Positive laws in fixed points

Let be an elementary abelian group of order at least acting on a finite -group in such a manner that satisfies a positive law of degree for any . It is proved that the entire group satisfies a positive law of degree bounded by a function of and only.

     

How to do a -descent on an elliptic curve

In this paper, we describe an algorithm that reduces the computation of the (full) -Selmer group of an elliptic curve over a number field to standard number field computations such as determining the (-torsion of) the -class group and a basis of the -units modulo th powers for a suitable set of primes. In particular, we give a result reducing this set of `bad primes' to a very small set, which in many cases only contains the primes above . As of today, this provides a feasible algorithm for performing a full -descent on an elliptic curve over , but the range of our algorithm will certainly be enlarged by future improvements in computational algebraic number theory. When the Galois module structure of is favorable, simplifications are possible and -descents for larger are accessible even today. To demonstrate how the method works, several worked examples are included.

     

The cohomology of certain Hopf algebras associated with -groups

We study the cohomology of a locally finite, connected, cocommutative Hopf algebra over . Specifically, we are interested in those algebras for which is generated as an algebra by and . We shall call such algebras semi-Koszul. Given a central extension of Hopf algebras with monogenic and semi-Koszul, we use the Cartan-Eilenberg spectral sequence and algebraic Steenrod operations to determine conditions for to be semi-Koszul. Special attention is given to the case in which is the restricted universal enveloping algebra of the Lie algebra obtained from the mod- lower central series of a -group. We show that the algebras arising in this way from extensions by of an abelian -group are semi-Koszul. Explicit calculations are carried out for algebras arising from rank 2 -groups, and it is shown that these are all semi-Koszul for .

     

Flat holomorphic connections on principal bundles over a projective manifold

Let be a connected complex linear algebraic group and its unipotent radical. A principal -bundle over a projective manifold will be called polystable if the associated principal -bundle is so. A -bundle over is polystable with vanishing characteristic classes of degrees one and two if and only if admits a flat holomorphic connection with the property that the image in of the monodromy of the connection is contained in a maximal compact subgroup of .

     

Asymptotic behavior of the solutions of linear and quasilinear elliptic equations on

We investigate the relationship between the decay at infinity of the right-hand side and solutions of an equation when is a second order elliptic operator on It is shown that when is Fredholm, inherits the type of decay of (for instance, exponential, or power-like). In particular, the generalized eigenfunctions associated with all the Fredholm eigenvalues of isolated or not, decay exponentially. No use is made of spectral theory. The result is next extended when is replaced by a Fredholm quasilinear operator. Various generalizations to other unbounded domains, higher order operators or elliptic systems are possible and briefly alluded to, but not discussed in detail.

     

Boundary correspondence of Nevanlinna counting functions for self-maps of the unit disc

Let be a holomorphic self-map of the unit disc . For every , there is a measure on (sometimes called Aleksandrov measure) defined by the Poisson representation . Its singular part measures in a natural way the ``affinity' of for the boundary value . The affinity for values inside is provided by the Nevanlinna counting function of . We introduce a natural measure-valued refinement of and establish that the measures are obtained as boundary values of the refined Nevanlinna counting function . More precisely, we prove that is the weak limit of whenever converges to non-tangentially outside a small exceptional set . We obtain a sharp estimate for the size of in the sense of capacity.

     

Stability of parabolic Harnack inequalities

Let be a graph with weights for which a parabolic Harnack inequality holds with space-time scaling exponent . Suppose is another set of weights that are comparable to . We prove that this parabolic Harnack inequality also holds for with the weights . We also give stable necessary and sufficient conditions for this parabolic Harnack inequality to hold.

     

On restrictions of modular spin representations of symmetric and alternating groups

Let be an algebraically closed field of characteristic and be an almost simple group or a central extension of an almost simple group. An important problem in representation theory is to classify the subgroups of and -modules such that the restriction is irreducible. For example, this problem is a natural part of the program of describing maximal subgroups in finite classical groups. In this paper we investigate the case of the problem where is the Schur's double cover or .

     

Newton polyhedra, unstable faces and the poles of Igusa's local zeta function

In this paper we examine when the order of a pole of Igusa's local zeta function associated to a polynomial is smaller than ``expected'. We carry out this study in the case that is sufficiently non-degenerate with respect to its Newton polyhedron , and the main result of this paper is a proof of one of the conjectures of Denef and Sargos. Our technique consists in reducing our question about the polynomial to the same question about polynomials , where are faces of depending on the examined pole and is obtained from by throwing away all monomials of whose exponents do not belong to . Secondly, we obtain a formula for Igusa's local zeta function associated to a polynomial , with unstable, which shows that, in this case, the upperbound for the order of the examined pole is obviously smaller than ``expected'.

     

On orbital partitions and exceptionality of primitive permutation groups

Let and be transitive permutation groups on a set such that is a normal subgroup of . The overgroup induces a natural action on the set of non-trivial orbitals of on . In the study of Galois groups of exceptional covers of curves, one is led to characterizing the triples where fixes no elements of ; such triples are called exceptional. In the study of homogeneous factorizations of complete graphs, one is led to characterizing quadruples where is a partition of such that is transitive on ; such a quadruple is called a TOD (transitive orbital decomposition). It follows easily that the triple in a TOD is exceptional; conversely if an exceptional triple is such that is cyclic of prime-power order, then there exists a partition of such that is a TOD. This paper characterizes TODs such that is primitive and is cyclic of prime-power order. An application is given to the classification of self-complementary vertex-transitive graphs.

     

On Diophantine definability and decidability in some infinite totally real extensions of

Let be a number field, and a set of its non-Archimedean primes. Then let . Let be a finite set of prime numbers. Let be the field generated by all the -th roots of unity as and . Let be the largest totally real subfield of . Then for any 0$">, there exist a number field , and a set of non-Archimedean primes of such that has density greater than , and has a Diophantine definition over the integral closure of in .

     

Complete hyperelliptic integrals of the first kind and their non-oscillation

Let be a real polynomial of degree , and be an oval contained in the level set . We study complete Abelian integrals of the form

where are real and is a maximal open interval on which a continuous family of ovals exists. We show that the -dimensional real vector space of these integrals is not Chebyshev in general: for any 1$">, there are hyperelliptic Hamiltonians and continuous families of ovals , , such that the Abelian integral can have at least zeros in . Our main result is Theorem 1 in which we show that when , exceptional families of ovals exist, such that the corresponding vector space is still Chebyshev.

     

Semi-linear homology -spheres and their equivariant inertia groups

This paper introduces an abelian group for all semi-linear homology -spheres, which corresponds to a known abelian group for all semi-linear homotopy -spheres, where is a compact Lie group and is a -representation with 0$">. Then using equivariant surgery techniques, we study the relation between both and when is finite. The main result is that under the conditions that -action is semi-free and with 0$">, the homomorphism defined by is an isomorphism if , and a monomorphism if . This is an equivariant analog of a well-known result in differential topology. Such a result is also applied to the equivariant inertia groups of semi-linear homology -spheres.

     

Poincaré's closed geodesic on a convex surface

We present a new proof for the existence of a simple closed geodesic on a convex surface . This result is due originally to Poincaré. The proof uses the -dimensional Riemannian manifold of piecewise geodesic closed curves on with a fixed number of corners, chosen sufficiently large. In we consider a submanifold formed by those elements of which are simple regular and divide into two parts of equal total curvature . The main burden of the proof is to show that the energy integral , restricted to , assumes its infimum. At the end we give some indications of how our methods yield a new proof also for the existence of three simple closed geodesics on .

     

Involutions fixing , II

This paper studies the equivariant cobordism classification of all involutions fixing a disjoint union of an odd-dimensional real projective space with its normal bundle nonbounding and a Dold manifold with a positive even and 0$">. The complete analysis of the equivariant cobordism classes of such involutions is given except that the upper and lower bounds on the codimension of may not be best possible. In particular, we find that there exist such involutions with nonstandard normal bundle to . Together with the results of part I of this title (Trans. Amer. Math. Soc. 354 (2002), 4539-4570), the argument for involutions fixing is finished.

     

Analysing finite locally -arc transitive graphs

We present a new approach to analysing finite graphs which admit a vertex intransitive group of automorphisms and are either locally -arc transitive for or -locally primitive. Such graphs are bipartite with the two parts of the bipartition being the orbits of . Given a normal subgroup which is intransitive on both parts of the bipartition, we show that taking quotients with respect to the orbits of preserves both local primitivity and local -arc transitivity and leads us to study graphs where acts faithfully on both orbits and quasiprimitively on at least one. We determine the possible quasiprimitive types for in these two cases and give new constructions of examples for each possible type. The analysis raises several open problems which are discussed in the final section.

     

Morse index and uniqueness for positive solutions of radial -Laplace equations

We study the positive radial solutions of the Dirichlet problem in , 0$"> in , on , where , 1$">, is the -Laplace operator, is the unit ball in centered at the origin and is a function. We are able to get results on the spectrum of the linearized operator in a suitable weighted space of radial functions and derive from this information on the Morse index. In particular, we show that positive radial solutions of Mountain Pass type have Morse index 1 in the subspace of radial functions of . We use this to prove uniqueness and nondegeneracy of positive radial solutions when is of the type and .

     

Cohomology operations for Lie algebras

If is a Lie algebra over and its centre, the natural inclusion extends to a representation of the exterior algebra of in the cohomology of . We begin a study of this representation by examining its Poincaré duality properties, its associated higher cohomology operations and its relevance to the toral rank conjecture. In particular, by using harmonic forms we show that the higher operations presented by Goresky, Kottwitz and MacPherson (1998) form a subalgebra of , and that they can be assembled to yield an explicit Hirsch-Brown model for the Borel construction associated to .

     

Symmetrization, symmetric stable processes, and Riesz capacities

Let be a symmetric -stable process killed on exiting an open subset of . We prove a theorem that describes the behavior of its transition probabilities under polarization. We show that this result implies that the probability of hitting a given set in the complement of in the first exit moment from increases when and are polarized. It can also lead to symmetrization theorems for hitting probabilities, Green functions, and Riesz capacities. One such theorem is the following: Among all compact sets in with given volume, the balls have the least -capacity ( ).