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

slopes of eigencurves over boundary disks let p be a prime number we study the slopes of u p eigenvalues on the subspace of modular forms that can be transferred to a definite quaternion algebra we give a sharp lower bound of the corresponding newton polygon the computation happens over a definite quaternion algebra by jacquet langlands correspondence it generalizes a prior work of jacobs slopes of compact hecke operators thesis university of london imperial college 2004 who treated the case of p 3 with a particular level in case when the modular forms have a finite character of conductor highly divisible by p we improve the lower bound to show that the slopes of u p eigenvalues grow roughly like arithmetic progressions as the weight k increases this is the first very positive evidence for buzzard kilford s conjecture on the behavior of the eigencurve near the boundary of the weight space that is proved for arbitrary p and general level we give the exact formula of a fraction of the slope sequence

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

on the density of modular points in universal deformation spaces based on comparison theorems for hecke algebras and universal deformation rings with strong restrictions at the critical prime l as provided by the results of wiles taylor diamond et al we prove under rather general conditions that the corresponding universal deformation spaces with no restrictions at l can be identified with certain hecke algebras of l adic modular forms as conjectured by gouvea thus generalizing previous work of gouvea and mazur along the way we show that the universal deformation spaces we consider are complete intersections flat over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i l of relative dimension three in which the modular points form a zariski dense subset furthermore the fibers above inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i l of these spaces are generically smooth

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

uniqueness and existence for anisotropic degenerate parabolic equations with boundary conditions on a bounded rectangle we study the comparison principle for anisotropic degenerate parabolic hyperbolic equations with initial and nonhomogeneous boundary conditions we prove a comparison theorem for any entropy sub and super solution which immediately deduces the l1 contractivity and therefore uniqueness of entropy solutions the method used here is based upon the kinetic formulation and the kinetic techniques developed by lions perthame and tadmor by adapting and modifying those methods to the case of dirichlet boundary problems for degenerate parabolic equations we can establish a comparison property moreover in the quasi isotropic case the existence of entropy solutions is proved

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

a new version of the liu storey conjugate gradient method in this paper the global convergence of a new version of the liu storey conjugate gradient method is discussed this method combines the liu storey conjugate gradient formula and a new inexact line search we prove that the new method is globally convergent some preliminary numerical results show that the corresponding algorithm is efficient

modularity of 2 adic barsotti tate representations we prove a modularity lifting theorem for two dimensional 2 adic potentially barsotti tate representations this proves hypothesis h of khare wintenberger and completes the proof of serre s conjecture the main new ingredient is a classification of connected finite flat group schemes over rings of integers of finite extensions of 2

geometric deformations of modular galois representations let f be a newform on 1 n and v f the 2 dimensional p adic galois representation attached to f let s be a finite set of primes containing the primes divisors of np and denote by adv f the adjoint of v f under some mild conditions on f we show that h1 g g s adv f 0 using this result we show that the universal deformation space of the residual representation attached to f is smooth and 3 dimensional at the point corresponding to f when f has finite slope one can also use this result to give a deformation theoretic description of the eigencurve of coleman mazur at the point corresponding to f

potentially semi stable deformation rings let k qp be a finite extension and gk gal k k the galois group of an alge braic closure k let f be a finite field of characteristic p and vf a finite dimensional f vector space equipped with a continuous action of gkthe study of the defor mation theory of galois representations was initiated by mazur ma who showed that if vf has no non trivial endomorphisms then it admits a universal deformation ring ryw after the work of wiles wi and the conjectures of fontaine mazur fm it became clear that for arithmetic applications it was important to understand cer tain quotients of ry corresponding to deformations satisfying certain conditions for example wiles uses deformations which arise from finite flat group schemes and the corresponding quotient of ryw was constructed by ramakrishna ra suppose that l k is a finite extension and let a b be integers it seems to be a kind of folklore conjecture that there should be a quotient of ry whose points in finite extensions of w f l p correspond to deformations of vf which become semi stable over l and have hodge tate weights in the interval a b this is closely related to the final conjecture of fo 3 special cases of this when vf is 2 dimensional are conjectured in the papers of fontaine mazur fm p 191 breuil conrad diamond taylor bcdt conj 1 1 1 and breuil m zard bm 2 2 2 4 the purpose of this paper is to prove such a result

on the weights of mod p hilbert modular forms we prove many cases of a conjecture of buzzard diamond and jarvis on the possible weights of mod p hilbert modular forms by making use of modularity lifting theorems and computations in p adic hodge theory

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

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

construction of some families of 2 dimensional crystalline representations we construct explicitly some analytic families of etale modules which give rise to analytic families of 2 dimensional crystalline representations as an application of our constructions we verify some conjectures of breuil on the reduction modulo p of those representations and extend some results of deligne edixhoven fontaine and serre on the representations arising from modular forms

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

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

actions infinitesimales dans la correspondance de langlands locale p adique let v be a two dimensional absolutely irreducible overline mathbb qp representation of rm gal overline mathbb qp mathbb qp and let prod v be the rm gl 2 mathbb qp banach representation associated by colmez s p adic langlands correspondence we establish a link between the action of the lie algebra of rm gl 2 mathbb qp on the locally analytic vectors prod v rm an of prod v the connection nabla on the varphi gamma module associated to v and the sen polynomial of v this answers a question of harris concerning the infinitesimal character of prod v rm an using this result we give a new proof of a theorem of colmez stating that prod v has nonzero locally algebraic vectors if and only if v is potentially semi stable with distinct hodge tate weights

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