Modular Form Congruences and Selmer Groups

Motivated by Cremona and Mazur's notion of visibility of elementsin Shafarevich–Tate groups [6, 27], there have been anumber of recent works which test its compatibility with theBirch and Swinnerton–Dyer conjecture and the Bloch–Katoconjecture. These conjectures provide formulas for the ordersof Shafarevich–Tate groups in terms of values of L-functions.For example, one may see recent work of Agashe, Dummigan, Steinand Watkins [1, 2, 10, 11]. In their examples, they find thatthe presence of visible elements agrees with the expected divisibilityproperties of the relevant L-values.     

ALPERIN'S CONJECTURE FOR ALGEBRAIC GROUPS

We prove analogues for reductive algebraic groups of some resultsfor finite groups due to Knörr and Robinson from ‘Someremarks on a conjecture of Alperin’, J. London Math. Soc(2) 39 (1989), 48–60, which play a central rôlein their reformulation of Alperin's conjecture for finite groups.     

An Invariant for Fraction Fields of McConnell Algebras

In their seminal work [1] on the fields of fractions of theenveloping algebra of an algebraic Lie algebra, Gel'fand andKirillov formulate the following conjecture. Assume that g isa finite-dimensional algebraic Lie algebra over a field of characteristiczero. Then D(g) is a Weyl skew-field over a purely transcendentalextension of the base field. They showed that neither the conjecture nor its negation holdsfor all non-algebraic algebras. In [2], A. Joseph gave a particularlyeasy non-algebraic counterexample devised by L. Makar-Limanov:this is a non-algebraic 5-dimensional solvable Lie algebra,providing a counterexample despite the fact that the centreis one-dimensional. Besides, he raised a question of generalizationof this method for any completely solvable Lie algebra. On the other hand, consider A(V, , ), the McConnell algebrafor the triple (V, , ) as defined in [4, 14.8.4] and below.McConnell in [3] described the completely prime quotients ofthe enveloping algebra of a solvable Lie algebra in terms ofA(V, , ), and found a complete set of invariants to separatethem. In [2], A. Joseph raised the question whether the fieldsof fractions of these McConnell algebras remain non-isomorphic.The purpose of this note is to extend the work of L. Makar-Limanovreported in [2, Section 6], and so provide an integer-valuedinvariant which, for McConnell algebras defined over Z, saysprecisely when this skew-field is isomorphic to a Weyl skew-field:this number has simply to be positive. This result thereforegives a large supply of skew-fields which ‘resemble’a Weyl skew-field very nearly, but nevertheless are not isomorphicto it. 1991 Mathematics Subject Classification 17B35.     

Burns' Equivariant Tamagawa Invariant T{Omega}loc(N/Q,1) for some Quaternion Fields

Inspired by the work of Bloch and Kato in [2], David Burns constructedseveral ‘equivariant Tamagawa invariants’ associatedto motives of number fields. These invariants lie in relativeK-groups of group-rings of Galois groups, and in [3] Burns gaveseveral conjectures (see Conjecture 3.1) about their values.In this paper I shall verify Burns' conjecture concerning theinvariant T^loc(N/Q,1) for some families of quaternion extensionsN/Q. Using the results of [9] I intend in a subsequent paperto verify Burns' conjecture for those families of quaternionfields which are not covered here.     

Locally Nilpotent p-Groups whose Proper Subgroups are Hypercentral or Nilpotent-by-Chernikov

Let G be a group and P be a property of groups. If every propersubgroup of G satisfies P but G itself does not satisfy it,then G is called a minimal non-P group. In this work we studylocally nilpotent minimal non-P groups, where P stands for ‘hypercentral’or ‘nilpotent-by-Chernikov’. In the first case weshow that if G is a minimal non-hypercentral Fitting group inwhich every proper subgroup is solvable, then G is solvable(see Theorem 1.1 below). This result generalizes [3, Theorem1]. In the second case we show that if every proper subgroupof G is nilpotent-by-Chernikov, then G is nilpotent-by-Chernikov(see Theorem 1.3 below). This settles a question which was consideredin [1–3, 10]. Recently in [9], the non-periodic case ofthe above question has been settled but the same work containsan assertion without proof about the periodic case. The main results of this paper are given below (see also [13]).     

The Ubiquity of Thompson's Group F in Groups of Piecewise Linear Homeomorphisms of the Unit Interval

In the 1960s, Richard J. Thompson introduced a triple of groupsF T G which, among them, supplied the first examples of infinite,finitely presented, simple groups [14] (see [6] for publisheddetails), a technique for constructing an elementary exampleof a finitely presented group with an unsolvable word problem[12], the universal obstruction to a problem in homotopy theory[8], and the first examples of torsion free groups of type FPand not of type FP [5]. In abstract measure theory, it has beensuggested by Geoghegan (see [3] or [9, Question 13]) that Fmight be a counterexample to the conjecture that any finitelypresented group with no non-cyclic free subgroup is amenable(admits a bounded, non-trivial, finitely additive measure onall subsets that is invariant under left multiplication). Recently,F has arisen in the theory of groups of diagrams over semigrouppresentations [10], and as the object of questions in the algebraof string rewriting systems [7]. For more extensive bibliographiesand more results on Thompson's groups and their generalizationssee [1, 4, 6]. A persistent peculiarity of Thompson's groups is their abilityto pop up in diverse areas of mathematics. This suggests thatthere might be something very natural about Thompson's groups.We support this idea by showing (Theorem 1.1 below) that PL_o(I),the group of piecewise linear (finitely many changes of slope),orientation-preserving, self-homeomorphisms of the unit interval,is riddled with copies of F: a very weak criterion implies thata subgroup of PL_o(I) must contain an isomorphic copy of F.     

Multiple Positive Solutions of Semilinear Differential Equations with Singularities

The existence of positive solutions of a second order differentialequation of the form z'+g(t)f(z)=0 (1.1) with the separated boundary conditions: z(0) – ßz'(0)= 0 and z(1)+z'(1) = 0 has proved to be important in physicsand applied mathematics. For example, the Thomas–Fermiequation, where f = z^3/2 and g = t^–1/2 (see [12, 13, 24]),so g has a singularity at 0, was developed in studies of atomicstructures (see for example, [24]) and atomic calculations [6].The separated boundary conditions are obtained from the usualThomas–Fermi boundary conditions by a change of variableand a normalization (see [22, 24]). The generalized Emden–Fowlerequation, where f = z^p, p > 0 and g is continuous (see [24,28]) arises in the fields of gas dynamics, nuclear physics,chemically reacting systems [28] and in the study of multipoletoroidal plasmas [4]. In most of these applications, the physicalinterest lies in the existence and uniqueness of positive solutions.     

On Fusion in Unipotent Blocks

The context of this note is as follows. One considers a connectedreductive group G and a Frobenius endomorphism F: G G definingG over a finite field of order q. One denotes by G^F the associated(finite) group of fixed points. Let l be a prime not dividing q. We are interested in the l-blocksof the finite group G^F. Such a block is called unipotent ifthere is a unipotent character (see, for instance, [6, Definition12.1]) among its representations in characteristic zero. Roughlyspeaking, it is believed that the study of arbitrary blocksof G^F might be reduced to unipotent blocks (see [2, Théorème2.3], [5, Remark 3.6]). In view of certain conjectures aboutblocks (see, for instance, [9]), it would be interesting tofurther reduce the study of unipotent blocks to the study ofprincipal blocks (blocks containing the trivial character).Our Theorem 7 is a step in that direction: we show that thelocal structure of any unipotent block of G^F is very close tothat of a principal block of a group of related type (notionof ‘control of fusion’, see [13, 49]). 1991 MathematicsSubject Classification 20Cxx.     

Weakly Type-Preserving Sequences and Strong Convergence

Jørgensen conjectured that if there are no new parabolics in the algebraic limit of a sequence of Kleinian groups then the sequence should converge strongly. We verify this conjecture in the case that the algebraic limit has non-empty domain of discontinuity. An immediate corollary under the same assumptions is that any sequence converging algebraically to a minimally parabolic limit converges strongly.     

Examples of Non-Homeomorphic Harmonic Maps Between Negatively Curved Manifolds

Let M and N be closed non-positively curved manifolds, and letf:MN be a smooth map. Results of Eells and Sampson [1] showthat f is homotopic to a harmonic map , and Hartman [6] showedthat this is unique when N is negatively curved and f_*(₁ M)is not cyclic. Lawson and Yau conjectured that if f:MN is ahomotopy equivalence, where M and N are negatively curved, thenthe unique harmonic map homotopic to f would be a diffeomorphism.Counterexamples to this conjecture appeared in [2], and laterin [7] and [5]. There remains the question of whether a ‘topological’Lawson–Yau conjecture holds. 1991 Mathematics SubjectClassification 53C20, 55P10, 57C25, 58E20.     

Beardon's Diophantine Equations and Non-Free Mobius Groups

Let µ be a real number. The Möbius group G_µis the matrix group generated by It is known that G_µ is free if |µ| 2 (see [1])or if µ is transcendental (see [3, 8]). Moreover, thereis a set of irrational algebraic numbers µ which is densein (–2, 2) and for which G_µ is non-free [2, p. 528].We may assume that µ > 0, and in this paper we considerrational µ in (0, 2). The following problem is difficult. Let G^nf denote the set of all rational numbers µ in (0,2) for which G_µ is non-free. In 1969 Lyndon and Ullman[8] proved that G^nf contains the elements of the forms p/(p²+ 1) and 1/(p + 1), where p = 1, 2, ..., and that if µ₀ G^nf, then µ₀/p G^nf for p = 1, 2, .... In 1993 Beardon[2] studied problem (P) by means of the words of the form A^rB^s A^t and A^r B^s A^t B^u A^v, and he obtained a sufficient conditionfor solvability of (P), included implicitly in [2, pp. 530–531],by means of the following Diophantine equations: 1991 Mathematics SubjectClassification 20E05, 20H20, 11D09.     

Splendid Derived Equivalences for Blocks of Finite Groups

A central issue in finite group modular representation theoryis the relationship between the p-local structure and the p-modularrepresentation theory of a given finite group. In [5], Brouéposes some startling conjectures. For example, he conjecturesthat if e is a p-block of a finite group G with abelian defectgroup D and if f is the Brauer correspondent block of e of thenormalizer, N_G(D), of D then e and f have equivalent derivedcategories over a complete discrete valuation ring with residuefield of characteristic p. Some evidence for this conjecturehas been obtained using an important Morita analog for derivedcategories of Rickard [11]. This result states that the existenceof a tilting complex is a necessary and sufficient conditionfor the equivalence of two derived categories. In [5], Brouéalso defines an equivalence on the character level between p-blockse and f of finite groups G and H that he calls a ‘perfectisometry’ and he demonstrates that it is a consequenceof a derived category equivalence between e and f. In [5], Brouéalso poses a corresponding perfect isometry conjecture betweena p-block e of a finite group G with an abelian defect groupD and its Brauer correspondent p-block f of N_G(D) and presentsseveral examples of this phenomena. Subsequent research hasprovided much more evidence for this character-level conjecture. In many known examples of a perfect isometry between p-blockse, f of finite groups G, H there are also perfect isometriesbetween p-blocks of p-local subgroups corresponding to e andf and these isometries are compatible in a precise sense. In[5], Broué calls such a family of compatible perfectisometries an ‘isotypy’. In [11], Rickard addresses the analogous question of defininga p-locally compatible family of derived equivalences. In thisimportant paper, he defines a ‘splendid tilting complex’for p-blocks e and f of finite groups G and H with a commonp-subgroup P. Then he demonstrates that if X is such a splendidtilting complex, if P is a Sylow p-subgroup of G and H and ifG and H have the same ‘p-local structure’, thenp-local splendid tilting complexes are obtained from X via theBrauer functor and ‘lifting’. Consequently, in thissituation, we obtain an isotypy when e and f are the principalblocks of G and H. Linckelmann [9] and Puig [10] have also obtained important resultsin this area. In this paper, we refine the methods and program of [11] toobtain variants of some of the results of [11] that have widerapplicability. Indeed, suppose that the blocks e and f of Gand H have a common defect group D. Suppose also that X is asplendid tilting complex for e and f and that the p-local structureof (say) H with respect to D is contained in that of G, thenthe Brauer functor, lifting and ‘cutting’ by blockindempotents applied to X yield local block tilting complexesand consequently an isotypy on the character level. Since thep-local structure containment hypothesis is satisfied, for example,when H is a subgroup of G (as is the case in Broué'sconjectures) our results extend the applicability of these ideasand methods.     

On the Group of Symplectic Matrices Over a Free Associative Algebra

Symplectic groups are well known as the groups of isometriesof a vector space with a non-singular bilinear alternating form.These notions can be extended by replacing the vector spaceby a module over a ring R, but if R is non-commutative, it willalso have to have an involution. We shall here be concernedwith symplectic groups over free associative algebras (witha suitably defined involution). It is known that the generallinear group GL_n over the free algebra is generated by the setof all elementary and diagonal matrices (see [1, Proposition2.8.2, p. 124]). Our object here is to prove that the symplecticgroup over the free algebra is generated by the set of all elementarysymplectic matrices. For the lowest order this result was obtainedin [4]; the general case is rather more involved. It makes useof the notion of transduction (see [1, 2.4, p. 105]). When thereis only a single variable over a field, the free algebra reducesto the polynomial ring and the weak algorithm becomes the familiardivision algorithm. In that case the result has been provedin [3, Anhang 5].     

Kazhdan-Lusztig Cells, q-Schur Algebras and James' Conjecture

We consider the Dipper–James q-Schur algebra S_q(n, r)_k,defined over a field k and with parameter q 0. An understandingof the representation theory of this algebra is of considerableinterest in the representation theory of finite groups of Lietype and quantum groups; see, for example, [6] and [11]. Itis known that S_q(n, r)_k is a semisimple algebra if q is nota root of unity. Much more interesting is the case when S_q(n,r)_k is not semisimple. Then we have a corresponding decompositionmatrix which records the multiplicities of the simple modulesin certain ‘standard modules’ (or ‘Weyl modules’).A major unsolved problem is the explicit determination of thesedecomposition matrices.     

Weakly Almost Periodic Functions on N with a Negative Base

Weakly almost periodic compactifications have been seriouslystudied for over 30 years. In the pioneering papers of de Leeuwand Glicksberg [4] and [5], the approach adopted was operator-theoretic.The current definition is more likely to be created from theperspective of universal algebra (see [1, Chapter 3]). For adiscrete group or semigroup S, the weakly almost periodic compactificationwS is the largest compact semigroup which (i) contains S asa dense subsemigroup, and (ii) has multiplication continuousin each variable separately (where largest means that any othercompact semigroup with the properties (i) and (ii) is a quotientof wS). A third viewpoint is to envisage wS as the Gelfand spaceof the C*-algebra of bounded weakly almost periodic functionson S (for the definition of such functions, see below). In this paper, we are concerned only with the simplest semigroup(N, +). The three approaches described above give three methodsof obtaining information about wN. An early striking resultabout wN, that it contains more than one idempotent, was obtainedby T. T. West using operator theory [13]. He considered theweak operator closure of the semigroup {T, T², T³, ...} of iteratesof a single operator T on the Hilbert space L²(µ) fora particular measure µ on [0, 1]. Brown and Moran, ina series of papers culminating in [2], used sophisticated techniquesfrom harmonic analysis to produce measures µ that permittedthe detection of further structure in wN; in particular, theyfound 2^cdistinct idempotents. However, for many years, no otherway of showing the existence of more than one idempotent inwN was found. The breakthrough came in 1991, and it was made by Ruppert [11].In his paper, he created a direct construction of a family ofweakly almost periodic functions which could detect 2^c differentidempotents in wN. His method was very ingenious (he used aunique variant of the p-adic expansion of integers) and rathercomplicated. Our main aim in this paper is to construct weaklyalmost periodic functions which are easy to describe and soappear more ‘natural’ than Ruppert's. We also showthat there are enough functions of our type to distinguish 2^cidempotentsin wN.     

An Optimal Representation Formula for Carnot-Caratheodory Vector Fields

The simplest example of the sort of representation formula thatwe shall study is the following familiar inequality for a smooth,real-valued function f(x) defined on a ball B in N-dimensionalEuclidean space R^N: [formula] where f denotes the gradient of f, f_B is the average |B|^–1_Bf(y)dy, |B| is the Lebesgue measure of B, and C is a constantwhich is independent of f, x and B. This formula can be found,for example, in [4] and [12]; see also the closely related estimatesin [20, pp. 228{231]. Indeed, such a formula holds in any boundedconvex domain. 1991 Mathematics Subject Classification 31B10,46E35, 35A22.     

Euler Characteristics in Relative K-Groups

Suppose that M is a finite module under the Galois group ofa local or global field. Ever since Tate's papers [17, 18],we have had a simple and explicit formula for the Euler–Poincarécharacteristic of the cohomology of M. In this note we are interestedin a refinement of this formula when M also carries an actionof some algebra A, commuting with the Galois action (see Proposition5.2 and Theorem 5.1 below). This refinement naturally takesthe shape of an identity in a relative K-group attached to A(see Section 2). We shall deduce such an identity whenever wehave a formula for the ordinary Euler characteristic, the keystep in the proof being the representability of certain functorsby perfect complexes (see Section 3). This representabilitymay be of independent interest in other contexts. Our formula for the equivariant Euler characteristic over Aimplies the ‘isogeny invariance’ of the equivariantconjectures on special values of the L-function put forwardin [3], and this was our motivation to write this note. Incidentally,isogeny invariance (of the conjectures of Birch and Swinnerton-Dyer)was also a motivation for Tate's original paper [18]. I am verygrateful to J-P. Serre for illuminating discussions on the subjectof this note, in particular for suggesting that I consider representability.I should also like to thank D. Burns for insisting on a mostgeneral version of the results in this paper. 2000 MathematicsSubject Classification 19A99, 18G99, 11R34.     

Tameness on the boundary and Ahlfors’ measure conjecture

Let N be a complete hyperbolic 3-manifold that is an algebraic limit of geometrically finite hyperbolic 3-manifolds. We show N is homeomorphic to the interior of a compact 3-manifold, or tame, if one of the following conditions holds:1. N has non-empty conformal boundary,2. N is not homotopy equivalent to a compression body, or3. N is a strong limit of geometrically finite manifolds.The first case proves Ahlfors measure conjecture for Kleinian groups in the closure of the geometrically finite locus: given any algebraic limit of geometrically finite Kleinian groups, the limit set of is either of Lebesgue measure zero or all of . Thus, Ahlfors conjecture is reduced to the density conjecture of Bers, Sullivan, and Thurston.     

Algebraic convergence of finitely generated Kleinian groups in all dimensions

In this paper we prove that for a sequence {G_i,r} of r-generator Kleinian groups acting on , if {G_i,r} satisfies Condition A, then its algebraic limit is also a Kleinian group. This generalizes the main result in [X. Wang, Algebraic convergence theorems of n-dimensional Kleinian groups, Israel J. Math. 162 (2007) 221-233].     

On the Analytic Order-Preserving Discrete-Time Dynamical Systems in Rn with every Fixed Point Stable

This paper studies the asymptotic behaviour of an analytic order-preservingdiscrete-time dynamical system in Rⁿ, which is usually generatedby a periodic cooperative system. The author proves that forsuch a dynamical system, if every fixed point is Liapunov stableand every positive semi-orbit has compact closure, then everypositive semi-orbit converges. This result does not requirethe assumption ‘strongly’ and gives an affirmativeanswer to the conjecture proposed by the author in [17] forthe analytic case.