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.

     

On the -Minkowski problem

A volume-normalized formulation of the -Minkowski problem is presented. This formulation has the advantage that a solution is possible for all , including the degenerate case where the index is equal to the dimension of the ambient space. A new approach to the -Minkowski problem is presented, which solves the volume-normalized formulation for even data and all .

     

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.

     

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 .

     

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.

     

Exponential sums on , II

We prove a vanishing theorem for the -adic cohomology of exponential sums on . In particular, we obtain new classes of exponential sums on that have a single nonvanishing -adic cohomology group. The dimension of this cohomology group equals a sum of Milnor numbers.

     

Parametrized principles

We will present a collection of guessing principles which have a similar relationship to as cardinal invariants of the continuum have to . The purpose is to provide a means for systematically analyzing and its consequences. It also provides for a unified approach for understanding the status of a number of consequences of and in models such as those of Laver, Miller, and Sacks.

     

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.

     

On the Iwasawa -invariants of real abelian fields

For a prime number and a number field , let denote the projective limit of the -parts of the ideal class groups of the intermediate fields of the cyclotomic -extension over . It is conjectured that is finite if is totally real. When is an odd prime and is a real abelian field, we give a criterion for the conjecture, which is a generalization of results of Ichimura and Sumida. Furthermore, in a special case where divides the degree of , we also obtain a rather simple criterion.

     

Cartan-decomposition subgroups of

For and , we give explicit, practical conditions that determine whether or not a closed, connected subgroup of has the property that there exists a compact subset of with . To do this, we fix a Cartan decomposition of , and then carry out an approximate calculation of for each closed, connected subgroup of .

     

Higher homotopy commutativity of -spaces and the permuto-associahedra

In this paper, we give a combinatorial definition of a higher homotopy commutativity of the multiplication for an -space. To give the definition, we use polyhedra called the permuto-associahedra which are constructed by Kapranov. We also show that if a connected -space has the finitely generated mod cohomology for a prime and the multiplication of it is homotopy commutative of the -th order, then it has the mod homotopy type of a finite product of Eilenberg-Mac Lane spaces s, s and s for .

     

On the -torsion of elliptic curves and elliptic surfaces in characteristic

We study the extension generated by the -coordinates of the -torsion points of an elliptic curve over a function field of characteristic . If is a non-isotrivial elliptic surface in characteristic with a -torsion section, then for 11$"> our results imply restrictions on the genus, the gonality, and the -rank of the base curve , whereas for such a surface can be constructed over any base curve . We also describe explicitly all occurring in the cases where the surface is rational or or the base curve is rational, elliptic or hyperelliptic.

     

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 .

     

A unified approach to improved Hardy inequalities with best constants

We present a unified approach to improved Hardy inequalities in . We consider Hardy potentials that involve either the distance from a point, or the distance from the boundary, or even the intermediate case where the distance is taken from a surface of codimension . In our main result, we add to the right hand side of the classical Hardy inequality a weighted norm with optimal weight and best constant. We also prove nonhomogeneous improved Hardy inequalities, where the right hand side involves weighted norms, .

     

Rationality, regularity, and -cofiniteness

We demonstrate that, for vertex operator algebras of CFT type, -cofiniteness and rationality is equivalent to regularity. For -cofinite vertex operator algebras, we show that irreducible weak modules are ordinary modules and -cofinite, is -cofinite, and the fusion rules are finite.

     

Simple birational extensions of the polynomial algebra

The Abhyankar-Sathaye Problem asks whether any biregular embedding can be rectified, that is, whether there exists an automorphism such that is a linear embedding. Here we study this problem for the embeddings whose image is given in by an equation , where and . Under certain additional assumptions we show that, indeed, the polynomial is a variable of the polynomial ring (i.e., a coordinate of a polynomial automorphism of ). This is an analog of a theorem due to Sathaye (1976) which concerns the case of embeddings . Besides, we generalize a theorem of Miyanishi (1984) giving, for a polynomial as above, a criterion for when .

     

On -adic intermediate Jacobians

For an algebraic variety of dimension with totally degenerate reduction over a -adic field (definition recalled below) and an integer with , we define a rigid analytic torus together with an Abel-Jacobi mapping to it from the Chow group of codimension algebraic cycles that are homologically equivalent to zero modulo rational equivalence. These tori are analogous to those defined by Griffiths using Hodge theory over . We compare and contrast the complex and -adic theories. Finally, we examine a special case of a -adic analogue of the Generalized Hodge Conjecture.

     

Examples for the mod motivic cohomology of classifying spaces

Let be the classifying space of a compact Lie group . Some examples of computations of the motivic cohomology are given, by comparing with , and .

     

A new variational characterization of -dimensional space forms

A Riemannian manifold is associated with a Schouten -tensor which is a naturally defined Codazzi tensor in case is a locally conformally flat Riemannian manifold. In this paper, we study the Riemannian functional defined on , where is the space of smooth Riemannian metrics on a compact smooth manifold and is the elementary symmetric functions of the eigenvalues of with respect to . We prove that if and a conformally flat metric is a critical point of with , then must have constant sectional curvature. This is a generalization of Gursky and Viaclovsky's very recent theorem that the critical point of with characterized the three-dimensional space forms.

     

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.