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

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

a continuum model for nonlinear lattices under large deformations abstract a continuum model is developed for hexagonal lattices composed of a set of masses connected by linear axial and angular springs with nonlinearity arising solely from geometric effects for a set of lattice parameters these lattices exhibit complex deformation patterns under uniform loading conditions due to instabilities a continuum model accounting for these instabilities is developed from explicit expressions of the potential energy functional of a unit cell this functional is non convex it captures the bistable nature of the lattice and is used to derive its effective constitutive behavior finite element simulations of continuum medium illustrate the formation of microstructural patterns with discontinuous displacement gradients similar to the features observed in nonlinear elasticity and finite deformation plasticity a comparison of discrete lattice simulations and finite element analysis under general loading conditions illustrates that the continuum model captures the effective behavior due to instabilities within the lattice

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

finding meaning in error terms four decades ago mikio sato and john tate predicted the shape of probability distributions to which certain error terms in number theory conform their prediction known as the sato tate conjecture has been verified for an important class of cases thanks to the recent work of laurent clozel michael harris and richard taylor 3 and of michael harris nicholas shepherd barron and richard taylor 16 combined with richard taylor s most recent 50 which establishes this advance in our understanding

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 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

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

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

an f and m riesz theorem for a system of vector fields this work extends the classical f and m riesz theorem for holomorphic functions to the continuous solutions of real analytic involutive structures

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

a local to global principle for deformations of galois representations given an absolutely irreducible galois representation ge gln k e a number eld k a nite eld of characteristic l 2 and a nite set of places q of e containing all places above l and1 and all where ramies there have been dened many functors representing strict equivalence classes of deformations of such a representation e g by mazur or wiles in 15 or 26 with various conditions on the behaviour of the deformations at the places in q and with the condition that the deformations are unramied outside q those functors are known to be representable for as above our goal is to present a rather general class of global deformation functors that satisfy local deformation conditions and to investigate for those under what conditions the global deformation functor is determined by the local deformation functors we will give precise conditions under which the local functors for all places in q are sucient to describe the global functor rst in a coarse form then in a rened form using auxiliary primes as done by taylor and wiles in 24 this has several consequences the strongest is that one can derive ring theoretic results for the universal deformation space by mazur if one uses results of diamond and wiles c f 11 and 26 and if one has a good understanding of all local situations furthermore it is easier to understand what happens under increasing the ramication

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

hecke stability and weight 1 modular forms the galois representations associated to weight 1 eigenforms over bar mathbb f p are remarkable in that they are unramified at p but the effective computation of these modular forms presents challenges one complication in this setting is that a weight 1 cusp form over bar mathbb f p need not arise from reducing a weight 1 cusp form over bar mathbb q in this article we propose a unified hecke stability method for computing spaces of weight 1 modular forms of a given level in all characteristics simultaneously our main theorems outline conditions under which a finite dimensional hecke module of ratios of modular forms must consist of genuine modular forms we conclude with some applications of the hecke stability method that are motivated by the refined inverse galois problem

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