Property A, partial translation structures, and uniform embeddings in groups

We define the concept of a partial translation structure ona metric space X and show that there is a natural C*-algebraC*() associated with it, which is a subalgebra of the uniformRoe algebra C*_u(X). We introduce a coarse invariant of the metricwhich provides an obstruction to embedding the space in a group.When the space is sufficiently group-like, as determined byour invariant, properties of the Roe algebra can be deducedfrom those of C*(). We also give a proof of the fact that theuniform Roe algebra of a metric space is a coarse invariantup to Morita equivalence.     

The Mathieu Group M11 and the Modular Curve X(11)

In this paper, we prove that the modular curve X(11) over afield of characteristic 3 admits the Mathieu group M₁₁ as anautomorphism group. We also examine some aspects of the geometryof the curve X(11) in characteristic 3. In particular, we showthat every point of the curve is a point of inflection, thecurve has 110 hyperflexes and there are no inflectional trianglesand 11232 inflectional pentagons, of which 144 are self-conjugate.The hyperflexes correspond to the supersingular elliptic curves.We comment on the relationship of Ward's quadrilinear invariantfor M₁₂ to our work and announce for the first time the equationsfor Klein's A-curve of level 11. We also comment on the relationof our work to some unpublished work of Bott and Tate. 1991Mathematics Subject Classification: 11F32, 11G20, 14G10, 14H10,14N10, 20B25, 20C34.     

Projection genericity of space curves

If is a smooth space curve, we consider the family of projectionsof from a variable point not on to a fixed plane. For a residualset of curves , this family versally unfolds those singularitiesthat occur in it. To obtain a family of curves which is open as well as densein the space of smooth maps, we must compactify the parameterspace, so we study curves in real projective space, and includeprojections from points of the curve itself. If is smoothlyembedded, the projection C_P of from is a well-defined smooth curve, and for generic the family C_P has generic singularities. However, when the point of projection moves off , the projectionvaries discontinuously. We define a family of plane curves,parametrised by the blow-up X of P³ along , such that for apoint in the exceptional locus lying over , we have the union of the projection C_P of from P and a straight line L through the image of the tangentat P. A key result asserts that this is a flat family. We givean explicit list of restrictions on the family C_P L (the keycondition is that the total contact order of C_P with L neverexceeds 2), and show that these hold for a dense open set ofcurves , and that if they do hold, there is a neighbourhoodU of , such that the family of projections from points of is generic. Combining this list of conditions with those obtained previouslygives a natural definition of a dense set of space curves ,for which the complete family of projections has generic singularities,and we show that this set is also open. Received September 6, 2007.     

Moduli of rank 4 symplectic vector bundles over a curve of genus 2

Let X be a complex projective curve which is smooth and irreducibleof genus 2. The moduli space ₂ of semistable symplectic vectorbundles of rank 4 over X is a variety of dimension 10. Afterassembling some results on vector bundles of rank 2 and odddegree over X, we construct a generically finite cover of ₂by a family of 5-dimensional projective spaces, and outlinesome applications.     

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.     

A Conformal Mapping Problem Arising in Elasticity, II

Suppose that B is an infinite right cylinder over a horizontalbase , which is a Jordan domain bounded by a smooth curve .We suppose that a normal pressure p₁(x, y) and a tangentialpressure p₂(x, y) are applied at every point of the boundaryof B, where p₁, p₂ lie in the plane of the cross-section andare independent of the vertical co-ordinate. It was shown by Cassisa (1982) that in this situation, givenan arbitrary normal pressure p₁, the tangential pressure p₂can be so chosen, that a purely plane stress tensor existswith the above hypotheses, provided that a certain 3 ? 3 matrixM, depending only on the curve , is non-singular. In a previous paper (Hayman, 1982) the author showed that thesingular case can occur and further that an arbitrary pointsymmetric curve can be approximated by singular point symmetriccurves. In the present paper it is shown that M can have rank1, 2 or 3. In the case of rank 2 p₁ must satisfy one additionalintegral condition and in the case of rank 1 two such conditions.If these conditions are satisfied p₂ can be chosen to dependon one or two arbitrary parameters respectively. Some geometrical conditions for the various cases are obtained.Thus for curves with line symmetry the rank can be two but notone, whereas curves with point symmetry can yield matrices ofrank 1, 2 or 3.     

On the denominators of rational points on elliptic curves

Let x(P) = A_P/B²_P denote the x-coordinate of the rational pointP on an elliptic curve in Weierstrass form. We consider whenB_P can be a perfect power or a prime. Using Faltings' theorem,we show that for a fixed f > 1, there are only finitely manyrational points P with B_P equal to an fth power. Where descentvia an isogeny is possible, we show that there are only finitelymany rational points P with B_P equal to a prime, that thesepoints are bounded in number in an explicit fashion, and thatthey are effectively computable. Finally, we prove a strongerversion of this result for curves in homogeneous form.     

How To Make Davies' Theorem Visible

We prove that for an arbitrary measurable set A R² and a -finiteBorel measure µ on the plane, there is a Borel set oflines L such that for each point in A, the set of directionsof those lines from L containing the point is a residual set,and, moreover, We show how this result may be used to characterise the sets of the planefrom which an invisible set is visible. We also characterisethe rectifiable sets C₁, C₂ for which there is a set which isvisible from C₁ and invisible from C₂.     

Formal Groups of Building Blocks Completely Defined Over Finite Abelian Extensions of Q

Let C be an elliptic curve defined over Q. We can associatetwo formal groups with C: the formal group (X, Y) determinedby the formal completion of the Néron model of C overZ along the zero section, and the formal group F_L(X, Y) of theL-series attached to l-adic representations on C of the absoluteGalois group of Q. Honda shows that F_L(X, Y) is defined overZ, and it is strongly isomorphic over Z to (X, Y). In this paperwe give a generalization of the result of Honda to buildingblocks over finite abelian extensions of Q. The difficulty isto define new matrix L-series of building blocks. Our generalizationcontains the generalization of Deninger and Nart to abelianvarieties of GL₂-type. It also contains the generalization ofour previous paper to Q-curves over quadratic fields. 2000 MathematicsSubject Classification 11G10 (primary), 11F11 (secondary).     

Hyperbolic Hexagons and Algebraic Curves in Genus 3

One of the consequences of the uniformization theorem of Koebeand Poincaré is that any smooth complex algebraic curveC of genus g > 1 is conformally equivalent to H/G, whereG PSL₂(R) is a Fuchsian group and is naturally endowed witha hyperbolic metric. Conversely, any compact hyperbolic surfaceis isomorphic to an algebraic curve. Hence any curve of genusg > 1 may be described in two ways, either by an equationor by a Fuchsian group.     

On the Structure of Modular Categories

For a braided tensor category C and a subcategory K there isa notion of a centralizer C_C K, which is a full tensor subcategoryof C. A pre-modular tensor category is known to be modular inthe sense of Turaev if and only if the center Z₂C C_CC (not tobe confused with the center Z₁ of a tensor category, relatedto the quantum double) is trivial, that is, consists only ofmultiples of the tensor unit, and dimC 0. Here , the X_i being the simple objects. We prove several structural properties of modular categories.Our main technical tool is the following double centralizertheorem. Let C be a modular category and K a full tensor subcategoryclosed with respect to direct sums, subobjects and duals. ThenC_CC_CK = K and dim K·dim C_CK = dim C. We give several applications. (1) If C is modular and K is a full modular subcategory,then L=C_CK is also modular and C is equivalent as a ribbon categoryto the direct product: . Thus every modular category factorizes (non-uniquely, in general)into prime modular categories. We study the prime factorizationsof the categories D(G)-Mod, where G is a finite abelian group. (2) If C is a modular *-category and K is a full tensorsubcategory then dim C dim K · dim Z₂K. We give exampleswhere the bound is attained and conjecture that every pre-modularK can be embedded fully into a modular category C with dim C=dimK·dim Z₂K. (3) For every finite group G there is a braided tensor*-category C such that Z₂CRep,G and the modular closure/modularization is non-trivial. 2000 MathematicsSubject Classification 18D10.     

Buckling of an axisymmetric vesicle under compression: the effects of resistance to shear

We consider the axisymmetric deformation of an initially spherical,porous vesicle with incompressible membrane having finite resistanceto in-plane shearing, as the vesicle is compressed between parallelplates. We adopt a thin-shell balance-of-forces formulationin which the mechanical properties of the membrane are describedby a single dimensionless parameter, C, which is the ratio ofthe membrane's resistance to shearing to its resistance to bending.This results in a novel free-boundary problem which we solvenumerically to obtain vesicle shapes as a function of plateseparation, h. For small deformations, the vesicle contactseach plate over a small circular area. At a critical value ofplate separation, h_TC, there is a transcritical bifurcationfrom which a new branch of solutions emerges, representing buckledvesicles which contact each plate along a circular curve. Forthe values of C investigated, we find that the transcriticalbifurcation is subcritical and that there is a further saddle-nodebifurcation (fold) along the branch of buckled solutions ath = h_SN (where h_SN > h_TC). The resulting bifurcation structureis commensurate with a hysteresis loop in which a sudden transitionfrom an unbuckled solution to a buckled one occurs as h is decreasedthrough h_TC and a further sudden transition, this time froma buckled solution to an unbuckled one, occurs as h is increasedthrough h_SN. We find that h_SN and h_TC increase with C, thatis, vesicles that resist shear are more prone to buckling.     

Schottky Uniformizations of Genus 3 and 4 Reflecting S4

Schottky uniformizations are provided of every closed Riemannsurface S of genus g {3,4} admitting the symmetric group S₄as group of conformal automorphisms. These Schottky uniformizationsreflect the group S₄ and permit concrete representations ofS₄ to be obtained in the respective symplectic group Sp_g(Z).Their corresponding fixed points, in the Siegel space, giveprincipally polarized Abelian varieties of dimension g. Forg = 3 and for some cases of g = 4 they turn out to be holomorphicallyequivalent to the product of elliptic curves.     

On the L-Function of the Curves y2 = x5 + A

Let C_A/Q be the curve y² = x⁵ + A, and let L(s, J_A) denote theL-series of its Jacobian. Under the assumption that the signin the functional equation for L(s, J_A) is +1, the criticalvalue L(1, J_A) is evaluated in terms of the value of a thetaseries for depending on Aat a complex multiplication point coming from Q(₅).     

LANGLANDS DUALITY AND G2 SPECTRAL CURVES

We first demonstrate how duality for the fibres of the so-calledHitchin fibration works for the Langlands dual groups Sp(2m)and SO(2m + 1). We then show that duality for G₂ is implementedby an involution on the base space which takes one fibre toits dual. A formula for the natural cubic form is given andshown to be invariant under the involution.     

5-Torsion Points on Curves of Genus 2

Let C be a smooth proper curve of genus 2 over an algebraicallyclosed field k. Fix a Weierstrass point in C(k) and identifyC with its image in its Jacobian J under the Albanese embeddingthat uses as base point. For any integer N1, we write J_N forthe group of points in J(k) of order dividing N and for the subset of J_N of points oforder N. It follows from the Riemann–Roch theorem thatC(k)J₂ consists of the Weierstrass points of C and that C(k) and C(k) are empty (see [3]). The purpose of this paper is to study curvesC with C(k) non-empty.     

Singularities of Orbit Closures in Module Varieties and Cones Over Rational Normal Curves

Let N be a point of an orbit closure _Min a module variety such that its orbit _N has codimension 2in _M. We show that under someadditional conditions the pointed variety (_M, N) is smoothly equivalent to a cone over a rationalnormal curve.     

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.     

POLYNOMIAL NUMERICAL INDEX FOR SOME COMPLEX VECTOR-VALUED FUNCTION SPACES

We study the relation between the polynomial numerical indicesof a complex vector-valued function space and the ones of itsrange space. It is proved that the spaces C(K, X) and L(µ,X) have the same polynomial numerical index as the complex Banachspace X for every compact Hausdorff space K and every -finitemeasure µ, which does not hold any more in the real case.We give an example of a complex Banach space X such that, forevery k 2, the polynomial numerical index of order k of X isthe greatest possible, namely 1, while the one of X** is theleast possible, namely k^k/(1–k). We also give new examplesof Banach spaces with the polynomial Daugavet property, namelyL(µ, X) when µ is atomless, and C_w(K, X), C_w*(K,X*) when K is perfect.     

The Free-boundary Problem for Gravity-driven Unidirectional Viscous Flows

If a layer of viscous fluid drains steadily under gravity, whilein contact with an upward-moving vertical cylinder of a generalcross section with boundary C_B its velocity w(x, y) satisfiesa Poisson equation w_xx+w_yy = g/v = constant. There are two boundaries,the given curve C_B on which w takes a prescribed constant value,and a free boundary C_p marking the edge of the layer, whoseshape we wish to determine. In order to do this, we need twoboundary conditions on C_F; one is the zero-stress conditionw/n = 0, and we assume that the other is that w is constantaround C_F. The latter condition is shown to be equivalent toa requirement that the volume flux in the layer is maximal.The properties of the resulting free-boundary problem are discussed,and numerical solutions obtained for some special cases. Forexample, a solution for the case when C_B is a 270? corner showsthat the layer thickness at the corner is reduced to 76% ofthe thickness on the flat part of the boundary. The presentanalysis is relevant to industrial coating problems, such asedge effects in continuous galvanizing of sheet steel.