varieties of nilpotent elements for simple lie algebras i good primes let g be a simple algebraic group over k or fp where p is good set g lie g given r and a faithful restricted representation g gl v one can define a variety of nilpotent elements nr g x g x r 0 in this paper we determine this variety when is an irreducible representation of minimal dimension or the adjoint representation

the varieties and the cohomology ring of a module let g be a finite group and let k be a field of characteristic p 0 recently several studies have focused on the homological invariants of finitely generated kg modules in l alperin proposed the study of the complexity of a module the complexity is related to the degree of the polynomial rate of growth of the terms in a projective resolution of the module see definition 2 4 in 2 alperin and evens proved that the complexity of a kg module m is equal to the maximum of the complexities of the restrictions of m to elementary abelian p subgroups of g one of the roots of the alperin evens theorem is quillen s dimension theorem see 21 or 22 which with some difficulty can be interpreted as saying the same thing for the special case of the trivial kg module alperin and evens 3 and avrunin 4 have further studied the annihilator in ext k k of the cohomology of m and have produced theorems of a similar nature these are related to the work in 9 on the structure of the cohomology ring ext m m the purpose of this paper is to investigate the relationship between these homological invariants and the structure of the module we concentrate on the case in which g is an elementary abelian p group however using such results as those mentioned above we can apply the present work to more general groups even as the earlier results indicate that much of the structure and cohomology of a module is revealed in the restrictions to elementary abelian p subgroups we show here that in the case of an elementary abelian group much of this information can be found by looking at the restrictions to cyclic p subgroups of the group of units of kg some of the results in this paper were announced in s let g xl xn be an elementary abelian group of order p suppose that k is algebraically closed to any finitely generated kg module m we

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

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

an integral equation method for electrostatics of anisotropic composites an interface integral equation is presented for the electrostatic problem in a composite where the components have anisotropic conductivity tensors the electric potential is represented by a single layer charge density on the component interfaces in a simple numerical example involving anisotropic discs the charge density is expanded in a fourier series the integral equation is then solved to high accuracy with a modest computational effort

cohomological finite generation for finite supergroup schemes abstract in this paper we compute extension groups in the category of strict polynomial superfunctors and thereby exhibit certain universal extension classes for the general linear supergroup some of these classes restrict to the universal extension classes for the general linear group exhibited by friedlander and suslin while others arise from purely super phenomena we then use these extension classes to show that the cohomology ring of a finite supergroup scheme equivalently of a finite dimensional cocommutative hopf superalgebra over a field is a finitely generated algebra implications for the rational cohomology of the general linear supergroup are also discussed

representations and cohomology of a family of finite supergroup schemes abstract we examine the cohomology and representation theory of a family of finite supergroup schemes of the form g a g a g a r z p s in particular we show that a certain relation holds in the cohomology ring and deduce that for finite supergroup schemes having this as a quotient both cohomology mod nilpotents and projectivity of modules is detected on proper sub supergroup schemes this special case feeds into the proof of a more general detection theorem for unipotent finite supergroup schemes in a separate work of the authors joint with iyengar and krause we also completely determine the cohomology ring in the smallest cases namely g a g a g a 1 and g a g a z p the computation uses the local cohomology spectral sequence for group cohomology which we describe in the context of finite supergroup schemes

on exponentiation and infinitesimal one parameter subgroups of reductive groups abstract let g be a reductive algebraic group over an algebraically closed field k of characteristic p 0 and assume p is good for g let p be a parabolic subgroup with unipotent radical u for r 1 denote by g a r the r th frobenius kernel of g a we prove that if the nilpotence class of u is less than p then any embedding of g a r in u lies inside a one parameter subgroup of u and there is a canonical way in which to choose such a subgroup applying this result we prove that if p is at least as big as the coxeter number of g then the cohomological variety of g r is homeomorphic to the variety of r tuples of commuting elements in n 1 g the p nilpotent cone of lie g

deformations and the rigidity method abstract in d rohrlich false division towers of elliptic curves j algebra 229 1 2000 249 279 d rohrlich a deformation of the tate module j algebra 229 1 2000 280 313 rohrlich proved rigidity for psl 2 z p t for p 5 obtained this group as a galois group over c t using modular function fields and derived from this interesting consequences for galois representations attached to the tate modules of elliptic curves furthermore in an unpublished preprint he established that the corresponding galois representation g c t gal c t alg c t psl 2 z p t is universal here we will turn things around we first provide a general framework for rigid deformations of projective representations of the absolute galois group of a function field in one variable over a separably closed base under natural rather general hypothesis we will determine the corresponding universal deformation ring if the residual representation is geometrically rigid which happens to be the case for many surjective representation to psl 2 f p p 2 which arise from belyi triples then certain universal deformations will be geometrically rigid too this will give new proofs for most of the results of rohrlich our method also applies to thompson tuples we then go on to give two further applications which are based on the example computed by rohrlich over f q t where q is a power of a prime l we construct infinite p adic galois extensions which have finite ramification and whose constant field is finite furthermore for p 5 and p 1 mod 4 we obtain a family of surjective galois representations gal q alg q 1 sl 2 z p 1 where the parameter runs over all p power roots of unity finally we exhibit a general class of rigid universal deformation rings which are finite flat over z p in particular this shows that the above examples of galois representations are not a singular event but a general phenomenon

projective representations of symmetric groups via sergeev duality in this article we determine the irreducible projective representations of the symmetric group sd and the alternating group ad over an algebraically closed field of characteristic p 6 2 these matters are well understood in the case p 0 thanks to the fundamental work of schur 24 in 1911 as well as the much more recent work of nazarov 19 20 sergeev 25 26 and others so the focus here is primarily on the case of positive characteristic where surprisingly little is known as yet in particular we obtain a natural combinatorial labelling of the irreducibles in terms of a certain set rpp d of restricted p strict partitions of d such partitions arose recently in work of kashiwara miwa peterson and yung 11 and leclerc and thibon 14 on the q deformed fock space of the affine kac moody algebra of type a p 1 leclerc and thibon proposed that rpp d should label the irreducible projective representations in some natural way and we show here how this can be done note that for p 3 5 the labelling problem was solved in 1 3 while if p 2 all projective representations of sd and ad are linear so do not need to be considered further here to be more precise recall that is a partition of d if 1 2 is a non increasing sequence of non negative integers summing to d call p strict if in addition

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

support varieties and hochschild cohomology rings we define a support variety for a finitely generated module over an artin algebra lambda over a commutative artinian ring k with lambda flat as a module over k in terms of the maximal ideal spectrum of the algebra rm hh lambda of lambda this is modelled on what is done in modular representation theory and the varieties defined in this way are shown to have many of the same properties as those for group rings in fact the notions of a variety in our sense and those for principal and non principal blocks are related by a finite surjective map of varieties for a finite dimensional self injective algebra over a field the variety is shown to be an invariant of the stable component of the auslander reiten quiver moreover we give information on nilpotent elements in rm hh lambda give a thorough discussion of the ring rm hh lambda on a class of nakayama algebras give a brief discussion of a possible notion of complexity and make a comparison with support varieties for complete intersections

support varieties and representation types for basic classical lie superalgebras abstract let be an algebraically closed field of characteristic p 3 and g a restricted lie superalgebra over we introduce the definition of restricted cohomology for g and show its cohomology ring is finitely generated provided g is a basic classical lie superalgebra as a consequence we show that the restricted enveloping algebra of a basic classical lie superalgebra g is always wild except g sl 2 or g osp 1 2 or g c 2

representation theoretic support spaces for finite group schemes we introduce the space p g of abelian p points of a finite group scheme over an algebraically closed field of characteristic p 0 we construct a homeomorphism g p g proj g from p g to the projectivization of the cohomology variety for any finite group g for an elementary abelian p group respectively an infinitesimal group scheme p g can be identified with the projectivization of the variety of cyclic shifted subgroups resp variety of 1 parameter subgroups for a finite dimensional g module m g restricts to a homeomorphism p g m proj g m thereby giving a representation theoretic interpretation of the cohomological support variety

representations of elementary abelian p groups and bundles on grassmannians abstract we initiate the study of representations of elementary abelian p groups via restrictions to truncated polynomial subalgebras of the group algebra generated by r nilpotent elements k t 1 t r t 1 p t r p we introduce new geometric invariants based on the behavior of modules upon restrictions to such subalgebras we also introduce modules of constant radical and socle type generalizing modules of constant jordan type and provide several general constructions of modules with these properties we show that modules of constant radical and socle type lead to families of algebraic vector bundles on grassmannians and illustrate our theory with numerous examples

the conductivity of a sheet containing inclusions with sharp corners we have written down an integral equation for the surface charge on the interface between an n sided regular polygonal inclusion with conductivity o 1 and a host with conductivity 0 the conductivity of a sheet with few inclusions can be determined from the induced dipole associated with this charge distribution the integral equation is solved numerically and detailed results are given for n 3 4 and 6 the circle limit is approached for large n we show that our results are consistent with the reciprocity theorem for two dimensional conducting media and we reproduce the known analytic results in the limits when the inclusion is either a hole or a superconducting region we also find the conductivity of a sheet containing a general n pointed star analytic fits to our numerical results are given which are used as input for effective medium theories the results given in this paper for the electrical conductivity apply also to the thermal conductivity and to dielectric media

complexity for modules over finite chevalley groups and classical lie algebras 1 1 let g be a connected reductive algebraic group defined over a finite prime field fp of p elements for a given rational g module m one can consider the restrictions of m to the restricted lie algebra lie g and to the finite group g fp of fp rational points even though there is no direct functorial connection between the categories of restricted lie g modules and g fp modules the ambient algebraic group g can serve as a bridge linking these two important module categories early results of curtis in the 1960s provided a one to one correspondence between the simple restricted modules over k fp of lie g and those of g fp this correspondence is given by simply restricting certain classes of simple g modules more importantly these results along with work of steinberg allow one to easily transfer the questions pertaining to the computation of irreducible characters among these three categories for over thirty years this approach has been one of the main ideas used to relate the representation and cohomology theory of algebraic groups lie algebras and finite groups of lie type in the defining characteristic see e g cpsvan jan2 hum3 and jan1 given the previous results in this area a natural course of investigation should be to find relationships between canonical modules like projective modules for lie g and g fp more precisely

a renormalization method for the evaluation of lattice sums a number of problems in solid state physics and materials science can be resolved by the evaluation of certain lattice sums sums over all sites of an infinite perfect lattice of some potential energy function one classical example the calculation of lattice sums of circular and spherical harmonics dates back to the last century to lord rayleigh s work on computing the effective conductivity of a simple composite while lord rayleigh presented an efficient asymptotic method for two dimensional problems he resorted to direct evaluation of the lattice sums in the three dimensional case more recent methods based on ewald summation have been developed by nijboer and de wette schmidt and lee and others in this article a fast method for evaluating lattice sums which is based on a new renormalization identity is described

a parallel fast multipole accelerated integral equation scheme for 3d stokes equations in this paper we discuss a numerical scheme for the stokes equations in three dimensions it uses an integral equation formulation and is accelerated by the new version of fast multipole method first introduced by greengard and rokhlin in 1997 acta numerica 1997 6 229 269 the code is parallelized to solve problems of extremely large size the resulting numerical solver can be applied to stokes flows in complex geometry and also serves as a building block for solving the navier stokes equations of low to moderate reynold s numbers copyright 2006 john wiley sons ltd

quillen stratification for modules let g be a finite group and k a fixed algebraically closed field of characteristic p o if p is odd let h be the subring of h g k consisting of elements of even degree following 20 221 we take h h g k if p 2 though one could just as well use the subring of elements of even degree for all p h is a finitely generated commutative k algebra 13 and we let v denote its associated affine variety max h if m is any finitely generated kg module then the cohomology variety v m of m may be defined as the support in v of the h module h g m if g is a p group and in general as the largest support of h g l m where l is any kg module 4 91 a module l with each irreducible kc module as a direct summand will serve d quillen 20 221 proved a number of beautiful results relating k to the varieties v associated with the various elementary abelian p subgroups e of g culminating in his stratification theorem 20 221 this theorem gives a piecewise description of v almost explicitly in terms of the subgroups e and their normalizers in g a well known corollary is that dim v max dim v where e

the variety of an indecomposable module is connected let g be a finite group and let k be an algebraically closed field of characteristic p 0 the connections between the modular representation theory of finite groups and cohomology theory is currently a topic of great interest in particular cohomology theory has been used to attach to each finitely generated kgmodule m a homogeneous affine variety v m if we let 17 m be the corresponding projective variety then our main result is the following

on cocommutative hopf algebras of finite representation type abstract let g be a finite algebraic group defined over an algebraically closed field k of characteristic p 0 such a group decomposes into a semidirect product g g 0 g red with a constant group g red and a normal infinitesimal subgroup g 0 if the principal block b 0 g of the group algebra h g has finite representation type then both constituents have the same property with at least one of them being semisimple we determine the structure of the infinitesimal constituent g 0 up to the classification of v uniserial groups

on rigid tuples in linear groups of odd dimension abstract we construct examples of linearly rigid tuples which lead to regular galois realizations over q for linear and unitary groups gl2m 1 q and u2m 1 q2 where q is odd and m q 1 resp m q 1 the notion of linear rigidity was introduced by strambach and volklein j reine angew math to appear as an analogue of katz physically rigid systems see n m katz rigid local systems annals of mathematics studies vol 139 princeton univ press princeton nj 1996

support varieties for infinitesimal group schemes the representation theory of a connected smooth affine group scheme over a field k of characteristic p 0 is faithfully captured by that of its family of frobenius kernels such frobenius kernels are examples of infinitesimal group schemes affine group schemes g whose coordinate hopf algebra k g is a finite dimensional local k algebra this paper presents a study of the cohomology algebra h g k of an arbitrary infinitesimal group scheme over k we provide a geometric determination of the cohomological support variety cgl spec h g k analogous to that given by d quillen for the cohomology of finite groups q we further study finite dimensional rational g modules m for arbitrary infinitesimal group schemes g over k in a manner initiated by j alperin and l evens a e and j carlson ci for finite groups we consider the variety igim c igi of the ideal im ker hev g k ext m m and provide a geometric description of this variety which is analogous to that given by g avrunin and l scott for finite dimensional modules for finite groups a s this paper is a continuation of our recent work establishing the finite generation of h g k f s and investigating the infinitesimal 1 parameter subgroups of g s f b earlier work of e friedlander and b parshall fp1 fp2 fp3 fp4 and j jantzen ji concerning the cohomology of restricted lie algebras are forerunners of the results presented here finite dimensional restricted lie algebras are in 1 1 correspondence with infinitesimal group schemes of height 1 our main theorems theorems 5 2 and 6 7 below when restricted to infinitesimal group schemes of height 1 significantly strengthen previously known cohomological information for restricted lie algebras an interesting aspect of our work is the extent to which infinitesimal 1 parameter subgroups v g a r g for infinitesimal group schemes g of height r play the role of elementary abelian p subgroups and their generalizations shifted subgroups for finite groups indeed much of our effort is dedicated to proving that cohomology classes are detected modulo nilpotence by such 1 parameter subgroups this is first done in 2 for unipotent infinitesimal group schemes using an induction argument made possible by a structure theorem presented in 1 this structure theorem is the analogue in our context of a theorem of j p serre characterizing elementary abelian p groups s

on nilpotent commuting varieties and cohomology of frobenius kernels abstract the paper studies the dimensions of irreducible components of commuting varieties of restricted nilpotent r tuples in a classical lie algebra g defined over an algebraically closed field k as applications we obtain some new results on the structure of the even cohomology ring of frobenius kernels g r for each r 1 where g is the simply connected simple algebraic group such that lie g g explicit calculations for rank two groups are also presented

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

on the structure of tame graded basic hopf algebras ii abstract in this paper firstly we give a complete list of tame local graded frobenius algebras then by the method of bosonization and this list we determine the structure of tame radically graded basic hopf algebras completely

on the cohomology of algebraic and related finite groups in the above theorem h gl m denotes the rational cohomology of the algebraic group gl with coefficients in m the rational gl module obtained from the adjoint module m m through twisting by the r th power of the frobenius endomorphism on gl the algebra structure on the cohomology is induced from the associative algebra structure on m on the other hand we provide the following systematic determination of certain rational cohomology groups h g v r where v r is the rational gmodule obtained from the rational g module v by applying the r th frobenius twist see 1 2 this is a somewhat sharpened version of theorem 4 of 36

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

modular representations of the ortho symplectic supergroups a chevalley type integral basis for the ortho symplectic lie superalgebra is constructed the simple modules of the ortho symplectic supergroup over an algebraically closed field of prime characteristic not equal to 2 are classified where a key combinatorial component comes from the mullineux conjecture on modular representations of the symmetric group a steinberg tensor product theorem for the ortho symplectic supergroup is also obtained

universal deformation rings and fusion abstract we study in this paper the extent to which one can detect fusion in certain finite groups from information about the universal deformation rings r v of absolutely irreducible f p modules v the we consider are extensions of either abelian or dihedral groups g of order prime to p by an elementary abelian p group of rank 2

the avrunin and scott theorem and a truncated polynomial algebra we prove an analogue of a theorem of avrunin and scott for truncated polynomial algebras m k x 1 x m x 2 i over an algebraically closed field of arbitrary characteristic the avrunin and scott theorem relates the support variety for a finite dimensional ke module to its rank variety where char k p and e is an elementary abelian p group the analogue of the avrunin and scott theorem relates the support variety for a finite dimensional m module using hochschild cohomology to its rank variety developed in k erdmann m holloway rank varieties and projectivity for a class of local algebras math z 247 2004 441 460 using clifford algebras along the way to proving our main result we provide a new proof of the avrunin and scott theorem for elementary abelian p group algebras which we are then able to generalise to the setting of m algebras

the deligne simpson problem a survey abstract the deligne simpson problem dsp respectively the weak dsp is formulated like this give necessary and sufficient conditions for the choice of the conjugacy classes c j gl n c or c j gl n c so that there exist irreducible respectively with trivial centralizer p 1 tuples of matrices m j c j or a j c j satisfying the equality m 1 m p 1 i or a 1 a p 1 0 the matrices m j and a j are interpreted as monodromy operators of regular linear systems and as matrices residua of fuchsian ones on riemann s sphere the present paper offers a survey of the results known up to now concerning the dsp

fake congruence modular curves and subgroups of the modular group abstract we construct a special class of noncongruence modular subgroups and curves analogous in some ways to the usual congruence ones the subgroups are obtained via the burau representation and the associated quotient curves have a natural moduli space interpretation in fact they are reduced hurwitz spaces corresponding to covers with 4 branch points and monodromy group equal to semi direct products of a cyclic and an abelian group furthermore they form a modular tower in the sense of fried we study representations on the cohomology of these fake congruence modular curves and also calculate the genera of certain quotient curves

cross products by braided groups and bosonization let h be a braided cocommutative hopf algebra in a braided monoidal category c and b a hopf algebra in c on which h acts we construct a cross product hopf algebra b formula h in c as an application we show that every b in a certain class can be converted to an equivalent ordinary hopf algebra by a process of bosonization the class includes for example all super hopf algebras the constructions respect any quasitriangular structures on b as a corollary we show that if h r is an ordinary quasitriangular hopf algebra then the smash product by the adjoint action of h on itself had formula h can be given the structure of a hopf algebra we prove a similar result for any hopf algebra to which a quasitriangular hopf algebra maps

complexes and vanishing of cohomology for group schemes abstract completely reducible homology complexes were originally introduced in the context of finite groups and used to study the question of vanishing of cohomology in this paper we study these complexes and the vanishing of cohomology for arbitrary finite dimensional cocommutative hopf algebras applications are later provided for infinitesimal group schemes of reductive and solvable algebraic groups

modular representations of the supergroup q n i the representation theory of the algebraic supergroup q n has been studied quite intensively over the complex numbers in recent years especially by penkov and serganova 18 19 20 culminating in their solution 21 22 of the problem of computing the characters of all irreducible finite dimensional representations ofq n the characters of one important family of irreducible representations the so called polynomial representations had been determined earlier by sergeev 24 exploiting an analogue of schur weyl duality connecting polynomial representations of q n to the representation theory of the double covers n of the symmetric groups in 2 we used sergeev s ideas to classify for the first time the irreducible representations of n over fields of positive characteristic p 2 in the present article and its sequel we begin a systematic study of the representation theory of q n in positive characteristic motivated by its close relationship to n let us briefly summarize the main facts proved in this article by purely algebraic techniques let g q n defined over an algebraically closed field k of characteristic p 6 2 see 2 3 for the precise definition in 4 we construct the superalgebra dist g of distributions on g by reduction modulo p from a kostant z form for the enveloping superalgebra of the lie superalgebra q n c this provides one of the main tools in the remainder of the paper there is an explicit equivalence between the category of representations of g and the category of integrable dist g supermodules see corollary 5 7 in 6 we classify the irreducible representations of g by highest weight theory they turn out to be parametrized by the set x p n 1 n z 1 n with i i 1 only if p i for x p n the corresponding irreducible representation is denoted l and is constructed naturally as the simple socle of an induced representation h indbu where b is a borel subgroup of g and u is a certain irreducible representation of b of dimension a power of 2 the main difficulty here is to show that h0 6 0 for x p n which we prove by exploiting the main result of 2 classifying the irreducible polynomial representations of g so ultimately the proof that h0 6 0 depends on a counting argument involving p regular conjugacy classes in n

infinitesimal 1 parameter subgroups and cohomology this is the first of two papers in which we determine the spectrum of the cohomology algebra of infinitesimal group schemes over a field k of characteristic p 0 whereas sfb is concerned with detection of cohomology classes the present paper introduces the graded algebra k vr g of functions on the scheme of infinitesimal 1 parameter subgroups of height r on an affine group scheme g and demonstrates that this algebra is essentially a retract of h g k provided that g is an infinitesimal group scheme of height kr this work is a continuation of f s in which the existence of certain universal extension classes was established thereby enabling the proof of finite generation of h g k for any finite group scheme g over k the role of the scheme of infinitesimal 1 parameter subgroups of g was foreshadowed in f p where h g 1 k was shown to be isomorphic to the coordinate algebra of the scheme of p nilpotent elements of g lie g for g a smooth reductive group g 1 the first frobenius kernel of g and p char k sufficiently large indeed p nilpotent elements of g correspond precisely to infinitesimal 1 parameter subgroups of g 1 much of our effort in this present paper involves the analysis of the restriction of the universal extension classes to infinitesimal 1 parameter subgroups in 1 we construct the affine scheme vr g of homomorphisms from ga r to g which we call the scheme of infinitesimal 1 parameter subgroups of height r in g in the special case r 1 this is the scheme of p nilpotent elements of the p restricted lie algebra lie g for various classical groups g vr g is the scheme of r tuples of p nilpotent pairwise commuting elements of lie g more generally an embedding g c gln determines a closed embedding of vr g into the scheme of r tuples of p nilpotent pairwise commuting elements of gl1 lie gln the relationship between k vr g the coordinate algebra of vr g and h g k is introduced in theorem 1 14 a natural homomorphism of graded k algebras

cohomology and support varieties for lie superalgebras unlike lie algebras the finite dimensional complex representations of a simple lie superalgebra are usually not semisimple as a consequence despite over thirty years of study these remain mysterious objects in this paper we introduce a new tool the notion of cohomological support varieties for the finite dimensional supermodules for a classical lie superalgebra g g 0 g 1 which are completely reducible over g 0 they allow us to provide a new functorial description of the previously combinatorial notions of defect and atypicality we also introduce the detecting subalgebra of g its role is analogous to the defect subgroup in the theory of finite groups in positive characteristic using invariant theory we prove that there are close connections between the cohomology and support varieties of g and the detecting subalgebra

the structure of hopf algebras with a projection the smash product algebra and the smash coproduct coalgebra are well known in the context of hopf algebras 1 5 and these notions can be viewed as being motivated by the semidirect product construction in the theory of groups and in the theory of afiine group schemes respectively let

computations of sheaves associated to the representation theory of sl2 abstract we explicitly compute examples of sheaves over the projectivization of the spectrum of the cohomology of sl 2 in particular we compute ker m for every indecomposable m and we compute f i m when m is an indecomposable weyl module and i p we also give a brief review of the classification of sl 2 modules and of the general theory of such sheaves in the case of a restricted lie algebra

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

pi supports for modules for finite group schemes we introduce the space g of equivalence classes of points of a finite group scheme g and associate a subspace g m to any g module m our results extend to arbitrary finite group schemes g over arbitrary fields k of positive characteristic and to arbitrarily large g modules the basic results about cohomological support varieties and their interpretation in terms of representation theory in particular we prove that the projectivity of any possibly infinite dimensional g module can be detected by its restriction along points of g unlike the cohomological support variety of a g module m the invariant m 7 g m satisfies good properties for all modules thereby enabling us to determine the thick tensor ideal subcategories of the stable module category of finite dimensional g modules finally using the stable module category of g we provide g with the structure of a ringed space which we show to be isomorphic to the scheme projh g k

support varieties for modules over chevalley groups and classical lie algebras let g be a connected reductive algebraic group over an algebraically closed field of characteristic p 0 g 1 be the first frobenius kernel and g f p be the corresponding finite chevalley group let m be a rational g module in this paper we relate the support variety of m over the first frobenius kernel with the support variety of m over the group algebra kg f p this provides an answer to a question of parshall applications of our new techniques are presented which allow us to extend results of alperin mason and janiszczak jantzen and to calculate the dimensions of support varieties for finite chevalley groups

rigid generators of classical groups we introduce the notion of thompson tuples and show they yield rigid generating systems of length 3 of certain classical groups we also give a new construction method for belyi triples and use this to obtain galois realizations over q for all orthogonal and symplectic groups over the field f2 let g1 gr be generators of a finite group g with g1 gr 1 and gi 6 z g the center of g we say they form a quasi rigid generating system if for any generators g 1 g r of g with g 1 g r 1 such that g i is conjugate gi for all i there is g g with g i g 1gig for all i if additionally z g 1 then the system is called rigid we say the system is rational if for each integer m prime to g there is a permutation of r letters such that gm i is conjugate g i for all i one version of the standard rigidity criterion e g v thm 3 25 says if g has a quasi rigid and rational system of r generators then g z g occurs as the galois group of a regular extension of q x with r branch points

support varieties for the steenrod algebra in recent years the spectrum of the cohomology ring has been a fundamental object of interest in the study of modular group cohomology quillen q1 2 proved that this variety can be expressed as a disjoint union of locally closed varieties involving elementary abelian subgroups avrunin and scott as later generalized quillen s result for varieties associated to modules defined by carlson c1 for restricted lie algebra cohomology no such stratification theorem exists friedlander and parshall fp1 2 have shown that the cohomological varieties in this context can be realized as subvarieties of the given lie algebra when viewed as an affine space fix a prime p and let a be the mod p steenrod algebra the cohomology of a exta fp fp is an object of much interest in algebraic topology one way to study the cohomology ring is to approximate a by finite dimensional sub hopf algebras and then try to compute their cohomology rings by a theorem of wilkerson w the cohomology of any of these finite dimensional hopf algebras is a finitely generated commutative fp algebra hence one can get a great deal of information as in the case of groups and lie algebras by studying the spectrum of this ring one of the goals of this paper will be to show how the aforementioned results for finite groups and restricted lie algebras can be used to study the cohomological varieties for finite dimensional subhopf algebras of the steenrod algebra given a finite dimensional sub hopf algebra b of a let b fp be the spectrum of extb fp fp for a finite dimensional b module m the

classical groups probabilistic methods and the 2 3 generation problem we study the probability that randomly chosen elements of prescribed type in a finite simple classical group g generate g in particular we prove a conjecture of kantor and lubotzky in this area the probabilistic approach is then used to determine the finite simple classical quotients of the modular group psl2 z up to finitely many exceptions