construction of high order runge kutta methods which preserve delay dependent stability of ddes this paper is concerned with the construction of some high order runge kutta methods which preserve delay dependent stability of delay differential equations the methods of the first kind are developed by extending the ideas of brugnano et al while the methods of the second kind are developed according to the structure of the stability matrix we show that the derived methods are 0 stable for delay differential equations meanwhile the runge kutta methods can own the same order of the accuracy as the radau methods or gauss methods if the parameters are adequately defined these results not only improve the order of accuracy of the methods investigated by huang but also open an interesting route of finding new 0 stable runge kutta methods finally numerical experiments are proposed to illustrate the theoretical results

an algebraic approach to invariant preserving integators the case of quadratic and hamiltonian invariants in this article conditions for the preservation of quadratic and hamiltonian invariants by numerical methods which can be written as b series are derived in a purely algebraical way the existence of a modified invariant is also investigated and turns out to be equivalent up to a conjugation to the preservation of the exact invariant a striking corollary is that a symplectic method is formally conjugate to a method that preserves the hamitonian exactly another surprising consequence is that the underlying one step method of a symmetric multistep scheme is formally conjugate to a symplectic p series when applied to newton s equations of motion

base change and a problem of serre we establish a version of level lowering for mod p galois representations arising from the reductions of representations associated to hilbert modular forms in particular we show that level lowering can be easily achieved if one replaces the base field with a suitable solvable extension this is often enough for applications to proving the modularity of p adic representations

modular elliptic curves and fermat s last theorem when andrew john wiles was 10 years old he read eric temple bell s the last problem and was so impressed by it that he decided that he would be the first person to prove fermat s last theorem this theorem states that there are no nonzero integers a b c n with n 2 such that an bn cn the object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular fermat s last theorem follows as a corollary by virtue of previous work by frey serre and ribet

symplectic energy momentum preserving variational integrators the purpose of this paper is to develop variational integrators for conservative mechanical systems that are symplectic and energy and momentum conserving to do this a space time view of variational integrators is employed and time step adaptation is used to impose the constraint of conservation of energy criteria for the solvability of the time steps and some numerical examples are given

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

a note on the continuous stage runge kutta nystrom formulation of hamiltonian boundary value methods hbvms abstract in recent years the class of energy conserving methods named hamiltonian boundary value methods hbvms has been devised for numerically solving hamiltonian problems in this short note we study their natural formulation as continuous stage runge kutta nystrom methods which allows a deeper insight in the methods

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

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

analysis of energy and quadratic invariant preserving equip methods abstract in this paper we are concerned with the analysis of a class of geometric integrators at first devised in brugnano et al 2010 and brugnano et al 2012 which can be regarded as an energy conserving variant of gauss collocation methods with these latter they share the property of conserving quadratic first integrals but in addition they also conserve the hamiltonian function itself we here reformulate the methods in a more convenient way and propose a more refined analysis than that given in brugnano et al 2012 also providing as a by product a practical procedure for their implementation a thorough comparison with the original gauss methods is carried out by means of a few numerical tests solving hamiltonian and poisson problems

new energy preserving algorithms for nonlinear hamiltonian wave equation equipped with neumann boundary conditions abstract in this paper using the blend of spatial discretization by second order or fourth order finite difference methods fdm and time integration by the generalized average vector field gavf method or the generalized adapted average vector field gaavf method we propose and analyze novel energy preserving algorithms for solving the nonlinear hamiltonian wave equation equipped with homogeneous neumann boundary conditions firstly two kinds of finite difference methods are considered to discretize the spatial derivative which can be of order two and order four respectively in all the spatial grid points the conservation laws of the discrete energy are established after the semi discretization a hamiltonian system of odes is derived whose hamiltonian can be regarded as the approximate energy of the original continuous system then the gavf formula and the gaavf formula are developed and applied to the derived hamiltonian odes to yield some novel and efficient algorithms which can exactly preserve the discrete energy the numerical simulation is implemented and the numerical results demonstrate the spatial and temporal accuracy and the remarkable energy preserving property of the new algorithms presented in this paper

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

structure preserving algorithms for the two dimensional sine gordon equation with neumann boundary conditions abstract this paper presents two kinds of strategies to construct structure preserving algorithms with homogeneous neumann boundary conditions for the sine gordon equation while most existing structure preserving algorithms are only valid for zero or periodic boundary conditions the first strategy is based on the conventional second order central difference quotient but with a cell centered grid while the other is established on the regular grid but incorporated with summation by parts sbp operators both the methodologies can provide conservative semi discretizations with different forms of hamiltonian structures and the discrete energy however utilizing the existing sbp formulas schemes obtained by the second strategy can directly achieve higher order accuracy while it is not obvious for schemes based on the cell centered grid to make accuracy improved easily further combining the implicit midpoint method and the scalar auxiliary variable sav approach we construct symplectic integrators and linearly implicit energy preserving schemes for the two dimensional sine gordon equation respectively extensive numerical experiments demonstrate their effectiveness with the homogeneous neumann boundary conditions

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

energy and quadratic invariants preserving equip multi symplectic methods for hamiltonian wave equations abstract it is well known that a numerical method which is at the same time geometric structure preserving and physical property preserving cannot exist in general for hamiltonian partial differential equations motivated by equip methods proposed in brugnano et al 2012 13 in this paper we present a novel class of parametric multi symplectic runge kutta methods called equip multi symplectic methods for hamiltonian wave equations which can also conserve energy simultaneously in a weaker sense with a suitable parameter the existence of such a parameter which enforces the energy preserving property is proved under certain assumptions on the fixed step sizes and the fixed initial condition we compare the proposed method with the classical multi symplectic runge kutta method in numerical experiments which shows the remarkable energy preserving property of the proposed method and illustrates the validity of theoretical results these theoretical and numerical results show that equip methods can be well adapted to handle hamiltonian partial differential equations

a new energy and momentum conserving algorithm for the non linear dynamics of shells a numerical time integration scheme for the dynamics of non linear elastic shells is presented that simultaneously and independent of the time step size inherits exactly the conservation laws of total linear total angular momentum as well as total energy the proposed technique generalizes to non linear shells recent work of the authors on non linear elastodynamics and is ideally suited for long term large scale simulations the algorithm is second order accurate and can be immediately extended with no modification to a fourth order accurate scheme the property of exact energy conservation induces a strong notion of non linear numerical stability which manifests itself in actual simulations

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

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

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

on the first exterior p harmonic steklov eigenvalue abstract in this short paper we study the sobolev function property of the rayleigh s quotient q inf u e 1 p u u p u q u p as a function of q 1 p for p p n 1 n p when p 1 n as well as the asymptotic behavior of positive solutions with minimal energy of the following problem p u 0 in u subject to u p 2 u u q 2 u on u to the first p harmonic steklov eigenpair on an exterior region u r n when n 3

on icosahedral artin representations ii we prove that some new infinite families of odd two dimensional icosahedral representations of the absolute galois group of inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i are modular and hence satsify the artin conjecture we also give an account of work of ramakrishna on lifting mod l galois representations to characteristic zero

langlands base change for gl 2 let f be a totally real galois number eld we prove the existence of base change relative to the extension f q for every holomorphic newform of weight at least 2 and odd level under simple local assumptions on the eld f

line integral methods which preserve all invariants of conservative problems recently the class of hamiltonian boundary value methods hbvms has been introduced with the aim of preserving the energy associated with polynomial hamiltonian systems and more in general with all suitably regular hamiltonian systems however many interesting problems admit other invariants besides the hamiltonian function it would be therefore useful to have methods able to preserve any number of independent invariants this goal is achieved by generalizing the line integral approach which hbvms rely on thus obtaining a number of generalizations which we collectively name line integral methods in fact it turns out that this approach is quite general so that it can be applied to any numerical method whose discrete solution can be suitably associated with a polynomial such as a collocation method as well as to any conservative problem in particular a completely conservative variant of both hbvms and gauss collocation methods is presented numerical experiments confirm the effectiveness of the proposed methods

on serre s conjecture for 2 dimensional mod p representations of gal q q we prove the existence in many cases of minimally ramified p adic lifts of 2 dimensional continuous odd absolutely irreducible mod p representations of the absolute galois group of q it is predicted by serre s conjecture that such representations arise from newforms of optimal level and weight

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

some bounds for ramification of p n torsion semi stable representations let p be an odd prime k a finite extension of q p g k gal kbar k its absolute galois group and e e k q p its absolute ramification index suppose that t is a p n torsion representation of g k that is isomorphic to a quotient of g k stable z p lattices in a semi stable representation with hodge tate weights in 0 r we prove that there exists a constant mu depending only on n e and r such that the upper numbering ramification group g k mu acts on t trivially

energy conservation issues in the numerical solution of the semilinear wave equation in this paper we discuss energy conservation issues related to the numerical solution of the semilinear wave equation as is well known this problem can be cast as a hamiltonian system that may be autonomous or not depending on the prescribed boundary conditions we relate the conservation properties of the original problem to those of its semi discrete version obtained by the method of lines subsequently we show that the very same properties can be transferred to the solutions of the fully discretized problem obtained by using energy conserving methods in the hbvms hamiltonian boundary value methods class similar arguments hold true for different types of hamiltonian partial differential equations e g the nonlinear schrodinger equation

exponential collocation methods for conservative or dissipative systems abstract in this paper we propose and analyse a novel class of exponential collocation methods for solving conservative or dissipative systems based on exponential integrators and collocation methods it is shown that these novel methods can be of arbitrarily high order and exactly or approximately preserve first integrals or lyapunov functions we also consider order estimates of the new methods furthermore we explore and discuss the application of our methods in important stiff gradient systems and it turns out that our methods are unconditionally energy diminishing and strongly damped even for very stiff gradient systems practical examples of the new methods are derived and the efficiency and superiority are confirmed and demonstrated by three numerical experiments including a nonlinear schrodinger equation as a byproduct of this paper arbitrary order trigonometric rkn collocation methods are also presented and analysed for second order highly oscillatory general systems the paper is accompanied by numerical results that demonstrate the great potential of this work

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

energy and quadratic invariants preserving integrators based upon gauss collocation formulae we introduce a new family of symplectic integrators for canonical hamiltonian systems each method in the family depends on a real parameter alpha when alpha 0 we obtain the classical gauss collocation formula of order 2s where s denotes the number of the internal stages for any given non null alpha the corresponding method remains symplectic and has order 2s 2 hence it may be interpreted as an o h 2s 2 symplectic perturbation of the gauss method under suitable assumptions we show that the parameter alpha may be properly tuned at each step of the integration procedure so as to guarantee energy conservation in the numerical solution as well as to maintain the original order 2s as the generating gauss formula

the lack of continuity and the role of infinite and infinitesimal in numerical methods for odes the case of symplecticity when numerically integrating canonical hamiltonian systems the long term conservation of some of its invariants for example the hamiltonian function itself assumes a central role the classical approach to this problem has led to the definition of symplectic methods among which we mention gauss legendre collocation formulae indeed in the continuous setting energy conservation is derived from symplecticity via an infinite number of infinitesimal contact transformations however this infinite process cannot be directly transferred to the discrete setting by following a different approach in this paper we describe a sequence of methods sharing the same essential spectrum and then the same essential properties which are energy preserving starting from a certain element of the sequence on i e after a finite number of steps

overconvergent modular forms and the fontaine mazur conjecture we prove a conjecture of fontaine and mazur on modularity of representations of g which are potentially semi stable at p for representations coming from finite slope overconvergent eigenforms we also give an application of our technique to a question of gouvea on overconvergence of certain modular forms if is a suitable representation of g s on a two dimensional vector space over a finite field product of m and the s a finite set of primes we construct a rigid analytic subspace of the universal deformation space of which is defined by a purely representation theoretic condition and contains the eigencurve of coleman mazur conjecturally it is equal to this curve

deformations and rigidity of l adic sheaves abstract let x be a smooth connected projective curve over an algebraically closed field and let s be a finite nonempty closed subset in x we study deformations of lisse f l sheaves on x s the universal deformation space is a formal scheme its generic fiber has a rigid analytic space structure by a dimension counting of this rigid analytic space we prove a conjecture of katz which says that a lisse irreducible q l sheaf f on x s is rigid if and only if dim h 1 x j e n d f 2 g where j x s x is the open immersion and g is the genus of x

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

singularities of ordinary deformation rings let r mathrm univ be the universal deformation ring of a residual representation of a local galois group kisin showed that many loci in mathrm maxspec r mathrm univ 1 p of interest are zariski closed and gave a way to study the generic fiber of the corresponding quotient of r mathrm univ however his method gives little information about the quotient ring before inverting p we give a method for studying this quotient in certain cases and carry it out in the simplest non trivial case precisely suppose that v 0 is the trivial two dimensional representation and let r be the unique mathbf z p flat and reduced quotient of r mathrm univ such that mathrm maxspec r 1 p consists of ordinary representations with hodge tate weights 0 and 1 we describe the functor of points of a slightly modified version of r and show that the irreducible components of mathrm spec r are normal and cohen macaulay but not gorenstein as a consequence we find that certain global deformation rings are torsion free and cohen macaulay but not gorenstein

modularity lifting beyond the taylor wiles method we prove new modularity lifting theorems for p adic galois representations in situations where the methods of wiles and taylor wiles do not apply previous generalizations of these methods have been restricted to situations where the automorphic forms in question contribute to a single degree of cohomology in practice this imposes several restrictions one must be in a shimura variety setting and the automorphic forms must be of regular weight at infinity in this paper we essentially show how to remove these restrictions our most general result is a modularity lifting theorem which on the automorphic side applies to automorphic forms on the group mathrm gl n over a general number field it is contingent on a conjecture which in particular predicts the existence of galois representations associated to torsion classes in the cohomology of the associated locally symmetric space we show that if this conjecture holds then our main theorem implies the following if e is an elliptic curve over an arbitrary number field then e is potentially automorphic and satisfies the sato tate conjecture in addition we also prove some unconditional results for example in the setting of mathrm gl 2 over mathbf q we identify certain minimal global deformation rings with the hecke algebras acting on spaces of p adic katz modular forms of weight 1 such algebras may well contain p torsion moreover we also completely solve the problem for p odd of determining the multiplicity of an irreducible modular representation overline rho in the jacobian j 1 n where n is the minimal level such that overline rho arises in weight two

energy conserving methods for the nonlinear schrodinger equation in this paper we further develop recent results in the numerical solution of hamiltonian partial differential equations pdes brugnano et al 2015 by means of energy conserving methods in the class of line integral methods in particular the runge kutta methods named hamiltonian boundary value methods hbvms we shall use hbvms for solving the nonlinear schrodinger equation nlse of interest in many applications we show that the use of energy conserving methods able to conserve a discrete counterpart of the hamiltonian functional confers more robustness on the numerical solution of such a problem

a new class of energy preserving numerical integration methods the first ever energy preserving b series numerical integration method for ordinary differential equations is presented and applied to several hamiltonian systems related novel lie algebraic results are also discussed

a simple framework for the derivation and analysis of effective one step methods for odes in this paper we provide a simple framework to derive and analyse a class of one step methods that may be conceived as a generalization of the class of gauss methods the framework consists in coupling two simple tools firstly a local fourier expansion of the continuous problem is truncated after a finite number of terms and secondly the coefficients of the expansion are computed by a suitable quadrature formula different choices of the basis lead to different classes of methods even though we shall here consider only the case of an orthonormal polynomial basis from which a large subclass of runge kutta methods can be derived the obtained results are then applied to prove in a simplified way the order and stability properties of hamiltonian bvms hbvms a recently introduced class of energy preserving methods for canonical hamiltonian systems see 2 and references therein a few numerical tests are also included in order to confirm the effectiveness of the methods resulting from our analysis

integral models of hilbert modular varieties in the ramified case deformations of modular galois representations and weight one forms we prove the strong artin conjecture for continuous totally odd two dimensional representations of the absolute galois group of a totally real field f

construction of runge kutta type methods for solving ordinary differential equations in this paper we study runge kutta methods with continuous stage introduced by butcher in 1987 by setting the coefficients of this family of methods and choosing the appropriate numerical integration we derive new classes of runge kutta methods furthermore we extend the w transformation technique by permitting w to be a non square matrix this allows us to construct more high order implicit runge kutta methods with some geometric properties specially we provide runge kutta methods with continuous stage which are conjugate symplectic symmetric or energy preserving for solving hamiltonian systems we also present some numerical experiments to verify our results

p laplacian boundary value problems on exterior regions abstract this paper treats some variational principles for solutions of inhomogeneous p laplacian boundary value problems on exterior regions u r n with dimension n 3 existence uniqueness results when p 1 n are provided in a space e 1 p u of functions that contains w 1 p u functions in e 1 p u are required to decay at infinity in a measure theoretic sense various properties of this space are derived including results about equivalent norms traces and an l p imbedding theorem also an existence result for a general variational problem of this type is obtained

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

a note on the efficient implementation of hamiltonian bvms we discuss the efficient implementation of hamiltonian bvms hbvms a recently introduced class of energy preserving methods for canonical hamiltonian systems see brugnano et al 8 and references therein also sketching their blended formulation we also discuss the case of separable problems for which the structure of the problem can be exploited to gain efficiency

on a conjecture of conrad diamond and taylor we prove a conjecture of conrad diamond and taylor on the size of certain deformation rings parametrizing potentially barsotti tate ga lois representations to achieve this we extend results of breuil and m ezard classifying galois lattices in semistable representations in terms of strongly divisible modules to the potentially crystalline case in hodge tate weights 0 1 we then use these strongly divisible modules to compute the desired deformation rings as a corollary we obtain new results on the modularity of potentially barsotti tate representations

the taylor wiles construction and multiplicity one wiles proof 17 of the modularity of semistable elliptic curves over q relies on a construction of taylor and wiles 16 showing that certain hecke algebras are complete intersections these hecke algebras are defined by considering the action of hecke operators on spaces of modular forms of minimal level or equivalently on homology groups or jacobians of modular curves taylor and wiles proceed roughly speaking by patching algebras arising from forms of different levels one of the deep results used in their construction was the fact that the homology of the modular curve becomes a free module of rank two over the hecke algebra upon localization at certain maximal ideals this result known as a multiplicity one result is a generalization of a theorem of mazur 11 its proof relied on the q expansion principle of deligne rapoport and katz and the comparison of mod betti and de rham cohomologies see sect 2 1 of 17 multiplicity one was thought to be a crucial ingredient of the taylor wiles construction as well as other parts of wiles proof the purpose of this paper is to explain how to alter the arguments of 16 and 17 so that multiplicity one results are a byproduct rather than an ingredient the key conceptual change underlying this improvement is the following rather than prove that after localization the hecke algebra can be identified with the universal deformation ring of a mod galois representation we prove that the homology of the modular curve is a free module over this deformation ring to carry this out we change the taylor wiles construction1 by 1 patching the modules as well as the algebras and 2 applying the auslander buchsbaum

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