small linearly equivalent g sets and a construction of beaulieu abstract two g sets g a finite group are called linearly equivalent over a commutative ring k if the permutation representations k x and k y are isomorphic as modules over the group algebra kg pairs of linearly equivalent non isomorphic g sets have applications in number theory and geometry we characterize the groups g for which such pairs exist for any field and give a simple construction of these pairs if k is q these are precisely the non cyclic groups for any non cyclic group we prove that there exist g sets which are non isomorphic and linearly equivalent over q of cardinality 3 g 2 also we investigate a construction of p beaulieu which allows us to construct pairs of transitive linearly equivalent s n sets from arbitrary g sets for an arbitrary group g we show that this construction works over all fields and use it construct for each finite set p of primes s n sets linearly equivalent over a field k if and only if the characteristic of k lies in p

equivariant stable homotopy and segal s burnside ring conjecture analogous results were proved later for ko in the generality of compact lie groups by atiyah and segal 8 and for kfq the algebraic k theory spectrum associated to the finite field fq by rector 29 using quillen s 26 computation of it kfq in each case the answer involves an appropriate completed representation ring of g and the cohomology theory in question is constructed from the permutative category of finite dimensional vector spaces over a field see 31 if one considers cohomology theories constructed from other permutative categories one expects to find analogous computations in terms of a completed representation ring of g in the given category appropriately defined in particular stable cohomotopy ts is constructed from the category of finite sets 31 and for this category the analogue of the representation ring is a well known object the burnside ring a g 13 a g is a commutative ring with augmentation so one may speak of a g the completed burnside ring moreover there is

solving fermat type equations via modular ℚ curves over polyquadratic fields we solve the diophantine equations x 4 dy 2 z p for d 2 and d 3 and any prime p 349 and p 131 respectively the method consists in generalizing the ideas applied by frey ribet and wiles in the solution of fermat s last theorem and by ellenberg in the solution of the equation x 4 y 2 z p and we use q curves modular forms and inner twists in principle our method can be applied to solve this type of equations for other values of d

embedding problems over abelian groups and an application to elliptic curves conditions for the solvability of certain embedding problems can be given in terms of the existence of elements with certain norm properties a classical example due to witt 1936 crelle s j 174 237 245 is that of embedding a klein extension into a dihedral extension in construction de p extensions galoisiennes d un corps de caracteristique differente de p 1987 j algebra109 508 535 massy finds this type of condition for central p extensions of an abelian group of exponent p in this paper we generalize the results of massy to the case of central p extensions of any abelian group in the last section we discuss the reason for our interest in these problems they appear in the theory of elliptic q curves if one is interested in computing representatives with especially good arithmetic properties in the isogeny class of a given curve

towers and injective cohomology algebras since carlsson s paper 5 and the work of lannes and zarati 9 topo1ogists have learned to recognize many spaces y so h y is injective in u for example if g is a finite group with p sylow subgroup z p kn the h bg is injective further infinite examples include eb z p k and any of the many wedge summands of these spaces a complete classification of u injectives is given in 18 and from this it can be seen that the only such injectives which are not a unbounded are the brown gitler modules these have been adequately discussed elsewhere see 6 among many an immediate consequence of theorem a will be the following result suppose y satisfies the same hypotheses as in theorem a

q curves and their manin ideals let c be an elliptic curve over q and let denote its neron differential uniquely determined up to sign according to a conjecture of manin stevens there is a modular parametrization x 1 n c defined over q such that the so called manin constant c is equal to 1 the constant c is defined by the equality c f q dq q where f is a normalized newform of weight 2 on 1 n in this paper we propose a generalization of the manin constant as a certain ideal we call it the manin ideal attached to modular parametrizations of elliptic curves defined over number fields we conjecture that its value is 1 generalizing manin stevens and show the manin ideal to be an integral ideal involving only primes dividing twice the conductor motivated by the manin ideal considerations we first study several aspects of building blocks of modular abelian varieties and modular parametrizations of q curves some examples are included to provide numerical evidence of the generalized conjecture and the paper also contains the analogous items on the manin ideals for modular building blocks of higher dimension

companion forms and explicit computation of pgl2 number fields with very little ramification abstract in previous works 18 and 19 we described algorithms to compute the number field cut out by the mod l representation attached to a modular form of level n 1 in this article we explain how these algorithms can be generalised to forms of higher level n as an application we compute the galois representations attached to a few forms which are supersingular or admit a companion mod l with l 13 and l 41 and we obtain previously unknown number fields of degree l 1 whose galois closure has galois group pgl 2 f l and a root discriminant that is so small that it beats records for such number fields finally we give a formula to predict the discriminant of the fields obtained by this method and we use it to find other interesting examples which are unfortunately out of our computational reach

class numbers of imaginary quadratic fields the classical class number problem of gauss asks for a classification of all imaginary quadratic fields with a given class number n the first complete results were for n 1 by heegner baker and stark after the work of goldfeld and gross zagier the task was a finite decision problem for any n indeed after oesterle handled n 3 in 1985 serre wrote no doubt the same method will work for other small class numbers up to 100 say however more than ten years later after doing n 5 6 7 wagner remarked that the n 8 case seemed impregnable we complete the classification for all n 100 an improvement of four powers of 2 arguably the most difficult case over the previous best results the main theoretical technique is a modification of the goldfeld oesterle work which used an elliptic curve l function with an order 3 zero at the central critical point to instead consider dirichlet l functions with low height zeros near the real line though the former is still required in our proof this is numerically much superior to the previous method which relied on work of montgomery weinberger our method is still quite computer intensive but we are able to keep the time needed for the computation down to about seven months in all cases we find that there is no abnormally large exceptional modulus of small class number which agrees with the prediction of the generalised riemann hypothesis

the cohomology of the steenrod algebra and the mod p lannes zarati homomorphism abstract for any pointed space x the mod p lannes zarati homomorphism s h x is considered as a graded associated version of the mod p hurewicz map h q x h q x in the e 2 term of the adams spectral sequence in this paper we investigate the behavior of 3 f p and s h b z p s 1 for an odd prime p

certification of modular galois representations we show how the output of the algorithm to compute modular galois representations described in our previous article can be certified we have used this process to compute certified tables of such galois representations obtained thanks to an improved version of this algorithm including representations modulo primes up to 31 and representations attached to a newform with non rational but of course algebraic coefficients which had never been done before these computations take place in the jacobian of modular curves of genus up to 26 the resulting data are available on the author s webpage

liftings of projective 2 dimensional galois representations and embedding problems abstract in this paper we study liftings of 2 dimensional projective representations of the absolute galois group of a field k these liftings are related to solutions of certain galois embedding problems for representations with image isomorphic to one of the groups psl 2 q or pgl 2 q with q 3 mod 8 or to one of the groups psl 2 7 or psl 2 9 and k of characteristic different from 2 we compute the obstruction to the existence of a lifting of index 2 theorems 3 6 and 3 7 to make this computation we consider the natural representations of these groups as permutation groups and then apply serre s formula on the witt invariant of tr x2 given by serre comment math helv 59 1984 651 676 we also obtain from crespo c r acad sci paris315 1992 625 628 c4 extensions of sn as galois groups preprint barcelona 1993 central extensions of the alternating group as galois groups preprint barcelona 1993 galois realization of central extensions of the symmetric group with kernel a cyclic 2 group preprint barcelona 1993 a criterion for the existence of liftings of index 4 and 8 theorem 3 8 in the case k q we exploit a result by tate to obtain an easy computable criterion for the existence of a lifting with any given index proposition 4 1 and corollaries the same kind of criterion can be used for other embedding problems over q in theorem 4 4 we give a criterion for the existence of a solution to some embedding problems over the symmetric group section 1 contains preliminaries on projective representations and their liftings in section 2 we classify the cyclic central extensions of the groups psl 2 q and pgl 2 q and establish the relation to liftings of the projective representations we are interested in in section 3 we find which degree 2 extensions of the symmetric and alternating group correspond to each degree 2 extension of psl 2 q and pgl 2 q under the natural permutation representations and then apply serre s formula section 4 is devoted to the case k q in section 5 some numerical examples are given

finiteness obstructions and euler characteristics of categories we introduce notions of finiteness obstruction euler characteristic l2 euler characteristic and mobius inversion for wide classes of categories the finiteness obstruction of a category of type fpr is a class in the projective class group k0 r the functorial euler characteristic and functorial l2 euler characteristic are respectively its r rank and l2 rank we also extend the second author s k theoretic mobius inversion from finite categories to quasi finite categories our main example is the proper orbit category for which these invariants are established notions in the geometry and topology of classifying spaces for proper group actions baez and dolan s groupoid cardinality and leinster s euler characteristic are special cases of the l2 euler characteristic some of leinster s results on mobius rota inversion are special cases of the k theoretic mobius inversion

idempotent ideals and nonfinitely generated projective modules over integral group rings of polycyclic by finite groups abstract we prove that every nonfinitely generated projective module over the integral group ring of a polycyclic by finite group g is free if and only if g is polycyclic

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

the fermat type equations 2 or 3 solved through ℚ curves we solve the diophantine equations x 5 y 5 dz p with d 2 3 for a set of prime numbers of density 1 4 1 2 respectively the method consists in relating a possible solution to another diophantine equation and solving the later by using q curves and a generalized modular technique as in work of ellenberg and dieulefait jimenez along with some new techniques for eliminating newforms

compatible systems of symplectic galois representations and the inverse galois problem iii automorphic construction of compatible systems with suitable local properties this article is the third and last part of a series of three articles about compatible systems of symplectic galois representations and applications to the inverse galois problem this part proves the following new result for the inverse galois problem for symplectic groups for any even positive integer n and any positive integer d mathrm psp n mathbb f ell d or mathrm pgsp n mathbb f ell d occurs as a galois group over the rational numbers for a positive density set of primes ell the result depends on some work of arthur s which is conditional but expected to become unconditional soon the result is obtained by showing the existence of a regular algebraic self dual cuspidal automorphic representation of hbox gl n mathbb a mathbb q with local types chosen so as to obtain a compatible system of galois representations to which the results from part ii of this series apply

on the mod p lannes zarati homomorphism abstract for any pointed space x the mod p lannes zarati homomorphism s h x is considered as a graded associated version of the mod p hurewicz map h q x h q x in the e 2 term of adams spectral sequence therefore the study of the mod p lannes zarati homomorphism plays an important role in the determination of the image of the mod p hurewicz map in this paper we investigate the behavior of s s f p s h s 0 s 2 for an odd prime p

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

on the peterson hit problem let pk f2 x1 x2 xk pk f2 x1 x2 xk be the polynomial algebra over the prime field of two elements f2f2 in k variables x1 x2 xkx1 x2 xk each of degree 1 we study the hit problem set up by f peterson of finding a minimal set of generators for pkpk as a module over the mod 2 steenrod algebra aa in this paper we study a minimal set of generators for aa module pkpk in some so called generic degrees and apply these results to explicitly determine the hit problem for k 4k 4

quadratic mathbb q curves units and hecke l values we show that if k is a quadratic field and if there exists a quadratic mathbb q curve e k of prime degree n satisfying weak conditions then any unit u of o k satisfies a congruence u r equiv 1 pmod n where r mathrm g c d n 1 12 if k is imaginary quadratic we prove a congruence modulo a divisor of n between an algebraic hecke character tilde psi and roughly speaking the elliptic curve we show that this divisor then occurs in a critical value l tilde psi 2 by constructing a non zero element in a selmer group and applying a theorem of kato

generators for the mod 2 cohomology of the steinberg summand of thom spectra over b z 2 n abstract let l n k be the steinberg summand of the ideal generated by the k th power of the top dickson class of h b z 2 n z 2 the module l n k is the mod 2 cohomology of the steinberg summand of a thom spectrum over the classifying space b z 2 n in this paper using the kameko map kameko 1990 6 and short exact sequences induced by the cofibration sequences constructed by takayasu 1999 15 we construct a minimal generating set for l n k as a module over the mod 2 steenrod algebra this generalises the result of masateru inoue 2002 5 for the cases k 0 1

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

twisting commutative algebraic groups abstract if v is a commutative algebraic group over a field k o is a commutative ring that acts on v and i is a finitely generated free o module with a right action of the absolute galois group of k then there is a commutative algebraic group i o v over k which is a twist of a power of v these group varieties have applications to cryptography in the cases of abelian varieties and algebraic tori over finite fields and to the arithmetic of abelian varieties over number fields for purposes of such applications we devote this article to making explicit this tensor product construction and its basic properties

erratum to spherical classes and the algebraic transfer see 6 and have shown that these maps correspond to an associated graded of the hurewicz homomorphism the proof of this assertion is unpublished but it is sketched by j e lannes 5 and by p g goerss 1 the hopf invariant 1 and the kervaire invariant 1 classes are respectively represented by certain permanent cycles in ext4 f2 f2 and ext f2 f2 on which pl and 02 are non zero therefore we are led to the following conjecture

l 2 betti numbers and plancherel measure we compute l 2 betti numbers of postliminal locally compact unimodular groups in terms of ordinary dimensions of reduced cohomology with coefficients in irreducible unitary representations and the plancherel measure this allows us to compute the l 2 betti numbers for semi simple lie groups with finite center simple algebraic groups over local fields and automorphism groups of locally finite trees acting transitively on the boundary c 2013 elsevier inc all rights reserved

dimension theory of arbitrary modules over finite von neumann algebras and l 2 betti numbers ii applications to grothendieck groups l 2 euler characteristics and burnside groups

finiteness results for modular curves of genus at least 2 a curve x over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i is modular if it is dominated by x 1 n for some n if in addition the image of its jacobian in j 1 n is contained in the new subvariety of j 1 n then x is called a new modular curve we prove that for each g 2 the set of new modular curves over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i of genus g is finite and computable for the computability result we prove an algorithmic version of the de franchis severi theorem similar finiteness results are proved for new modular curves of bounded gonality for new modular curves whose jacobian is a quotient of j 0 n new with n divisible by a prescribed prime and for modular curves new or not with levels in a restricted set we study new modular hyperelliptic curves in detail in particular we find all new modular curves of genus 2 explicitly and construct what might be the complete list of all new modular hyperelliptic curves of all genera finally we prove that for each field k of characteristic zero and g 2 the set of genus g curves over k dominated by a fermat curve is finite and computable

spherical classes and the algebraic transfer we study a weak form of the classical conjecture which predicts that there are no spherical classes in qos0 except the elements of hopf invariant one and those of kervaire invariant one the weak conjecture is obtained by restricting the hurewicz homomorphism to the homotopy classes which are detected by the algebraic transfer let pk f2 xl xkl with lxii 1 the general linear group glk gl k f2 and the mod 2 steenrod algebra a act on pk in the usual manner we prove that the weak conjecture is equivalent to the following one the canonical homomorphism jk if2 9 plk if2 0 pk glk induced by the a 4 identity map on pk is zero in positive dimensions for k 2 in other words every dickson invariant i e element of p k of positive dimension belongs to a pk for k 2 where a denotes the augmentation ideal of a this conjecture is proved for k 3 in two different ways one of these two ways is to study the squaring operation sqo on p f2 09 pk the range of jk and glk to show it commuting through jk with kameko s sqo on f2 0 p pk the glk domain of jk we compute explicitly the action of sqo on p f2 09 pk for glk k 4

the hit problem for the modular invariants of linear groups abstract let the mod 2 steenrod algebra a and the general linear group glk gl k f 2 act on pk f 2 x1 xk with deg xi 1 in the usual manner we prove that for a family of some rather small subgroups g of glk every element of positive degree in the invariant algebra pkg is hit by a in pk in other words pkg a pk where pkg and a denote respectively the submodules of pkg and a consisting of all elements of positive degree this family contains most of the parabolic subgroups of glk it should be noted that the smaller the group g is the harder the problem turns out to be remarkably when g is the smallest group of the family the invariant algebra pkg is a polynomial algebra in k variables whose degrees are 8 and fixed while k increases it has been shown by hu ng trans amer math soc 349 1997 3893 3910 that for g glk the inclusion pkglk a pk is equivalent to a weak algebraic version of the long standing conjecture stating that the only spherical classes in q0s0 are the elements of hopf invariant 1 and those of kervaire invariant 1

the amenability of affine algebras abstract we introduce the notion of amenability for affine algebras we characterize amenability by folner sequences paradoxicality and the existence of finitely invariant dimension measures then we extend the results of rowen on ranks from affine algebras of subexponential growth to amenable affine algebras

the connes embedding property for quantum group von neumann algebras for a compact quantum group g of kac type we study the existence of a haar trace preserving embedding of the von neumann algebra l g into an ultrapower of the hyperfinite ii 1 factor the connes embedding property for l g we establish a connection between the connes embedding property for l g and the structure of certain quantum subgroups of g and use this to prove that the ii 1 factors l o n and l u n associated to the free orthogonal and free unitary quantum groups have the connes embedding property for all n 4 as an application we deduce that the free entropy dimension of the standard generators of l o n equals 1 for all n 4 we also mention an application of our work to the problem of classifying the quantum subgroups of o n

on the behavior of the algebraic transfer let trk f 2 ph i bv k ext k k i a f 2 f 2 be the alge gl k braic transfer which is defined by w singer as an algebraic version of the geometrical transfer tr k s bv k s s 0 it has been shown that the algebraic transfer is highly nontrivial and more precisely that tr k is an isomorphism for k 1 2 3 however singer showed that tr 5 is not an epimorphism in this paper we prove that tr 4 does not detect the nonzero element g s ext 4 12 2s a f 2 f 2 for every s 1 as a consequence the localized sq 0 1 tr 4 given by inverting the squaring operation sq is not an epimorphism this gives a negative answer to a prediction by minami

sheared algebra maps and operation bialgebras for mod 2 homology and cohomology the mod 2 steenrod algebraa and dyer lashof al gebra r have both striking similarities and dierences arising from their common origins in lower indexed algebraic operations these algebraic operations and their relations generate a bigraded bialgebrak whose module actions are equivalent to but quite dif ferent from those ofa andr the exact relationships emerge as sheared algebra bijections which also illuminate the role of the cohomology ofk as a bialgebra k has a particularly attractive and potentially useful structure providing a bridge between those ofa andr and suggesting possible applications to the miller spectral sequence and thea structure of dickson algebras

the a decomposability of the singer construction abstract let r s m denote the singer construction on an unstable module m over the steenrod algebra a at the prime two r s m is canonically a subobject of p s m where p s f 2 x 1 x s with generators of degree one and f 2 is the field with two elements passage to a indecomposables gives the natural transformation r s m f 2 a p s m which identifies with the dual of the composition of the singer transfer and the lannes zarati homomorphism the main result of the paper proves the weak generalized algebraic spherical class conjecture which was proposed by the first author namely this morphism is trivial on elements of positive degree when s 2 the condition s 2 is necessary as exhibited by the spherical classes of hopf invariant one and those of kervaire invariant one

constructing hilbert modular forms without exceptional primes in this paper we construct families of hilbert modular newforms without exceptional primes this is achieved by generalizing the notion of good dihedral primes introduced by khare and wintenberger in their proof of serre s modularity conjecture to totally real fields

the relative burnside module and the stable maps between classifying spaces of compact lie groups tom dieck s burnside ring of compact lie groups is generalized to the relative case for any g d n a compact lie group and its normal subgroup a g n is defined to be an appropriate set of the equivalence classes of compact g enr s with free n action in such a way that i a g d n 7g n so b n g where b n g is the classifying space of principal n g bundle under the product situation i e g f x k n k a f x k d k is also denoted by a f k as it turns out to be the usual a f k when both f and k are finite then a couple of applications are given to the study of stable maps between classifying spaces of compact lie groups a conceptual proof of feshbach s double coset formula and a density theorem on the map ap a l h a a bl bh p for any compact lie groups l k when p is odd some restriction is applied to l when p 2 this latter result may be regarded as the pushout of feshbach s density theorem and the theorem of may snaith zelewski over the celebrated carlsson solution of segal s burnside ring conjecture 0 introduction even before the arrival of carlsson s celebrated paper c the segal conjecture for finite groups was known to imply the complete calculation of the stable maps between classifying spaces of finite groups agm lmm more recently snaith s2 got interested in the stable maps between classifying spaces of compact lie groups while studying his explicit brauer induction problem see s2 which is a nice reference on this topic snaith s research led to the further generalization of the segal conjecture by msz theorem a msz let f and j be finite groups and let k be a compact lie group then a induces the isomorphism a a f x j k a j fxj bfj bfk f where a f x j k is the grothendieck a f x j module of principal f x j k bundles over finite f x j sets and aj fxj is the completion of received by the editors june 23 1993 1991 mathematics subject classification primary 55p42 55r35 57s15 secondary 55q91 55r9 1

the irreducibility of some level 1 hecke polynomials let t p k x be the characteristic polynomial of the hecke operator tp acting on the space of level 1 cusp forms s k 1 we show that t p k x is irreducible and has full galois group over q for k 2000 and p 2000 p prime

on modular forms and the inverse galois problem in this article new cases of the inverse galois problem are established the main result is that for a fixed integer n there is a positive density set of primes p such that psl2 fpn occurs as the galois group of some finite extension of the rational nu mbers these groups are obtained as projective images of residual modular galois representations moreover families of modular forms are constructed such that the images of all their resid ual galois representations are as large as a priori possible both results essentially use khare s a nd wintenberger s notion of gooddihedral primes particular care is taken in order to exclud e nontrivial inner twists 2000 mathematics subject classification 11f80 primary 12f12 11f11

the negative answer to kameko s conjecture on the hit problem abstract the goal of this paper is to give a negative answer to kameko s conjecture on the hit problem stating that the cardinal of a minimal set of generators for the polynomial algebra p k considered as a module over the steenrod algebra a is dominated by an explicit quantity depending on the number of the polynomial algebra s variables k the conjecture was shown by kameko himself for k 3 in his phd thesis in the johns hopkins university in 1990 and recently proved by us and kameko for k 4 however we claim that it turns out to be wrong for any k 4 in order to deny kameko s conjecture we study a minimal set of generators for a module p k in some so call generic degrees what we mean by generic degrees is a bit different from that of other authors in the fields such as crabb hubbuck nam repka selick wood we prove an inductive formula for the cardinal of the minimal set of generators in these generic degrees when the number of the variables k increases as an immediate consequence of this inductive formula we recognize that kameko s conjecture is no longer true for any k 4

compatible systems of symplectic galois representations and the inverse galois problem i images of projective representations this article is the first part of a series of three articles about compatible systems of symplectic galois representations and applications to the inverse galois problem in this first part we determine the smallest field over which the projectivisation of a given symplectic group representation satisfying some natural conditions can be defined the answer only depends on inner twists we apply this to the residual representations of a compatible system of symplectic galois representations satisfying some mild hypothesis and obtain precise information on their projective images for almost all members of the system under the assumption of huge residual images by which we mean that a symplectic group of full dimension over the prime field is contained up to conjugation finally we obtain an application to the inverse galois problem msc 2010 11f80 galois representations 20c25 projective representations and multipliers 12f12 inverse galois theory

on arithmetically equivalent fields with distinct p class numbers we show that for each odd prime number p two number fields with the same zeta function but distinct p class numbers have degree at least 2p 2 moreover two such number fields of degree 2p 2 have a common galois closure with galois group gl2 fp f 2 p

the weak conjecture on spherical classes let mathcal a be the mod 2 steenrod algebra we construct a chain level representation of the dual of singer s algebraic transfer which maps singer s invariant theoretic model of the dual of the lambda algebra gamma k wedge to bf f 2 x 1 pm 1 ldots x k pm 1 and is the inclusion of the dickson algebra d k subset gamma k wedge into bf f 2 x 1 ldots x k this chain level representation allows us to confirm the weak conjecture on spherical classes see 9 assuming the truth of 1 either the conjecture that the dickson invariants of at least k 3 variables are homologically zero in tor k mathcal a bf f 2 bf f 2 2 or a conjecture on mathcal a decomposability of the dickson algebra in gamma k wedge we prove the conjecture in item 1 for k 3 and also show a weak form of the conjecture in item 2

the cohomology of the steenrod algebra and representations of the general linear groups let tr k be the algebraic transfer that maps from the coinvariants of certain gl k representations to the cohomology of the steenrod algebra this transfer was defined by w singer as an algebraic version of the geometrical transfer tr k s bv k s s 0 it has been shown that the algebraic transfer is highly nontrivial more precisely that tr k is an isomorphism for k 1 2 3 and that tr k tr k is a homomorphism of algebras in this paper we first recognize the phenomenon that if we start from any degree d and apply sq repeatedly at most k 2 times then we get into the region in which all the iterated squaring operations are isomorphisms on the coinvariants of the gl k representations as a consequence every finite sq 0 family in the coinvariants has at most k 2 nonzero elements two applications are exploited the first main theorem is that tr k is not a n isomorphism for k 5 furthermore for every k 5 there are infinitely many degrees in which tr k is not an isomorphism we also show that if tr detects a nonzero element in certain degrees of ker sq 0 then it is not a monomorphism and further for each k tr k is not a monomorphism in infinitely many degrees the second main theorem is that the elements of any sq 0 family in the cohomology of the steenrod algebra except at most its first k 2 elements are either all detected or all not detected by tr k for every k applications of this study to the cases k 4 and 5 show that tr 4 does not detect the three families g d 3 and p and that tr 5 does not detect the family h n 1g n n 1

curves of genus 2 with group of automorphisms isomorphic to d_8 or d_ 12 the classification of curves of genus 2 over an algebraically closed field was studied by clebsch and bolza using invariants of binary sextic forms and completed by igusa with the computation of the corresponding three dimensional moduli variety m 2 the locus of curves with group of automorphisms isomorphic to one of the dihedral groups d 8 or d 12 is a one dimensional subvariety in this paper we classify these curves over an arbitrary perfect field k of characteristic chark 2 in the ds case and char k 2 3 in the d 12 case we first parameterize the k isomorphism classes of curves defined over k by the k rational points of a quasi affine one dimensional subvariety of m 2 then for every curve c k representing a point in that variety we compute all of its k twists which is equivalent to the computation of the cohomology set h 1 g k aut c the classification is always performed by explicitly describing the objects involved the curves are given by hyperelliptic models and their groups of automorphisms represented as subgroups of gl 2 k in particular we give two generic hyperelliptic equations depending on several parameters of k that by specialization produce all curves in every k isomorphism class