Cohomology Actions and Centralisers in Unitary Reflection Groups

In an earlier work, the second author proved a general formulafor the equivariant Poincaré polynomial of a linear transformationg which normalises a unitary reflection group G, acting on thecohomology of the corresponding hyperplane complement. Thisformula involves a certain function (called a Z-function below)on the centraliser CG(g), which was proved to exist only incertain cases, for example, when g is a reflection, or is G-regular,or when the centraliser is cyclic. In this work we prove theexistence of Z-functions in full generality. Applications includereduction and product formulae for the equivariant Poincarépolynomials. The method is to study the poset L(C_G(g)) of subspaceswhich are fixed points of elements of C_G(g). We show that thisposet has Euler characteristic 1, which is the key propertyrequired for the definition of a Z-function. The fact aboutthe Euler characteristic in turn follows from the ‘join-atom’property of L(C_G(g)), which asserts that if [X₁,..., X_k} isany set of elements of L(C_G(g)) which are maximal (set theoretically)then their setwise intersection lies in L(C_G(g)). 2000 Mathematical Subject Classification:primary 14R20, 55R80; secondary 20C33, 20G40.     

On Decomposition Numbers and Branching Coefficients for Symmetric and Special Linear Groups

In this paper we find the multiplicities dim L()_– where is an arbitrary root and L() is an irreducible SL_n-module withhighest weight . We provide different bases of the correspondingweight spaces and outline some applications to the symmetricgroups. In particular we describe certain composition multiplicitiesin the modular branching rule. 1991 Mathematics Subject Classification:20C05, 20G05.     

Supplementary Note on 'Rational Invariants of Certain Orthogonal and Unitary Groups'

Given a field k and a finite group G acting on the rationalfunction field k(X₁, ..., X_n) as a group of k-automorphisms,an important Noether's problem asks whether the invariant subfield [forumal] is purely transcendental over k. 1991 Mathematics Subject Classification12F20, 20G40.     

Filtration-Closed Auslander-Reiten Components for Wild Hereditary Algebras

Let H=kQ be a finite-dimensional connected wild hereditary pathalgebra, over some field k. Denote by H-reg the category offinite-dimensional regular H-modules, that is, the categoryof modules M with for all integers m, where _H denotes the Auslander–Reiten translation.Call a filtration of a regular H-module M a regular filtration if all subquotients M_i/M_i+1are regular. Call a regular filtration (*) a regular compositionseries if it is strictly decreasing and has no proper refinement.A regular component C in the Auslander–Reiten quiver (H) of H-mod is called filtration closed if, for each M addC, the additive closure of C, and each regular filtration (*)of M, all the subquotients M_i/M_i+1 are also in add C. We showthat most wild hereditary algebras have filtration-closed Auslander–Reitencomponents. Moreover, we deduce from this that there are alsoalmost serial components, that is regular components C, suchthat any indecomposable XC has a unique regular compositionseries. This composition series coincides with the Auslander–Reitenfiltration of X, given by the maximal chain of irreducible monosending at X. 1991 Mathematics Subject Classification: 16G70,16G20, 16G60, 16E30.     

The Symmetric Genus of the Mathieu Groups

The (symmetric) genus of a finite group may be defined as thesmallest genus of those closed orientable surfaces on whichG acts faithfully as a group of automorphisms. In this paperthe genus of each of the five Mathieu groups M₁₁, M₁₂, M₂₂,M₂₃ and M₂₄ is determined, with the help of some computer calculationsand a little-known of Ree on permutations.     

Weierstrass Points on X0(pM) and Supersingular j-Invariants

Weierstrass points are special points on a Riemann surface thatcarry a lot of information. Ogg studied such points on X₀(pM)(for M such that X₀(M) has genus zero and p prime with p M),and he proved that if Q is a Q-rational Weierstrass point onX₀(pM), then its reduction modulo p is supersingular. The papershows that, for square-free M on the list, all supersingularj-invariants are obtained in this way. Furthermore, for mostcases where M is prime, the explicit correspondence betweenWeierstrass points and supersingular j-invariants in characteristicp is described. Along the way, a useful formula of Rohrlichfor computing a certain Wronskian of modular forms modulo pis generalized.     

Toroidal and Annular Dehn Fillings

Suppose that M is a hyperbolic 3-manifold which admits two Dehnfillings M(r₁) and M(r₂) such that M(r₁) contains an essentialannulus, and M(r₂) contains an essential torus. It is knownthat = (r₁, r₂) 5. We will show that if = 5 then M is theWhitehead sister link exterior, and if = 4 then M is the exteriorof either the Whitehead link or the 2-bridge link associatedto the rational number . There are infinitely many examples with = 3. 1991 Mathematics SubjectClassification: 11D25, 11G05, 14G05.     

Stability of the Picard Bundle

Let X be a non-singular algebraic curve of genus g 2, n 2an integer, a line bundle over X of degree d > 2n(g –1) with (n,d) = 1 and M the moduli space of stable bundles ofrank n and determinant over X. It is proved that the Picardbundle W is stable with respect to the unique polarisation ofM. 2000 Mathematics Subject Classification 14H60, 14J60.     

The Weierstrass Subgroup of a Curve has Maximal Rank

We show that the Weierstrass points of the generic curve ofgenus g over an algebraically closed field of characteristic0 generate a group of maximal rank in the Jacobian. 2000 MathematicsSubject Classification 11G30, 14H40, 14H10, 14H55, 14Q05.     

Maximal Subgroups of Large Rank in Exceptional Groups of Lie Type

Let G = G(q) be a finite almost simple exceptional group ofLie type over the field of q elements, where q = p^a and p isprime. The main result of the paper determines all maximal subgroupsM of G(q) such that M is an almost simple group which is alsoof Lie type in characteristic p, under the condition that rank(M)> rank(G). The conclusion is that either M is a subgroupof maximal rank, or it is of the same type as G over a subfieldof F_q, or (G, M) is one of (, F₄(q)), (, C₄(q)), (E₇(q),³D₄(q)). This completes work of the first author with Saxl andTesterman, in which the same conclusion was obtained under someextra assumptions.     

The Manin-Mumford Conjecture: A Brief Survey

The Manin–Mumford conjecture asserts that if K is a fieldof characteristic zero, C a smooth proper geometrically irreduciblecurve over K, and J the Jacobian of C, then for any embeddingof C in J, the set C(K)J(K)_tors is finite. Although the conjecturewas proved by Raynaud in 1983, and several other proofs haveappeared since, a number of natural questions remain open, notablyconcerning bounds on the size of the intersection and the completedetermination of C(K)J(K)_tors for special families of curvesC. The first half of this survey paper presents the Manin–Mumfordconjecture and related general results, while the second describesrecent work mostly dealing with the above questions. 1991 MathematicsSubject Classification 11G10, 11G30, 11G35, 14G25, 14H25, 14H40.     

Modular Branching Rules and the Mullineux Map for Hecke Algebras of Type A

We prove the quantum version - for Hecke algebras H A_n of typeA at roots of unity - of Kleshchev's modular branching rulefor symmetric groups. This result describes the socle of therestriction of an irreducible H A_n-module to the subalgebraH A_n–1. As a consequence, we describe the quantum versionof the Mullineux involution describing the irreducible moduleobtained on twisting an irreducible module with the sign representation.1991 Mathematics Subject Classification: 20C05, 20G05.     

S-adic L-Functions Attached to the Symmetric Square of a Newform

In order to apply the ideas of Iwasawa theory to the symmetricsquare of a newform, we need to be able to define non-archimedeananalogues of its complex L-series. The interpolated p-adic L-functionis closely connected via a "Main Conjecture" with certain Selmergroups over the cyclotomic Z_p-extension of Q. In the p-ordinarycase these functions are well understood. In this article we extend the interpolation to an arbitraryset S of good primes (not necessarily satisfying ordinarityconditions). The corresponding S-adic functions can be characterisedin terms of certain admissibility criteria. We also allow interpolationat particular primes dividing the level of the newform. One interesting application is to the symmetric square of amodular elliptic curve E defined over Q. Our constructions yieldp-adic L-functions at all primes of stable or semi-stable reduction.If p is ordinary or multiplicative the corresponding analyticfunction is bounded; if p is supersingular our function behaveslike log²(1 + T). 1991 Mathematics Subject Classification: 11F67,11F66, 11F33, 11F30     

Numerical Radius Norms on Operator Spaces

We introduce a numerical radius operator space (X, W_n). Theconditions to be a numerical radius operator space are weakerthan Ruan's axiom for an operator space (X, O_n). Let w(·)be the numerical radius on B(H). It is shown that, if X admitsa norm W_n(·) on the matrix space M_n(X) which satisfiesthe conditions, then there is a complete isometry, in the senseof the norms W_n(·) and w_n(·), from (X, W_n) into(B(H), w_n). We study the relationship between the operator space(X, O_n) and the numerical radius operator space (X, W_n). Thecategory of operator spaces can be regarded as a subcategoryof numerical radius operator spaces.     

Topologie Des (M - 2)-Courbes Reelles Symetriques

Let X be a non-singular real algebraic curve in CP² of evendegree. In this paper a refinement is proved of a theorem ofKharlamov about (M – 2)-curves that are invariants underthe projective involution. In particular, if the (M –2)-symmetric curve X satisfies the Arnold congruence, then eitherX or its twin is a separating curve. 2000 Mathematics SubjectClassification 14P25.     

On the non-existence of exceptional automorphisms on Shimura curves

We study the group of automorphisms of Shimura curves X₀(D,N) attached to an Eichler order of square-free level N in anindefinite rational quaternion algebra of discriminant D>1.We prove that, when the genus g of the curve is greater thanor equal to 2, Aut (X₀(D, N)) is a 2-elementary abelian groupwhich contains the group of Atkin–Lehner involutions W₀(D,N) as a subgroup of index 1 or 2. It is conjectured that Aut(X₀(D, N))=W₀(D, N) except for finitely many values of (D, N)and we provide criteria that allow us to show that this is indeedoften the case. Our methods are based on the theory of complexmultiplication of Shimura curves and the Cerednik–Drinfeldtheory on their rigid analytic uniformization at primes p| D.     

Cm Elliptic Curves and Bicolorings of Steiner Triple Systems

It is shown that a necessary condition for the existence ofa bicolored Steiner triple system of order n is that n can bewritten in the form A²+3B² for integers A and B. In the casewhen n=q is either a prime congruent to 1 mod 3, or the squareof a prime congruent to 2 mod 3, it is shown that the numbersof colored vertices in the triple system would be unique, andare given by the number of points on specific twists of theCM elliptic curve y²=x³–1 over the finite field F_q. 2000Mathematics Subject Classification 05B07, 11G20, 14G15 (primary);11G15, 14K22 (secondary).     

The finite images of finitely generated groups

Given any sequence of non-abelian finite simple primitive permutationgroups S_n, we construct a finitely generated group G whose profinitecompletion is the infinite permutational wreath product ...S_n S_n–1 ... S₀. It follows that the upper compositionfactors of G are exactly the groups S_n. By suitably choosingthe sequence S_n we can arrange that G has any one of a continuousrange of slow, non-polynomial subgroup growth types. We alsoconstruct a 61-generator perfect group that has every non-abelianfinite simple group as a quotient. 2000 Mathematics SubjectClassification: 20E07, 20E08, 20E18, 20E32.     

Brauer Group Invariants Associated to Orthogonal Epsilon-Constants

In this paper, the theory of -constants associated to tame finitegroup actions on arithmetic surfaces is used to define a Brauergroup invariant µ(X, G, V) associated to certain symplecticmotives of weight one. The relationship between this invariantand w₂() (the Galois-theoretic invariant associated to tamecovers of surfaces defined by Cassou-Noguès, Erez andTaylor) is also discussed. 2000 Mathematics Subject Classification11G35 (primary), 11G40, 14G40 (secondary).