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.

     

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.

     

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 .

     

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.

     

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 .

     

An elementary invariant problem and general linear group cohomology restricted to the diagonal subgroup

Conjecturally, for an odd prime and a certain ring of -integers, the stable general linear group and the étale model for its classifying space have isomorphic mod cohomology rings. In particular, these two cohomology rings should have the same image with respect to the restriction map to the diagonal subgroup. We show that a strong unstable version of this last property holds for any rank if is regular and certain homology classes for vanish. We check that this criterion is satisfied for as evidence for the conjecture.

     

Lattice invariants and the center of the generic division ring

Let be a finite group, let be a -lattice, and let be a field of characteristic zero containing primitive roots of 1. Let be the quotient field of the group algebra of the abelian group . It is well known that if is quasi-permutation and -faithful, then is stably equivalent to . Let be the center of the division ring of generic matrices over . Let be the symmetric group on symbols. Let be a prime. We show that there exist a split group extension of by a -elementary group, a -faithful quasi-permutation -lattice , and a one-cocycle in such that is stably isomorphic to . This represents a reduction of the problem since we have a quasi-permutation action; however, the twist introduces a new level of complexity. The second result, which is a consequence of the first, is that, if is algebraically closed, there is a group extension of by an abelian -group such that is stably equivalent to the invariants of the Noether setting .

     

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 .

     

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.

     

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.

     

Koszul homology and extremal properties of Gin and Lex

For every homogeneous ideal in a polynomial ring and for every we consider the Koszul homology with respect to a sequence of of generic linear forms. The Koszul-Betti number is, by definition, the dimension of the degree part of . In characteristic , we show that the Koszul-Betti numbers of any ideal are bounded above by those of the gin-revlex of and also by those of the Lex-segment of . We show that iff is componentwise linear and that and iff is Gotzmann. We also investigate the set of all the gin of and show that the Koszul-Betti numbers of any ideal in are bounded below by those of the gin-revlex of . On the other hand, we present examples showing that in general there is no is such that the Koszul-Betti numbers of any ideal in are bounded above by those of .

     

Backward stability for polynomial maps with locally connected Julia sets

We study topological dynamics on unshielded planar continua with weak expanding properties at cycles for which we prove that the absence of wandering continua implies backward stability. Then we deduce from this that a polynomial with a locally connected Julia set is backward stable outside any neighborhood of its attracting and neutral cycles. For a conformal measure this easily implies that one of the following holds: 1. for -a.e. , ; 2. for -a.e. , for a critical point depending on .

     

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.

     

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 .

     

The periodic isoperimetric problem

Given a discrete group of isometries of , we study the -isoperimetric problem, which consists of minimizing area (modulo ) among surfaces in which enclose a -invariant region with a prescribed volume fraction. If is a line group, we prove that solutions are either families of round spheres or right cylinders. In the doubly periodic case we prove that for most rank two lattices, solutions must be spheres, cylinders or planes. For the remaining rank two lattices we show, among other results, an isoperimetric inequality in terms of the topology of the isoperimetric surfaces. Finally, we study the case where (the group of symmetries of the integer rank three lattice ) and other crystallographic groups of cubic type. We prove that isoperimetric solutions must be spheres if the prescribed volume fraction is less than , and we give an isoperimetric inequality for -invariant regions that, for instance, implies that the area (modulo ) of a surface dividing the three space in two -invariant regions with equal volume fractions, is at least (the conjectured solution is the classical Schwarz triply periodic minimal surface whose area is ). Another consequence of this isoperimetric inequality is that -symmetric surfaces (other than families of spheres) cannot be isoperimetric for the lattice group .

     

Character degree graphs and normal subgroups

We consider the degrees of those irreducible characters of a group whose kernels do not contain a given normal subgroup . We show that if and is not perfect, then the common-divisor graph on this set of integers has at most two connected components. Also, if is solvable, we obtain bounds on the diameters of the components of this graph and, in the disconnected case, we study the structure of and of .

     

Uniqueness of varieties of minimal degree containing a given scheme

We prove that if has dimension and it is -Buchsbaum with \max{(\operatorname{codim}{X}-k,0)}$">, then is contained in at most one variety of minimal degree and dimension .

     

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 .

     

Essential cohomology and extraspecial -groups

Let be an odd prime number and let be an extraspecial -group. The purpose of the paper is to show that has no non-zero essential mod- cohomology (and in fact that is Cohen-Macaulay) if and only if and .

     

On the divergence of the Rogers-Ramanujan continued fraction on the unit circle

This paper studies ordinary and general convergence of the Rogers-Ramanujan continued fraction.

Let the continued fraction expansion of any irrational number be denoted by and let the -th convergent of this continued fraction expansion be denoted by . Let

where . Let . It is shown that if , then the Rogers-Ramanujan continued fraction diverges at . is an uncountable set of measure zero. It is also shown that there is an uncountable set of points such that if , then does not converge generally.

It is further shown that does not converge generally for 1$">. However we show that does converge generally if is a primitive -th root of unity, for some . Combining this result with a theorem of I. Schur then gives that the continued fraction converges generally at all roots of unity.