interior regularity estimates for elliptic systems of difference equations we prove interior regularity estimates for a large class of difference approximations for elliptic systems of partial differential equations these estimates are analogous to those for the systems of differential equations from these regularity estimates we obtain estimates on the convergence of the solutions of the difference equations a theory of pseudo difference operators with order is used to prove the regularity results we also comment on factors to be considered in choosing a difference scheme for computations

constructing semisimple p adic galois representations with prescribed properties consider a characteristic p representation of the absolute galois group of the rational numbers in this paper we show how to deform this representation to the p adics while guaranteeing that the characteristic polynomials of frobenius at a density one set of primes are algebraic and pure of specified weight the resulting representation is ramified at an infinite density zero set of primes as a consequence of the technique of proof we show that one can compatibly lift a mod pq representation again the resulting representation is ramified at infinitely many primes

on the modularity of elliptic curves over q wild 3 adic exercises in this paper building on work of wiles wi and of wiles and one of us r t tw we will prove the following two theorems see 2 2 theorem a if e q is an elliptic curve then e is modular theorem b if gal q q gl2 f5 is an irreducible continuous representation with cyclotomic determinant then is modular we will first remind the reader of the content of these results and then briefly outline the method of proof if n is a positive integer then we let 1 n denote the subgroup of sl2 z consisting of matrices that modulo n are of the form 1 0 1

symmetrizable finite difference operators we introduce the notion of a symmetrizable finite difference oper ator and prove that such operators are stable we then present some sufficient conditions for symmetrizability one of these extends h o kreiss theorem on dissipative difference schemes for hyperbolic equations to a more general case with full jc invariable coefficients

on mixed finite element methods for first order elliptic systems a physically based duality theory for first order elliptic systems is shown to be of central importance in connection with the galerkin finite element solution of these systems using this theory in conjunction with a certain hypothesis concerning approximation spaces optimal error estimates for galerkin type approximations are demonstrated an example of a grid which satisfies the hypothesis is given and numerical examples which illustrate the theory are provided

variation of iwasawa invariants in hida families let gq gl2 k be an absolutely irreducible modular galois representation over a finite field k of characteristic p assume further that is p ordinary and p distinguished in the sense that the restriction of to a decomposition group at p is reducible and non scalar the hida family h of is the set of all p ordinary p stabilized newforms f with mod p galois representation isomorphic to if is unramified at p then one must also fix an unramified line in and require that the ordinary line of f reduces to this fixed line these newforms are a dense set of points in a certain p adic analytic space of overconvergent eigenforms consisting of an intersecting system of branches i e irreducible components t a indexed by the minimal primes a of a certain hecke algebra to each modular form f h one may associate the iwasawa invariants an f an f alg f and alg f the analytic resp algebraic invariants are the number of zeroes of the p adic l function resp of the characteristic power series of the dual of the selmer group of f while the invariants are the exponents of the powers of p dividing the same objects in this paper we prove the following results on the behavior of these iwasawa invariants as f varies over h

stability instability and error of the force based quasicontinuum approximation due to their algorithmic simplicity and high accuracy force based model coupling techniques are popular tools in computational physics for example the force based quasicontinuum qcf approximation is the only known pointwise consistent quasicontinuum approximation for coupling a general atomistic model with a finite element continuum model in this paper we present a detailed stability and error analysis of this method our optimal order error estimates provide a theoretical justification for the high accuracy of the qcf approximation they clearly demonstrate that the computational efficiency of continuum modeling can be utilized without a significant loss of accuracy if defects are captured in the atomistic region the main challenge we need to overcome is the fact that the linearized qcf operator is typically not positive definite moreover we prove that no uniform inf sup stability condition holds for discrete versions of the w1 p w1 q duality pairing with 1 p 1 q 1 if 1 p however we were able to establish an inf sup stability condition for a discrete version of the w1 w1 1 duality pairing which leads to optimal order error estimates in a discrete w1 norm

stability of a force based hybrid method with planar sharp interface we study a force based hybrid method that couples an atomistic model with a cauchy born elasticity model with sharp transition interface we identify stability conditions that guarantee the convergence of the hybrid scheme to the solution of the atomistic model with second order accuracy as the ratio between lattice parameter and the characteristic length scale of the deformation tends to zero convergence is established for hybrid schemes with planar sharp interface for systems without defects with general finite range atomistic potential and simple lattice structure the key ingredients of the proof are regularity and stability analysis of elliptic systems of difference equations we apply the results to atomistic to continuum scheme for a two dimensional triangular lattice with planar interface

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

finite size corrections to the free energy of coulomb systems with a periodic boundary condition classical coulomb systems ind dimensions d 2 with a periodic boundary condition periodw in the directionx d are considered with the other directions of the confining volume of lengthl it is shown that if the system is in a conducting phase then the strip free energyktf w f w liml l d 1 log z has the large w expansion f w sim wf infty frac d 2 1 gamma d 2 1 pi d 2 w d 1 zeta d o left frac 1 w d 1 right wherektf is the bulk free energy per unit volume x denotes the riemann zeta function and x denotes the gamma function with 1 w identified askt this result is precisely the low temperature behavior of the free energy of a d 1 dimensional debye solid this fact is explained in terms of an equivalence between the coulomb gas and quantum fields also the expansion is verified for some exactly solved models of coulomb systems in two dimensions

regularity estimates up to the boundary for elliptic systems of difference equations this paper proves regularity estimates up to the boundary for solutions of elliptic systems of finite difference equations the regularity estimates obtained for boundary fitted coordinate systems on domains with smooth boundary involve discrete sobolev norms and are proved using pseudodifference operators to treat systems with variable coefficients the elliptic systems of difference equations and the boundary conditions that are considered are very general in form it is proved that regularity of a regular elliptic system of difference equations is equivalent to the nonexistence of eigensolutions the regularity estimates obtained are analogous to those in the theory of elliptic systems of partial differential equations and to the results of gustafsson kreiss and sundstrom math comp 26 1972 pp 649 686 and others for hyperbolic difference equations

an extension of the kreiss matrix theorem a new condition is shown to be equivalent to the other conditions of the kreiss matrix theorem for power bounded families of matrices this new condition is important for the application of the theory of pseudodifference operators to stability estimates for variable coefficient finite difference equations as an example of the usefulness of this new condition it is used to prove stability of the leapfrog scheme for hyperbolic equations with variable coefficients

finite size corrections for coulomb systems in the debye huckel regime it has been argued that for a finite two dimensional classical coulomb system of characteristic size r in its conducting phase as r the total free energy times the inverse temperature admits an expansion of the form where is the euler characteristic of the manifold where the system exists the first two terms represent the bulk free energy and the surface free energy respectively the coefficients a and b are non universal but the coefficient of ln r is universal it does not depend on the detail of the microscopic constitution of the system particle densities temperature etc by doing the usual legendre transform this universal finite size correction is also present in the grand potential the explicit form of the expansion has been checked for some exactly solvable models for a special value of the coulombic coupling in this paper we present a method for obtaining these finite size corrections in the debye huckel regime it is based on the sine gordon field theory to find an expression for the grand canonical partition function in terms of the spectrum of the laplace operator as an example we find explicitly the grand potential expansion for a coulomb system confined in a disc and in an annulus with ideal conductor walls

analysis of a multigrid strokes solver model problem analysis or local mode analysis is used to investigate the two level convergence rates for multigrid strokes solvers the solvers are based on the discretization on the staggered grid and the distributive relaxation of the continuity equation the influence of the grid geometry is analyzed by also considering the poisson equation on the semistaggered grid various grid transfer operators are compared for an isotropic h x h y discretization simple tranfer operators are sufficient to obtain optimal rates but not in the anisotropic case the theoretical results are compared with observed multigrid convergence rates

a trickiness of the high temperature limit for number density correlation functions in classical coulomb fluids the debye huckel theory describes rigorously the thermal equilibrium of classical coulomb fluids in the high temperature 0 regime denotes the inverse temperature it is generally believed that the debye huckel theory and the systematic high temperature expansion provide an adequate description also in the region of small strictly positive values of 0 this hypothesis is tested in the present paper on a two dimensional coulomb gas of pointlike unit charges interacting via a logarithmic potential which is equivalent to an integrable sine gordon field model in particular we apply a form factor method to obtain the exact asymptotic large distance behavior of particle correlation functions considered in the charge and number density combinations we first determine the general forms of the leading and subleading asymptotic terms at strictly positive 0 and then evaluate their high temperature 0 forms in the case of the charge correlation function the leading asymptotic term at a strictly positive 0 is also the leading one in the high temperature 0 regime on the contrary the 0 behavior of the number density correlation function is accompanied by an interference between the first two asymptotic terms consequently the large distance behavior of this function exhibits a discontinuity when going from strictly positive values of 0 to the debye huckel limit 0 this is the crucial conclusion of the paper the large distance asymptotics and the high temperature limit do not commute for the density correlation function of the two dimensional coulomb gas

modularity of certain potentially barsotti tate galois representations where is the reduction of p these results were subject to hypotheses on the local behavior of p at x i e the restriction of p to a decomposition group at x and to irreducibility hypotheses on 1p in this paper we build on the methods of 461 451 and 121 and relax the hypotheses on local behavior in particular we treat certain adic representations which are not semistable at x but potentially semistable we do this using results of 61 generalizing a theorem of ramakrishna 321 see fontaine mazur 22 131 for a slightly different point of view the results in 61 show that certain potentially barsotti tate deformation problems are smooth allowing us to define certain universal deformations for with the necessary galoistheoretic properties to apply wiles method to carry out the proof that these deformations are indeed realized in the cohomology of modular curves i e that the universal deformation rings are hecke algebras we need to identify the corresponding cohomology groups and prove they have the modular theoretic properties needed to apply wiles method as in 151 and 121 the identification is made by matching local behavior of automorphic representations and galois representations via the local langlands correspondence together with fontaine s theory at the prime we work directly with cohomology of modular curves instead of

general considerations on the finite size corrections for coulomb systems in the debye huckel regime we study the statistical mechanics of classical coulomb systems in a low coupling regime debye huckel regime in a confined geometry with dirichlet boundary conditions for the electric potential we use a method recently developed by the authors which relates the grand partition function of a coulomb system in a confined geometry with a certain regularization of the determinant of the laplacian on that geometry with dirichlet boundary conditions we study several examples of fully confining geometry in two and three dimensions and semi confined geometries where the system is confined only in one or two directions of the space we also generalize the method to study systems confined in arbitrary geometries with smooth boundary we find a relation between the expansion for small argument of the heat kernel of the laplacian and the large size expansion of the grand potential of the coulomb system this allow us to find the finite size expansion of the grand potential of the system in general we recover known results for the bulk grand potential in two and three dimensions and the surface tension for two dimensional systems we find the surface tension for three dimensional systems for two dimensional systems our general calculation of the finite size expansion gives a proof of the existence a universal logarithmic finite size correction predicted some time ago at least in the low coupling regime for three dimensional systems we obtain a prediction for the curvature correction to the grand potential of a confined system

convergence of a force based hybrid method in three dimensions we study a force based hybrid method that couples an atomistic model with the cauchy born elasticity model we show that the proposed scheme converges to the solution of the atomistic model with second order accuracy since the ratio between lattice parameter and the characteristic length scale of the deformation tends to 0 convergence is established for the three dimensional system without defects with general finite range atomistic potential and simple lattice structure the proof is based on consistency and stability analysis general tools for stability analysis are developed in the framework opseudodifference operators in arbitrary dimensions 2012 wiley periodicals inc

cauchy born rule and the stability of crystalline solids static problems we study the connection between atomistic and continuum models for the elastic deformation of crystalline solids at zero temperature we prove under certain sharp stability conditions that the correct nonlinear elasticity model is given by the classical cauchy born rule in the sense that elastically deformed states of the atomistic model are closely approximated by solutions of the continuum model with stored energy functionals obtained from the cauchy born rule the analysis is carried out for both simple and complex lattices and for this purpose we develop the necessary tools for performing asymptotic analysis on such lattices our results are sharp and they also suggest criteria for the onset of instabilities of crystalline solids

on modular representations of bar q q arising from modular forms where g is the galois group gai i q and f is a finite field of characteristic i 3 suppose that p is modular of level n i e that it arises from a weight 2 newform of level dividing n and trivial nebentypus character then p is an odd representation the matrix p c where c is a complex conjugation in g has eigenvalues 1 1 since 1 and 1 are distinct in f p is absolutely irreducible and has a model over every subfield of f containing the set trace p we assume that f has been chosen so that it is generated by this set assume that p is a prime which exactly divides n we write p i n and restrict p to a decomposition group of g for the prime p view the restriction as a representation p p of gal t p qp we say that p is finite at p 34 or 35 w if there is a finite flat f vector space scheme h over zp for which the action of gal t p qp on the f vector space h 1 p gives pp i f p 9 is finite at p if and only if pp is unramified it is clear that p is finite at p whenever p is already modular of level n p in w of 34 cf 35 serre conjectured the converse i e that if p is finite then p is modular of level n p soon thereafter mazur 1 20 proved this conjecture in the case

ring theoretic properties of certain hecke algebras the purpose of this article is to provide a key ingredient of w2 by establishing that certain minimal hecke algebras considered there are complete intersections as is recorded in w2 a method going back to mazur m allows one to show that these algebras are gorenstein but for the complete intersection property a new approach is required the methods of this paper are related to those of chapter 3 of w2 the methods of section 3 of this paper are based on a previous approach of one of us a w we would like to thank henri darmon fred diamond and gerd faltings for carefully reading the first version of this article gerd faltings has also suggested a simplification of our argument as well as of the argument of chapter 3 of w2 and we would like to thank him for allowing us to reproduce these in the appendix to this paper r t would like to thank a w for his invitation to collaborate and for sharing his many insights into the questions considered r t would also like to thank princeton university universite de paris 7 and harvard university for their hospitality during this collaboration a w was supported by an nsf grant

another derivation of a sum rule for the two dimensional two component plasma abstract in a two dimensional two component plasma the second moment of the number density correlation function has the simple value 12 1 4 2 1 where is the dimensionless coupling constant this result is derived directly by using diagrammatic methods

force based atomistic continuum blending for multilattices we formulate the blended force based quasicontinuum method for multilattices and develop rigorous error estimates in terms of the approximation parameters choice of atomistic region blending region and continuum finite element mesh balancing the approximation parameters yields a convergent atomistic continuum multiscale method for multilattices with point defects including a rigorous convergence rate in terms of the computational cost the analysis is illustrated with numerical results for a stone wales defect in graphene