valued ray class character of the cyclotomic field q p is attached to a hecke eigenclass in the mod p cohomology of a torsion free congruence subgroup m of gl p 1 z the method involves constructing an action of the hecke algebra on the fundamental exact sequence of smith theory arising from the action by conjugation of an element of order p in gl p 1 z on m

strong and weak convexity of sets and functions in this paper we study two classes of sets strongly and weakly convex sets for each class we derive a series of properties which involve either the concept of supporting ball an obvious extension of the concept of supporting hyperplane or the normal cone to the set we also study a class of functions denoted convex which satisfy for arbitrary points x1 and x2 and any value 0 1 the classical inequality of convex functions up to a term 1 x1 x2 2 depending on the sign of the constant the function is said to be strongly or weakly convex we provide characteristic properties of this class of sets and we relate it to strongly and weakly convex sets via the epigraph and the level sets finally we give three applications a separation theorem a sufficient condition for global optimum of a nonconvex programming problem and a sufficient geometrical condition for a set to be a manifold

pre invex functions in multiple objective optimization and called such functions convexlike if s is a convex set and if f is a convex function then clearly f is convexlike elster and nehse obtained a saddlepoint optimality condition for convexlike mathematical programs hayashi and komiya 9 also considered convexlike functions and developed a gordan type theorem of the alternative involving convexlike functions and in addition considered lagrangian duality for convexlike programs hanson 7 considered differentiable functions f s r for which there exists an n dimensional vector function q q u such that for all x u e s

on a conjecture of ash abstract in this paper we prove a particular case of a conjecture of ash that states the existence of galois representations associated to hecke eigenclasses in cohomology

doubly cuspidal cohomology for principal congruence subgroups of gl 3 z the cohomology of arithmetic groups is made up of two pieces the cuspidal and noncuspidal parts within the cuspidal cohomology is a subspacethe f cuspidal cohomology spanned by the classes that generate representations of the associated finite lie group which are cuspidal in the sense of finite lie group theory few concrete examples of f cuspidal cohomology have been computed geometrically outside the cases of rational rank 1 or where the symmetric space has a hermitian structure this paper presents new computations of the f cuspidal cohomology of principal congruence subgroups f p of gl 3 2 of prime level p we show that the f cuspidal cohomology of f p vanishes for all p 19 with p 11 but that it is nonzero for p 11 we give a precise description of the f cuspidal cohomology for jt 11 in terms of the f cuspidal representations of the finite lie group gl 3 e 11 we obtained the result ultimately by proving that a certain large complex matrix m is rank deficient computation with the svd algorithm gave strong evidence that m was rank deficient but to prove it we mixed ideas from numerical analysis with exact computation in algebraic number fields and finite fields

rigidity of p adic cohomology classes of congruence subgroups of gl n ℤ the main result of ash stevens 97 describes a framework for constructing padic analytic families of p ordinary arithmetic hecke eigenclasses in the cohomology of congruence subgroups of gl n q where the hecke eigenvalues vary p adic analytically as functions of the weight unanswered in that paper was the question of whether or not every p ordinary arithmetic eigenclass can be deformed in a positive dimensional family of arithmetic eigenclasses in this paper we make precise the notions of deformation and rigidity and investigate their properties rigidity corresponds to the non existence of positive dimensional deformations other than those coming from twisting by the powers of the determinant formal definitions are given in 3 when n 3 we will give a necessary and sufficient condition for a given p ordinary arithmetic eigenclass to be p adically rigid and we will use this criterion to give examples of p adically rigid eigenclasses on gl 3 more precisely we define a hecke eigenpacket to be a map from the hecke algebra to a coefficient ring which gives the hecke eigenvalues attached to a hecke eigenclass in the cohomology we then consider p adic analytic deformations of the hecke eigenpacket there are several advantages to this point of view as opposed to deforming the cohomology classes themselves for one thing a surjective map of coefficient modules need not induce a surjective map on the cohomology standard tools of algebra and geometry can be used to study the hecke algebra and the deformations of the eigenpackets moreover it is the hecke eigenvalues that carry the interesting arithmetic information attached to a cohomological eigenclass our main theoretical tool is the cohomology of congruence subgroups gl n z with coefficients in the space of bounded measures on the big cell of gl n see 2 in 4 we define the ring rord as a quotient of the hecke algebra acting on the p ordinary part of this cohomology in theorem 4 2 we prove that rord is universal in the sense that every p ordinary arithmetic hecke eigenpacket factors through rord we show that rord is a complete semilocal noetherian ring and study the geometry of its associated p adic rigid analytic space xord this leads