computing finite galois groups arising from automorphic forms abstract we study modulo p reduction of compatible systems of p adic representations of the absolute galois group of q arising from an algebraic automorphic representation in particular we prove that there is a field extension of q with the galois group g 2 p ramified at 5 and p only for a set of primes p of density one

mode coupling approach to the ideal glass transition of molecular liquids linear molecules the mode coupling theory mct for the ideal liquid glass transition which was worked out for simple liquids mainly by gotze sjogren and their co workers is extended to a molecular liquid of linear and rigid molecules by use of the projection formalism of zwanzig and mori an equation of motion is derived for the correlators s sub lm l sup prime m sup prime bold q t of the tensorial one particle density rho sub lm bold q t which contains the orientational degrees of freedom for l greater than 0 application of the mode coupling approximation to the memory kernel results into a closed set of equations for s sub lm l sup prime m sup prime bold q t which requires the static correlators s sub lm l sup prime m sup prime bold q as the only input quantities the corresponding mct equations for the nonergodicity parameters f sub l sup m q equivalent f sub lm lm q bold e sub 3 are solved for a system of dipolar hard spheres by restricting the values for l to 0 and 1 depending on the packing fraction cursive phi and on the temperature t three different phases exist a liquid phase where translational tdof s l 0 and orientational odof s l 1 degrees of freedom are ergodic a phase where the tdof are frozen into a nonergodic glassy state whereas the odof s remain ergodic and finally a glassy phase where both tdof s and odof s are nonergodic from the nonergodicity parameters f sub 0 sup 0 q and f sub 1 sup 1 q for q 0 we may conclude that the corresponding relaxation strength of the alpha peak of the compressibility can be much smaller than the corresponding strength of the dielectric function

numerical analysis of a finite element volume penalty method we present here some contributions to the numerical analysis of the penalty method in the finite element context we are especially interested in the ability provided by this approach to use cartesian non boundary fitted meshes to solve elliptic problems in complicated domain in the spirit of fictitious domains the initial problem is replaced by a penalized one posed over a simply shaped domain which covers the original one this method relies on two parameters namely h space discretization parameter and varepsilon penalty parameter we propose here a general strategy to estimate the error in both parameters and we present how it can be applied to various situations we pay special attention to a scalar version of the rigid motion constraint for fluid particle flows

unitary shimura correspondences for split real groups we find a relationship between certain complementary series representations for nonlinear coverings of split simple groups and spherical complementary series for different linear groups the main technique is barbasch s method of calculating some intertwining operators purely in terms of the weyl group

deformations of galois representations and exceptional monodromy for any simple algebraic group g of exceptional type we construct geometric ell adic galois representations with algebraic monodromy group equal to g in particular producing the first such examples in types mathrm f 4 and mathrm e 6 to do this we extend to general reductive groups ravi ramakrishna s techniques for lifting odd two dimensional galois representations to geometric ell adic representations

ornstein zernike equation and percus yevick theory for molecular crystals we derive the ornstein zernike equation for molecular crystals of axially symmetric particles and apply the percus yevick approximation the one particle orientational distribution function ensuremath rho 1 ensuremath omega has a nontrivial dependence on the orientation ensuremath omega in contrast to a liquid and is needed as an input despite some differences the ornstein zernike equation for molecular crystals has a similar structure as for liquids we solve both equations numerically for hard ellipsoids of revolution on a simple cubic lattice compared to molecular liquids the orientational correlators in direct and reciprocal space exhibit less structure however depending on the lengths a and b of the rotation axis and the perpendicular axes of the ellipsoids respectively different behavior is found for oblate and prolate ellipsoids with b ensuremath gtrsim 0 35 in units of the lattice constant damped oscillations in distinct directions of direct space occur for some of the orientational correlators they manifest themselves in some of the correlators in reciprocal space as a maximum at the brillouin zone edge accompanied by a maximum at the zone center for other correlators the oscillations indicate alternating orientational fluctuations while the maxima at the zone center originate from ferrorotational fluctuations for a ensuremath lesssim 2 5 and b ensuremath lesssim 0 35 the oscillations are weaker leading to no marked maxima at the brillouin zone edge for a ensuremath gtrsim 3 0 and b ensuremath lesssim 0 35 no oscillations occur any longer for many of the orientational correlators in reciprocal space an increase of a at fixed b or vice versa leads to a divergence at the zone center mathbf q mathbf 0 consistent with the formation of ferrorotational long range fluctuations and for some oblate and prolate systems with b ensuremath lesssim 1 0 a simultaneous tendency to divergence of few other correlators at the zone edge is observed comparison of the orientational correlators with those from monte carlo simulations shows satisfactory agreement from these simulations we also obtain a phase boundary in the a text ensuremath b plane for order disorder transitions

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

molecular mode coupling theory applied to a liquid of diatomic molecules we study the molecular mode coupling theory for a liquid of diatomic molecules the equations for the critical tensorial nonergodicity parameters f m ll q and the critical amplitudes of the beta relaxation h m ll q are solved up to a cutoff l co 2 without any further approximations here l m are indices of spherical harmonics contrary to previous studies where additional approximations were applied we find in agreement with simulations that all molecular degrees of freedom vitrify at a single temperature t c the theoretical results for the nonergodicity parameters and the critical amplitudes are compared with those from simulations the qualitative agreement is good for all molecular degrees of freedom to study the influence of the cutoff on the nonergodicity parameter we also calculate the nonergodicity parameters for an upper cutoff l co 4 in addition we also propose a method for the calculation of the critical nonergodicity parameter from the liquid side of transition

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

mod l representations of arithmetic fundamental groups i an analog of serre s conjecture for function fields there is a well known conjecture of serre that any continuous irreducible representation gq gl2 f that is odd arises from a newform here gq is the absolute galois group of q and f is an algebraic closure of the finite field f of elements we formulate such a conjecture for n dimensional mod representations of 1 x for any positive integer n and where x is a geometrically irreducible smooth curve over a finite field k of characteristic p p and we prove this conjecture in a large number of cases in fact a proof of all cases of the conjecture for 2 follows from a result announced conditionally by gaitsgory in g the methods are different

optimal isoparametric finite elements and error estimates for domains involving curved boundaries we describe a practical procedure for the triangulation of arbitrary n dimensional regular bounded domains by isoparametric simplicial elements preserving the optimal order of accuracy along similar lines we prove general error estimates for the finite element solution of second order elliptic problems various kinds of nonhomogeneous boundary conditions can be dealt with such as dirichlet neumann and robin boundary conditions

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

rational rigidity for f4 p abstract we prove the existence of certain rationally rigid triples in f 4 p for good primes p i e p 3 thereby showing that these groups occur as regular galois groups over q t and so also over q we show that these triples give rise to rigid triples in the algebraic group and prove that they generate an interesting subgroup in characteristic 0

finite element approximations of the lame system with penalized ideal contact boundary conditions we consider finite element approximations of the lame system of elasticity with ideal contact boundary conditions imposed with the penalty method for a polygonal or polyhedral boundary we prove convergence estimates in terms of both the penalty and discretization parameters in the case of a smooth curved boundary we show through a numerical two dimensional example that convergence may not hold due to a babuska s type paradox we also propose and test numerically several remedies

infinitely ramified galois representations in this paper we show how to construct for most p 5 two types of surjective representations p gq gal q q gl2 zp that are ramified at an infinite number of primes the image of inertia at almost all of these primes will be torsion free the first construction is unconditional the catch is that we cannot say whether p igp gal qp q is crystalline or even potentially semistable the second construction assumes the generalized riemann hypothesis grh with this assumption we can further arrange that p igp is crystalline at p we remark that infinitely ramified reducible representations have been previously constructed by more elementary means

construction of central elements in the affine hecke algebra via nearby cycles 0 1 overview let g be a connected reductive group over a finite field fq and let g k be the corresponding group over the local field k fq t let g o g k be a maximal compact subgroup of g k here o fq t and let hsph denote the hecke algebra of g k with respect to g o in other words hsph as a vector space consists of compactly supported bi g o invariant functions g k q and the product is defined by

the inverse galois problem for mathrm psl _ 2 mathbb f _ p we show that the simple group psl2 fp occurs as the galois group of an extension of the rationals for all primes p 5 we obtain our galois extensions by studying the galois action on the second etale cohomology groups of a specific elliptic surface

global springer theory abstract we generalize springer representations to the context of groups over a global function field the global counterpart of the grothendieck simultaneous resolution is the parabolic hitchin fibration we construct an action of the graded double affine hecke algebra daha on the direct image complex of the parabolic hitchin fibration in particular we get representations of the degenerate graded daha on the cohomology of parabolic hitchin fibers providing the first step towards a global springer theory

penalty finite element approximation of stokes equations with slip boundary conditions we consider the finite element approximation of the stationary stokes equations with slip boundary conditions on a domain with a smooth curved boundary the slip boundary condition is imposed weakly with the penalty method on polygonal domains approaching the smooth domain for taylor hood elements we derive error estimates which depend on the penalty parameter varepsilon the disctretization parameter h h and the approximation error of the normal to the boundary in particular if in the penalty term we use the normal to the polygonal boundary the best convergence order is 2 3 2 3 and it is obtained with varepsilon c h 2 3 c h 2 3 this convergence result shows that babuska s paradox associated to stokes equations with slip boundary conditions is circumvented a numerical example illustrates the theoretical results notably that regularized normal approximations give better approximations and convergence orders

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

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

deformations of certain reducible galois representations ii we examine the deformation theory of two dimensional mod p reducible indecomposable galois representations removing or relaxing many of the hypotheses of part i we are able to prescribe local conditions on our deformations by allowing ramification at a set of primes congruent to 1 mod p not 1 mod p 2 and satisfying some other splitting conditions the new hypotheses admit irreducible ordinary characteristic zero lifts of many mod p representations which by skinner wiles allows us to conclude the modularity of such representations

towards optimal finite element error estimates for the penalized dirichlet problem in a domain with curved boundary we consider the finite element approximation of an elliptic problem with homogeneous dirichlet boundary conditions on a curved boundary and imposed using the penalty method we establish optimal h 1 error estimates with suitable assumptions on the penalty parameter e as a function of the elements size h our focus is on establishing these results with least restrictive assumptions possible on this dependency

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

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

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

enhanced functional evaluation for the finite element penalty method abstract we analyze the impact of using the penalty method on the estimation of a prescribed solution functional or quantity of interest qoi specifically we consider the use of penalty methods to enforce dirichlet boundary constraints focusing our attention on boundary fluxes as qois we propose an enhanced estimator of the boundary flux that includes a term involving the derivative of the flux with respect to the penalty parameter we show that the new estimator reduces the error arising from the use of the penalty method from o to o a well posed adjoint problem associated with the boundary flux is also proposed following an analysis of the penalty method errors in the enhanced flux estimator are then controlled using adjoint based techniques several numerical experiments are presented to demonstrate that the enhanced estimator in combination with adjoint error estimation allows one to efficiently control both the discretization and penalty errors in target qois

the main conjecture of iwasawa theory for totally real fields let p be an odd prime let mathcal g be a compact p adic lie group with a quotient isomorphic to p we give an explicit description of k 1 of the iwasawa algebra of mathcal g in terms of iwasawa algebras of abelian subquotients of mathcal g we also prove a result about k 1 of a certain canonical localisation of the iwasawa algebra of mathcal g which occurs in the formulation of the main conjectures of noncommutative iwasawa theory these results predict new congruences between special values of artin l functions which we then prove using the q expansion principle of deligne ribet as a consequence we prove the noncommutative main conjecture for totally real fields assuming a suitable version of iwasawa s conjecture about vanishing of the cyclotomic invariant in particular we get an unconditional result for totally real pro p p adic lie extension of abelian extensions of

motives with exceptional galois groups and the inverse galois problem we construct motivic l adic representations of textup gal overline mathbb q mathbb q into exceptional groups of type e 7 e 8 and g 2 whose image is zariski dense this answers a question of serre the construction is uniform for these groups and is inspired by the langlands correspondence for function fields as an application we solve new cases of the inverse galois problem the finite simple groups e 8 mathbb f ell are galois groups over mathbb q for large enough primes l

even galois representations and the fontaine mazur conjecture we prove under mild hypotheses that there are no irreducible two dimensional ordinary even galois representations of mathrm gal overline mathbf q mathbf q with distinct hodge tate weights this is in accordance with the fontaine mazur conjecture if k q is an imaginary quadratic field we also prove again under certain hypotheses that mathrm gal overline mathbf q k does not admit irreducible two dimensional ordinary galois representations of non parallel weight

quantitative level lowering we give a quantitative version of ribet s famous level lowering result for modular forms specifically we measure how certain congruence ideals change as we vary the level by studying the deformation theory of the galois representation attached to the modular form we can use the numerical criterion of wiles to relate congruence ideals to selmer groups and thereby reduce the problem to a calculation in galois cohomology

a rigid irregular connection on the projective line in this paper we construct a connection on the trivial g bundle on p 1 for any simple complex algebraic group g which is regular outside of the points 0 and has a regular singularity at the point 0 with principal unipotent monodromy and has an irregular singularity at the point with slope 1 h the reciprocal of the coxeter number of g the connection which admits the structure of an oper in the sense of beilinson and drinfeld appears to be the characteristic 0 counterpart of a hypothetical family of l adic representations which should parametrize a specific automorphic representation under the global langlands correspondence these l adic representations and their characteristic 0 counterparts have been constructed in some cases by deligne and katz our connection is constructed uniformly for any simple algebraic group and characterized using the formalism of opers it provides an example of the geometric langlands correspondence with wild ramification we compute the de rham cohomology of our connection with values in a representation v of g and describe the differential galois group of as a subgroup of g

kloosterman sheaves for reductive groups deligne constructed a remarkable local system on p 1 f 0 1g attached to a family of kloosterman sums katz calculated its monodromy and asked whether there are kloosterman sheaves for general reductive groups and which automorphic forms should be attached to these local systems under the langlands correspondence motivated by work of gross and frenkel gross we nd an explicit family of such automorphic forms and even a simple family of automorphic sheaves in the framework of the geometric langlands program we use these automorphic sheaves to construct adic kloosterman sheaves for any reductive group in a uniform way and describe the local and global monodromy of these kloosterman sheaves in particular they give motivic galois representations with exceptional monodromy groups g2 f4 e7 and e8 this also gives an example of the geometric langlands correspondence with wild ramication for any reductive group

geometric langlands duality and representations of algebraic groups over commutative rings as such it can be viewed as a first step in the geometric langlands program the connected complex reductive groups have a combinatorial classification by their root data in the root datum the roots and the co roots appear in a symmetric manner and so the connected reductive algebraic groups come in pairs if g is a reductive group we write g for its companion and call it the dual group g the notion of the dual group itself does not appear in satake s paper but was introduced by langlands together with its various elaborations in li l2 and is a cornerstone of the langlands program it also appeared later in physics mo gno in this paper we discuss the basic relationship between g and g we begin with a reductive g and consider the affine grassmannian qx the grassmannian for the loop group of g for technical reasons we work with formal algebraic loops the affine grassmannian is an infinite dimen sional complex space we consider a certain category of sheaves the spherical perverse sheaves on r these sheaves can be multiplied using a convolution product and this leads to a rather explicit construction of a hopf algebra by what has come to be known as tannakian formalism the resulting hopf algebra turns out to be the ring of functions on g in this interpretation the spherical perverse sheaves on the affine grassman nian correspond to finite dimensional complex representations of g thus instead of defining g in terms of the classification of reductive groups we pro vide a canonical construction of g starting from g we can carry out our construction over the integers the spherical perverse sheaves are then those with integral coefficients but the grassmannian remains a complex algebraic object

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

congruence properties of zariski dense subgroups i this paper deals with the following general situation we are given an algebraic group g defined over a number field k and a subgroup f of the group g k of k rational points of g then what should it mean for f to be a large subgroup we might require f to be a lattice in g to be arithmetic to contain many elements of a specific kind to have a large closure in some natural topology et cetera there are many theorems proving implications between conditions of size of this kind we shall consider the case of g a k simple group usually q simple the zariski topology of g for which the closed sets are those defined by the vanishing of polynomial functions is very coarse on the other hand the various valuations v of k each give rise to a strong topology on g k if v is non archimedean and kv is the completion of k with respect to v then g kv is a non archimedean lie group and if is the ring of integers of kv the open subgroup g 5 j of g kv of integral points is defined for all but finitely many v the zariski closure f of a subgroup f of g k is a c algebraic subgroup of g while the strong closure fy is a lie subgroup of g kv for elementary reasons the dimension of tv is at most the dimension of f our results are in the opposite direction we show under suitable conditions on g that if f g then for almost all v we have that tv contains g ov to illustrate this with a specific example if ml 5 mk are matrices in sl2 q generating a group f which is zariskidense in sl2 then for all sufficiently large prime numbers p we have f p sl2 zp further this result is effective in the sense that we could in principle check the hypothesis for given mx mk and derive a bound on p beyond which the conclusion holds it is perhaps worth remarking that this special case which is the simplest application of our results may be proved with much less effort than the general theorem

unobstructed modular deformation problems let be a newform of weight k 3 with fourier coefficients in a number field k we show that the universal deformation ring of the mod a galois representation associated to is unobstructed and thus isomorphic to a power series ring in three variables over the witt vectors for all but finitely many primes a of k we give an explicit bound on such a for the 6 known cusp forms of level 1 trivial character and rational fourier coefficients we also prove a somewhat weaker result for weight 2

epipelagic l packets and rectifying characters we provide an explicit construction of the local langlands correspondence for general tamely ramified reductive p adic groups and a class of wildly ramified langlands parameters furthermore we verify that our construction satisfies many expected properties of such a correspondence more precisely we show that each l packet we construct admits a parameterization in terms of the langlands dual group contains a unique generic element for a fixed whittaker datum satisfies the formal degree conjecture is compatible with central and cocentral characters provides a stable virtual character and satisfies the expected endoscopic character identities moreover we show that in the case of mathrm gl n our construction coincides with the established local langlands correspondence our techniques provide a general approach to the construction of the local langlands correspondence for tamely ramified groups and regular supercuspidal parameters

epipelagic representations and invariant theory letg be a reductive p adic group we give a new construction of small depth epipelagic supercuspidal representations ofg k using stable orbits in geometric invariant theory git in contrast to previously known methods this construction works without any restrictions on the residue characteristic p we construct appropriate langlands parameters for these representations with some restrictions on p the git arises from graded moy prasad filtrations which we show are isomorphic to the graded lie algebras whose git was analyzed by vinberg and levy this leads to a classification of stable orbits in moy prasad filtrations as well as epipelagic supercuspidal representations in terms of z regular elliptic automorphisms of the absolute root system of g transferred to the l group of g generates the image of tame inertia under the corresponding wild langlands parameter for unramified groups and sufficiently large p we also classify the moy prasad filtrations which have semi stable orbits which solves the long outstanding problem of classifying non degenerate k types for g k

mode coupling theory of the glass transition for confined fluids we present a detailed derivation of a microscopic theory for the glass transition of a liquid enclosed between two parallel walls relying on a mode coupling approximation this geometry lacks translational invariance perpendicular to the walls which implies that the density profile and the density density correlation function depends explicitly on the distances to the walls we discuss the residual symmetry properties in slab geometry and introduce a symmetry adapted complete set of two point correlation functions since the currents naturally split into components parallel and perpendicular to the walls the mathematical structure of the theory differs from the established mode coupling equations in bulk we prove that the equations for the nonergodicity parameters still display a covariance property similar to bulk liquids

finite element approximation of the dirichlet problem using the boundary penalty method summary this paper considers a finite element approximation of the dirichlet problem for a second order self adjoint elliptic equation a u f in a region f2cn n 2 or 3 by the boundary penalty method if the finite element space defined over d h a union of elements has approximation power h in the l 2 norm then i for f2 d h convex polyhedral we show that choosing the penalty parameter e h x with 2 k yields optimal h 1 and l 2 error bounds if u hk i q

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

variations of hodge structure and orbits in flag varieties abstract period domains the classifying spaces for pure polarized hodge structures and more generally mumford tate domains arise as open g r orbits in flag varieties g p we investigate hodge theoretic aspects of the geometry and representation theory associated with these flag varieties in particular we relate the griffiths yukawa coupling to the variety of lines on g p under a minimal homogeneous embedding construct a large class of polarized g r orbits in g p and compute the associated hodge theoretic boundary components an emphasis is placed throughout on adjoint flag varieties and the corresponding families of hodge structures of levels two and four

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

fluids of hard ellipsoids phase diagram including a nematic instability from percus yevick theory an important aspect of molecular fluids is the relation between orientation and translation parts of the two particle correlations especially a detailed knowledge of the influence of orientation correlations is needed to explain and calculate in detail the occurrence of a nematic phase the simplest model system that shows both orientation and translation correlations is a system of hard ellipsoids we investigate an isotropic fluid formed of hard ellipsoids with the percus yevick theory solving the percus yevick equations self consistently and accurately in the high density regime gives contrary to previous works a clear criterion for a nematic instability we calculate in detail the equilibrium phase diagram for a fluid of hard ellipsoids of revolution our results compare well with monte carlo simulations and density functional theory

deformations of galois representations and exceptional monodromy ii raising the level building on lifting results of ramakrishna khare and ramakrishna proved a purely galois theoretic level raising theorem for odd representations bar rho mathrm gal overline mathbb q mathbb q rightarrow mathrm gl 2 overline mathbb f ell in this paper we generalize these techniques from type a1 to general semi simple groups we then strengthen our previous results on constructing geometric galois representations with exceptional monodromy groups achieving such constructions for almost all ell rather than a density one set and achieving greater flexibility in the hodge numbers of the lifts the latter improvement requires the new level raising result