on the eisenstein cohomology of arithmetic groups the cohomology h gamma e of an arithmetic subgroup gamma of a connected reductive algebraic group g defined over some algebraic number field f can be interpreted in terms of the automorphic spectrum of gamma with this framework in place there is a sum decomposition of the cohomology into the cuspidal cohomology i e classes represented by cuspidal automorphic forms for g and the so called eisenstein cohomology constructed as the span of appropriate residues or derivatives of eisenstein series attached to cuspidal automorphic forms on the levi components of proper parabolic f subgroups of g the main objective of this paper is to isolate a specific structural part in the eisenstein cohomology this pertains to regular eisenstein cohomology classes attached to cuspidal automorphic representations whose archimedean component is tempered it is shown that the cohomological degree of these classes is bounded from below by the constant q 0 g mathbb r 1 2 dim x g mathbb r rk g mathbb r rk k mathbb r where k mathbb r denotes a maximal compact subgroup of the real lie group g mathbb r where x g mathbb r is the associated symmetric space this investigation has various applications one of these is a vanishing result for the cohomology in the generic case i e where the representation determining the coefficient system e has regular highest weight in degrees below q 0 g mathbb r this is a sharp bound depending only on the underlying real lie group g mathbb r corollary 5 6 proposition 5 8 this result is supplemented by a qualitative structural result in the description of the cohomology in higher degrees by means of regular eisenstein cohomology classes

compatibility of local and global langlands correspondences we prove the compatibility of local and global langlands correspondences for gln which was proved up to semisimplification in ht more precisely for the ndimensional l adic representation rl of the galois group of an imaginary cm field l attached to a conjugate self dual regular algebraic cuspidal automorphic representation of gln al which is square integrable at some finite place we show that frobenius semisimplification of the restriction of rl to the decomposition group of a place v of l not dividing l corresponds to v by the local langlands correspondence if v is square integrable for some finite place v 6 l we deduce that rl is irreducible we also obtain conditional results in the case v l

on the cuspidal cohomology of arithmetic groups it is the main objective of this paper to show a vanishing result for cuspidal cohomology of arithmetic groups in classical groups g defined over some number field k our approach relies on the fact that cuspidal automorphic forms are nonsingular in the sense of howe this result puts a strong constraint on the archimedean components of irreducible cuspidal automorphic representations of g that can possibly contribute to the cuspidal cohomology combining this with the classification of vogan zuckerman of unitary representations with nonzero cohomology provides a constant r 0 g k only depending on g k below which the cuspidal cohomology vanishes we will give a formula for this constant for each classical group of type i in the classification scheme due to weil we conclude with making this result explicit for some split classical groups over a totally real algebraic number field

compact subgroups of linear algebraic groups abstract the general problem underlying this article is to give a qualitative classification of all compact subgroups gl n f where f is a local field and n is arbitrary it is natural to ask whether is an open compact subgroup of h e where h is a linear algebraic group over a closed subfield e f we show that indeed has this form up to finite index and a finite number of abelian subquotients when is zariski dense in a connected semisimple group we give a precise openness result for the closure of the commutator group of in the case char f 0 the answers have long been known by results of chevalley and weyl the motivation for this work comes from the positive characteristic case where such results are needed to study galois representations associated to function fields we also derive openness results over a finite number of local fields

degenerate lower dimensional tori in reversible systems abstract in this paper we prove the persistence of lower dimensional invariant tori with prescribed frequencies and singular normal matrices in reversible systems the normal variable is two dimensional and the unperturbed nonlinear terms in the differential equation for this variable have a special structure

galois representations arising from some compact shimura varieties our aim is to establish some new cases of the global langlands correspondence for glm along the way we obtain a new result on the description of the cohomology of some compact shimura varieties let f be a cm eld with complex conjugation c and

level raising and symmetric power functoriality ii we apply automorphy lifting techniques to establish new cases of symmetric power functoriality for hilbert modular forms of regular algebraic weight the proof is based on a novel application of an automorphy lifting theorem for residually reducible galois representations

a family of calabi yau varieties and potential automorphy we prove potential modularity theorems for l adic representations of any dimension from these results we deduce the sato tate conjecture for all elliptic curves with nonintegral j invariant defined over a totally real field

maximality of galois actions for compatible systems in the sense of serre 19 of course the image of is not generally zariski dense in gln q so the maximality condition must be formulated relative to the zariski closure g of gal k k one might hope that the image of is a maximal compact subgroup of g q but this is too optimistic the center causes problems and we do not even know a priori that g is reductive to formulate a maximality conjecture which avoids such problems we introduce maps g g g

vanishing theorems for torsion automorphic sheaves on general pel type shimura varieties abstract we consider the interior cohomology and the hodge graded pieces in the case of the de rham realization of general not necessarily compact pel type shimura varieties with coefficients in the local systems corresponding to sufficiently regular algebraic representations of the associated reductive group for primes p bigger than an effective bound we prove that the f p and z p cohomology groups are concentrated in the middle degree that the z p cohomology groups are free of p torsion and that every f p cohomology class lifts to a z p cohomology class

on torsion in the cohomology of locally symmetric varieties the main result of this paper is the existence of galois representations associated with the mod p or mod p m cohomology of the locally symmetric spaces for gln over a totally real or cm eld proving conjectures of ash and others following an old suggestion of clozel recently realized by harris lan taylor thorne for characteristic 0 cohomology classes one realizes the cohomology of the locally symmetric spaces for gln as a boundary contribution of the cohomology of symplectic or unitary shimura varieties so that the key problem is to understand torsion in the cohomology of shimura varieties thus we prove new results on the p adic geometry of shimura varieties of hodge type namely the shimura varieties become perfectoid when passing to the inverse limit over all levels at p and a new period map towards the ag variety exists on them called the hodge tate period map it is roughly analogous to the embedding of the hermitian symmetric domain which is roughly the inverse limit over all levels of the complex points of the shimura variety into its compact dual the hodge tate period map has several favorable properties the most important being that it commutes with the hecke operators away from p for the trivial action of these hecke operators on the ag variety and that automorphic vector bundles come via pullback from the ag variety

automorphy of some residually dihedral galois representations during the period this research was conducted jack thorne served as a clay research fellow

the sato tate conjecture for hilbert modular forms we prove the sato tate conjecture for hilbert modular forms more precisely we prove the natural generalisation of the sato tate conjecture for regular algebraic cuspidal automorphic representations of gl2 af f a totally real field which are not of cm type the argument is based on the potential automorphy techniques developed by taylor et al but makes use of automorphy lifting theorems over ramified fields together with a topological argument with local deformation rings in particular we give a new proof of the conjecture for modular forms which does not make use of potential automorphy theorems for non ordinary n dimensional galois representations

higher coherent cohomology and p adic modular forms of singular weights we investigate the p adic properties of higher coherent cohomology of automorphic vector bundles of singular weights on siegel threefolds

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

on the modularity of 2 adic potentially semi stable deformation rings using p adic local langlands correspondence for text gl 2 mathbb q 2 and an ordinary r mathbb t theorem we prove that the support of patched modules for quaternionic forms meet every irreducible component of the potentially semi stable deformation ring this gives a new proof of the breuil mezard conjecture for 2 dimensional representations of the absolute galois group of mathbb q 2 which is new in the case overline r a twist of an extension of the trivial character by itself as a consequence a local restriction in the proof of fontaine mazur conjecture in pask nas algebra number theory 10 6 1301 1358 2016 is removed

automorphy lifting for residually reducible l adic galois representations we prove automorphy lifting theorems for residually reducible galois representations in the setting of unitary groups over cm fields our methods are inspired by those of skinner wiles in the setting of gl2

efficient algorithm to compute mutually connected components in interdependent networks mutually connected components mccs play an important role as a measure of resilience in the study of interdependent networks despite their importance an efficient algorithm to obtain the statistics of all mccs during the removal of links has thus far been absent here using a well known fully dynamic graph algorithm we propose an efficient algorithm to accomplish this task we show that the time complexity of this algorithm is approximately o n 1 2 for random graphs which is more efficient than o n 2 of the brute force algorithm we confirm the correctness of our algorithm by comparing the behavior of the order parameter as links are removed with existing results for three types of double layer multiplex networks we anticipate that this algorithm will be used for simulations of large size systems that have been previously inaccessible

on l independence of algebraic monodromy groups in compatible systems of representations consider a profinite group g and a collection of continuous representations g gln q indexed by a set l of rational primes suppose that g is endowed with a dense subset of frobenius elements f a the system is called a compatible system of adic representations if for every a the characteristic polynomial of f has coefficients in the field of rational numbers and does not depend on in 6 5 we will give a more precise and less restrictive definition which allows us to throw out some bad pairs in order to accomodate ramification our notion slightly generalizes serre s original definition 14 to recover serre s definition we take g to be the galois group of a number field k and f to be frobenius representatives for primes of k let g be the zariski closure of g in gln q this is the algebraic monodromy group at our question is the following how does g vary with one hopes for some kind of independence at best there can exist a global algebraic group g gln q such that every g is conjugate to g q q unfortunately this does not always happen in the abstract setting in which we work so we must settle for weaker independence results we first recall what is already known in this direction the compatibility condition bears only on the semisimple part of the elements f so we lose no information by assuming all to be semisimple representations alternatively we could take arbitrary representations and define g as the quotient of the zariski closure of g by its unipotent radical thus g is a family of reductive groups serre has proved

universal eigenvarieties trianguline galois representations and p adic langlands functoriality using the overconvergent cohomology modules introduced by ash and stevens we construct eigenvarieties associated with reductive groups and establish some basic geometric properties of these spaces building on work of ash stevens urban and others we also formulate a precise modularity conjecture linking trianguline galois representations with overconvergent cohomology classes in the course of giving evidence for this conjecture we establish several new instances of p adic langlands functoriality our main technical innovations are a family of universal coefficients spectral sequences for overconvergent cohomology and a generalization of chenevier s interpolation theorem

power law distributions in empirical data power law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man made phenomena unfortunately the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution the part of the distribution representing large but rare events and by the difficulty of identifying the range over which power law behavior holds commonly used methods for analyzing power law data such as least squares fitting can produce substantially inaccurate estimates of parameters for power law distributions and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all here we present a principled statistical framework for discerning and quantifying power law behavior in empirical data our approach combines maximum likelihood fitting methods with goodness of fit tests based on the kolmogorov smirnov ks statistic and likelihood ratios we evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches we also apply the proposed methods to twenty four real world data sets from a range of different disciplines each of which has been conjectured to follow a power law distribution in some cases we find these conjectures to be consistent with the data while in others the power law is ruled out

potential automorphy and the leopoldt conjecture in the period during which this research was conducted jack thorne served as a clay research fellow chandrashekhar khare was supported by nsf grant dms 1161671 and by a humboldt research award and thanks the tata instititute of fundamental research mumbai for hospitality during the period in which most of the work on this paper was done

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

quasi periodic stability of normally resonant tori we study quasi periodic tori under a normal internal resonance possibly with multiple eigenvalues two non degeneracy conditions play a role the first of these generalizes invertibility of the floquet matrix and prevents drift of the lower dimensional torus the second condition involves a kolmogorov like variation of the internal frequencies and simultaneously versality of the floquet matrix unfolding we focus on the reversible setting but our results carry over to the hamiltonian and dissipative contexts

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

local global compatibility and the action of monodromy on nearby cycles we strengthen the local global compatibility of langlands correspondences for gln in the case when n is even and l 6d p let l be a cm field and let be a cuspidal automorphic representation of gln al which is conjugate self dual assume that 1 is cohomological and not slightly regular as defined by shin in this case chenevier and harris constructed an l adic galois representationrl and proved the local global compatibility up to semisimplification at primes v not dividing l we extend this compatibility by showing that the frobenius semisimplification of the restriction of rl to the decomposition group at v corresponds to the image of v via the local langlands correspondence we follow the strategy of taylor and yoshida where it was assumed that is square integrable at a finite place to make the argument work we study the action of the monodromy operator n on the complex of nearby cycles on a scheme which is locally etale over a product of strictly semistable schemes and we derive a generalization of the weight spectral sequence in this case we also prove the ramanujan petersson conjecture for as above

involutions of reductive lie algebras in positive characteristic abstract let g be a reductive group over a field k of characteristic 2 let g lie g let be an involutive automorphism of g and let g k p be the associated symmetric space decomposition for the case of a ground field of characteristic zero the action of the isotropy group g on p is well understood since the well known paper of kostant and rallis b kostant s rallis orbits and representations associated with symmetric spaces amer j math 93 1971 753 809 such a theory in positive characteristic has proved more difficult to develop here we use an approach based on some tools from geometric invariant theory to establish corresponding results in good positive characteristic among other results we prove that the variety n of nilpotent elements of p has a dense open orbit and that the same is true for every fibre of the quotient map p p g however we show that the corresponding statement for g conjectured by richardson is not true we provide a new mostly calculation free proof of the number of irreducible components of n extending a result of sekiguchi for k c finally we apply a theorem of skryabin to describe the infinitesimal invariants k p k

persistence of lower dimensional hyperbolic tori for reversible system abstract in this paper we consider the persistence of hyperbolic lower dimensional invariant tori with prescribed frequency for reversible system and prove that if the frequency mapping has nonzero topological degree at some diophantine frequency then the hyperbolic lower invariant dimensional torus with this frequency persists under small perturbations

hybrid phase transition into an absorbing state percolation and avalanches interdependent networks are more fragile under random attacks than simplex networks because interlayer dependencies lead to cascading failures and finally to a sudden collapse this is a hybrid phase transition hpt meaning that at the transition point the order parameter has a jump but there are also critical phenomena related to it here we study these phenomena on the erd s renyi and the two dimensional interdependent networks and show that the hybrid percolation transition exhibits two kinds of critical behaviors divergence of the fluctuations of the order parameter and power law size distribution of finite avalanches at a transition point at the transition point global or infinite avalanches occur while the finite ones have a power law size distribution thus the avalanche statistics also has the nature of a hpt the exponent m of the order parameter is 1 2 under general conditions while the value of the exponent m characterizing the fluctuations of the order parameter depends on the system the critical behavior of the finite avalanches can be described by another set of exponents a and a these two critical behaviors are coupled by a scaling law 1 m a

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

level raising and symmetric power functoriality iii we apply automorphy lifting techniques to establish new cases of symmetric power functoriality for hilbert modular forms of regular algebraic weight the proof is based on a novel application of an automorphy lifting theorem for residually reducible galois representations

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

hat g local systems on smooth projective curves are potentially automorphic dfg grants fg 1920 and spp 1489 erc grant 290766 aamot nsf grant dms 1404769 nsf grant dms 1161671 humboldt research award clay mathematics institute erc grant no 714405 gmlp

robustness of a network formed by n interdependent networks with a one to one correspondence of dependent nodes many real world networks interact with and depend upon other networks we develop an analytical framework for studying a network formed by n fully interdependent randomly connected networks each composed of the same number of nodes n the dependency links connecting nodes from different networks establish a unique one to one correspondence between the nodes of one network and the nodes of the other network we study the dynamics of the cascades of failures in such a network of networks non caused by a random initial attack on one of the networks after which a fraction p of its nodes survives we find for the fully interdependent loopless non that the final state of the non does not depend on the dynamics of the cascades but is determined by a uniquely defined mutual giant component of the non which generalizes both the giant component of regular percolation of a single network n 1 and the recently studied case of the mutual giant component of two interdependent networks n 2 we also find that the mutual giant component does not depend on the topology of the non and express it in terms of generating functions of the degree distributions of the network our results show that for any n 2 there exists a critical p p c 0 below which the mutual giant component abruptly collapses from a finite nonzero value for p p c to zero for p 2 a rr non is stable for any n with p c 1 this results arises from the critical role played by singly connected nodes which exist in an er non and enhance the cascading failures but do not exist in a rr non

reversing symmetries in dynamical systems dynamical systems may possess in addition to symmetries that leave the equations of motion invariant reversing symmetries that invert the equations of motion such dynamical systems are called weakly reversible some consequences of the existence of reversing symmetries for dynamical systems with discrete time mappings are discussed a reversing symmetry group is introduced and it is shown that every weakly reversible mapping l can be decomposed into two mappings k0 and k1 of the same order 2l limit l to infinity included such that k02 k12 i some applications are discussed briefly

the residual eisenstein cohomology of sp_ 4 over a totally real number field let g sp4 k be the k split symplectic group of k rank 2 where k is a totally real numbereld in this paper we compute the eisenstein cohomology of g with respect to anynite dimensional irreducible k rational representation e of g1 rk qg r where rk q denotes the restriction of scalars from k to q the approach is based on the work of schwermer regarding the eisenstein cohomology for sp4 q kim s description of the residual spectrum of sp4 and the frankeltration of the space of automorphic forms in fact taking the representation theoretic point of view we write for the group g the frankeltration with respect to the cuspidal support and give a precise description of theltration quotients in terms of induced representations this is then used as a prerequisite for the explicit computation of the eisenstein cohomology the special focus is on the residual eisenstein cohomology under a certain compatibility condition for the coefficient system e and the cuspidal support we prove the existence of non trivial residual eisenstein cohomology classes which are not square integrabl e that is represented by a non square integrabl residue of an eisenstein series

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

universality and scaling laws in the cascading failure model with healing cascading failures may lead to dramatic collapse in interdependent networks where the breakdown takes place as a discontinuity of the order parameter in the cascading failure cf model with healing there is a control parameter which at some value suppresses the discontinuity of the order parameter however up to this value of the healing parameter the breakdown is a hybrid transition meaning that besides this first order character the transition shows scaling too in this paper we investigate the question of universality related to the scaling behavior recently we showed that the hybrid phase transition in the original cf model has two sets of exponents describing respectively the order parameter and the cascade statistics which are connected by a scaling law in the cf model with healing we measure these exponents as a function of the healing parameter we find two universality classes in the wide range below the critical healing value the exponents agree with those of the original model while above this value the model displays trivial scaling meaning that fluctuations follow the central limit theorem

bifurcations in two dimensional reversible maps we give a treatment of the non resonant bifurcations involving asymmetric fixed points with jacobian j 1 in reversible mappings of the plane these bifurcations include the saddle node bifurcation not in the neighbourhood of a fixed point with j 1 as well as the so called transcritical bifurcations and generalized rimmer bifurcations taking place at a fixed point with jacobian j 1 the bifurcations are illustrated by some simple examples of model maps the rimmer type of bifurcation with e g a center point with j 1 changing into a saddle with jacobian j 1 an attractor and a repeller occurs under more general conditions i e also in non reversible mappings if only a certain order of local reversibility is satisfied these rimmer bifurcations are important in connection with the emergence of dissipative features in non measure preserving reversible dynamical systems

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

eigenvarieties for reductive groups we develop the theory of overconvergent cohomology introduced by g stevens and we use it to give a construction of eigenvarieties associated to any reductive group g over q such that g r has discrete series we prove that the so called eigenvarieties are equidimensional and generically at over the weight space

on the cohomology groups of local systems over hilbert modular varieties via higgs bundles let x be a hilbert modular variety and v a non trivial local system over x in this paper we study deligne saito s mixed hodge structure mhs on the cohomology groups h k x v using the method of higgs bundles among other results we give a dimension formula for the hodge numbers and show that the mixed hodge structure is split over r these results are analogous to 18 in the cocompact case and complement the results in 8 for constant coefficients

on the quasi periodic d fold degenerate bifurcation abstract this paper analyses the d fold degenerate bifurcation of invariant quasi periodic tori of normal dimension one it is shown that for integrable families of diffeomorphisms unfolding a d fold degenerate invariant torus versally invariant diophantine tori in the family persist under a small non integrable perturbation as a consequence a subset of positive measure with respect to an interpolating smooth manifold of the bifurcation set persists as well the proof is a consequence of the translated torus theorem of russmann and herman

a 2 adic automorphy lifting theorem for unitary groups over cm fields we prove a minimal type automorphy lifting theorem for 2 adic galois representations of unitary type over imaginary cm fields we use this to improve an automorphy lifting theorem of kisin for mathrm gl 2

rational structures on automorphic representations this paper proves the existence of global rational structures on spaces of cusp forms of general reductive groups we identify cases where the constructed rational structures are optimal which includes the case of mathrm mathrm gl n as an application we deduce the existence of a natural set of periods attached to cuspidal automorphic representations of mathrm mathrm gl n this has consequences for the arithmetic of special values of l functions that we discuss in januszewski on period relations for automorphic l functions i pp 1 46 arxiv 1504 06973 2015 on period relations for automorphic l functions ii pp 1 65 arxiv 1604 04253 2015 in the course of proving our results we lay the foundations for a general theory of harish chandra modules over arbitrary fields of characteristic 0 in this context a rational character theory translation functors and an equivariant theory of cohomological induction are developed we also study descent problems for harish chandra modules in quadratic extensions where we obtain a complete theory over number fields

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

on the images of the galois representations attached to genus 2 siegel modular forms we address the problem of the determination of the images of the galois representations attached to genus 2 siegel cusp forms of level 1 having multiplicity one these representations are symplectic we prove that the images are as large as possible for almost every prime if the siegel cusp form is not a maass spezialform and verifies two easy to check conditions mathematics subject classification 11f80 11f46

versal deformations and normal forms for reversible and hamiltonian linear systems the problem of this article is the characterization of equivalence classes and their versal deformations for reversible and reversible hamiltonian matrices in both cases the admissible transformations form a subgroup g of gl m therefore the gl m orbits of a given matrix may split into several g orbits these orbits are characterized by signs for each sign we have a normal form and a corresponding versal deformation the main tool in the characterization is reduction to the semi simple case c 1996 academic press inc

eisenstein cohomology for congruence subgroups of so n 2 the automorphic cohomology of a connected reductive algebraic group defined over mathbb q decomposes as a direct algebraic sum of cuspidal and eisenstein cohomology the present paper investigates the eisenstein cohomology for congruence subgroups of a rational form g of mathbb q rank 2 of so n 2 in the generic case the main results provide a description of the internal structure of the summands in the eisenstein cohomology corresponding to the associate classes of maximal parabolic mathbb q subgroups

potential automorphy and change of weight we prove an automorphy lifting theorem for l adic representations where we impose a new condition at l which we call potentential diagonalizability this result allows for change of weight and seems to be substantially more exible than previous theorems along the same lines we derive several applications for instance we show that any irreducible totally odd essentially self dual regular weakly compatible system of l adic representations of the absolute galois group of a totally real eld is potentially automorphic and hence is pure and its l function has meromorphic continuation to the whole complex plane and satises the expected functional equation

perturbations of lower dimensional tori in the resonant zone for reversible systems consider a nearly integrable system which is reversible with respect to an involution of n n type for a given l resonant surface 1 l n we show that there is a subset of it with positive measure such that in each resonant torus corresponding to a point of this set at least one dimensional invariant submanifold can survive any small perturbations

percolation transitions in the survival of interdependent agents on multiplex networks catastrophic cascades and solid on solid surface growth we present an efficient algorithm for simulating percolation transitions of mutually supporting viable clusters on multiplex networks also known as catastrophic cascades on interdependent networks this algorithm maps the problem onto a solid on solid type model we use this algorithm to study interdependent agents on duplex erdos renyi er networks and on lattices with dimensions 2 3 4 and 5 we obtain surprising results in all these cases and we correct statements in the literature for er networks and for two dimensional lattices in particular we find that d 4 is the upper critical dimension and that the percolation transition is continuous for d 4 but at least for d 3 not in the universality class of ordinary percolation for er networks we verify that the cluster statistics is exactly described by mean field theory but find evidence that the cascade process is not for d 5 we find a first order transition as for er networks but we find also that small clusters have a nontrivial mass distribution that scales at the transition point finally for d 2 with intermediate range dependency links we propose a scenario that differs from that proposed in w li et al phys rev lett 108 228702 2012

percolation of partially interdependent scale free networks we study the percolation behavior of two interdependent scale free sf networks under random failure of 1 p fraction of nodes our results are based on numerical solutions of analytical expressions and simulations we find that as the coupling strength between the two networks q reduces from 1 fully coupled to 0 no coupling there exist two critical coupling strengths q 1 and q 2 which separate three different regions with different behavior of the giant component as a function of p i for q q 1 an abrupt collapse transition occurs at p p c ii for q 2 3 q 1 decreases with increasing here is the scaling exponent of the degree distribution p k proportionality k b in the hybrid transition at the q 2 0 for 3 thus the known theoretical p c 0 for a single network with 3 is expected to be valid also for strictly partial interdependent networks

reversing k symmetries in dynamical systems we generalize the concept of reversing symmetries of a dynamical system i e we study dynamical systems that possess symmetry properties only if considered on a proper time scale in particular considering dynamical systems with discrete time the kth iterate of a map may possess more reversing symmetries than the map itself in this way the concepts of reversing symmetries and reversing symmetry groups are generalized to reversing k symmetries and reversing k symmetry groups furthermore a method is studied for finding orbits that are k symmetric with respect to reversing k symmetries firstly an existing method for finding orbits that are symmetric with respect to one reversing symmetry is extended to the case of more than one reversing symmetry and secondly a generalization of this method to the case of reversing k symmetries is introduced some physically relevant examples of dynamical systems possessing reversing k symmetries are discussed briefly

normal linear stability of quasi periodic tori abstract we consider families of dynamical systems having invariant tori that carry quasi periodic motions our interest is the persistence of such tori under small nearly integrable perturbations this persistence problem is studied in the dissipative the hamiltonian and the reversible setting as part of a more general kam theory for classes of structure preserving dynamical systems this concerns the parametrized kam theory as initiated by moser j k moser on the theory of quasiperiodic motions siam rev 8 2 1966 145 172 j k moser convergent series expansions for quasi periodic motions math ann 169 1967 136 176 and further developed in g b huitema unfoldings of quasi periodic tori phd thesis university of groningen 1988 h w broer g b huitema f takens unfoldings of quasi periodic tori mem amer math soc 83 421 1990 1 82 h w broer g b huitema unfoldings of quasi periodic tori in reversible systems j dynam differential equations 7 1 1995 191 212 the corresponding nondegeneracy condition involves certain trans versality conditions on the normal linear leading part at the invariant tori we show that as a consequence a cantor family of diophantine tori with positive hausdorff measure is persistent under nearly integrable perturbations this result extends the above references since presently the case of multiple floquet exponents is included our leading example is the normal 1 1 resonance which occurs a lot in applications both hamiltonian and reversible as an illustration of this we briefly describe the lagrange top coupled to an oscillator

the quasi periodic centre saddle bifurcation abstract nearly integrable families of hamiltonian systems are considered in the neighbourhood of normally parabolic invariant tori in the integrable case such tori bifurcate into normally elliptic and normally hyperbolic invariant tori with a kam theoretic approach it is shown that both the normally parabolic tori and the bifurcation scenario survive a non integrable perturbation parametrised by pertinent large cantor sets these results are applied to rigid body dynamics