Let $k\ge 3$ be an integer. We study the possible existence of finite sets of positive integers such that the product of any two of them increased by 1 is a $k$th power.

On page 209 of his lost notebook, Ramanujan records an unusual product formula, reminiscent of a product of theta functions. The formula involves hypergeometric functions and has a connection with elliptic functions. In this paper, we prove the formula, offer some generalizations, and indicate some further connections with Ramanujan's work.

The aim of this paper is to answer a question of Coates and Greenberg: let $F$ be a commutative $m$-dimensional formal group over the ring of integers of a local field $k$, and let $K$ be an algebraic extension of $k$ with infinite ramification index. Denote by ${\cal M}_{\Mbar}$ the maximal ideal in the ring of integers of the separable closure of $K$. Suppose that the height of $F$ is greater than $m$. Does $H^1 (K, F ({\cal M}{}^m_{\Mbar}))=0$ imply that $K$ is deeply ramified? The answer is positive.

In this paper we study Iwasawa modules arising from Galois cohomology over general $p$-adic Lie extensions both in the local and global case. In particular we calculate their $\Lambda$-ranks. We then apply the results to abelian varieties.

Let ${\bb D}_p$ be a central simple ${\bb Q}_p$-division algebra of index 2, with maximal ${\bb Z}_p$-order $\Delta_p$. We give an explicit formula for the number of subalgebras of any given finite index in the ${\bb Z}_p$-Lie algebra $\mathcal{L}\colone \spl_1(\Delta_p)$. From this we obtain a closed formula for the zeta function $\zeta_\mathcal{L}(s) \colone \sum_{M \leq \mathcal{L}} |\mathcal{L}:M|^{-s}$. The results are extended to the $p$-power congruence subalgebras of $\mathcal{L}$, and as an application we obtain the zeta functions of the corresponding congruence subgroups of the uniform pro-$p$ group $\SL_1^2(\Delta_p)$.

Let $G$ be a universal Chevalley group defined over an algebraically closed field $F$ of arbitrary characteristic. In this paper we investigate the restrictions to parabolic subgroups of $G$ of the irreducible $FG$-modules corresponding to minuscule highest weights via a combinatorial scheme of “chamber systems” indexed over certain cosets in the Weyl group.

For the dual object of any compact, non-abelian group, the theory of $I_0$ sets (sets of interpolation by Fourier transforms of discrete measures) is quite different from that for abelian compact groups. In general, infinite $I_0$ sets need not exist. However, when the dual object contains an infinite (local) Sidon set, we prove that this set itself has an infinite $I_0$ subset.

The proof is constructive and includes some key examples of $I_0$ sets: certain sets of representations of bounded degree and, for products of simple Lie groups, the set of self-representations of the factor groups and their conjugates (the FTR sets).

The main aim of this paper is to give a description for the structure of the (small) quantum cohomology ring of the toric variety $X={\bb P}(\oplus _{i=1}^r{\cal O}_{{\bb P}^1}(a_i))$ with $\sum_{i=1}^ra_i=\epsilon +k r$ and $\epsilon \in \{0,1 \}$. As we explain later, the importance of this result relies on the fact that, unless $k =0$, $X$ is a non-Fano toric variety and the fact that we determine not only a presentation of the quantum cohomology ring $QH^*(X;{\bb Z})$ but also all quantum products $\alpha* \beta$ with $\alpha, \beta \in H^*(X;{\bb Z})$ or, equivalently, all three-point genus-0 Gromov–Witten invariants.

A $genus$ (in the sense of Hirzebruch [4]) is a multiplicative invariant of cobordism classes of manifolds. Classical examples include the numerical invariants given by the signature and the $\widehat{A}$- and Todd genera. More recently genera were introduced which take as values modular forms on the upper half-plane, $\frak{h}=\{\,\tau\;|\;\mathrm{Im}(\tau)>0\,\}$. The main examples are the elliptic genus $\phi_{ell}$ and the Witten genus $\phi_W$; we refer the reader to the texts [7] or [9] for details.

In this paper, we give a positive answer for a special case of a problem posed by McGibbon [4]. We prove that a connected space with all homotopy groups finite has an H-space structure if and only all Postnikov approximations carry some H-space structure.

We study the connections among the mapping class group of the twice punctured torus, the cyclic branched coverings of (1, 1)-knots and the cyclic presentations of groups. We give the necessary and sufficient conditions for the existence and uniqueness of the $n$-fold strongly-cyclic branched coverings of (1, 1)-knots, through the elements of the mapping class group. We prove that every $n$-fold strongly-cyclic branched covering of a (1, 1)-knot admits a cyclic presentation for the fundamental group, arising from a Heegaard splitting of genus $n$. Moreover, we give an algorithm to produce the cyclic presentation and illustrate it in the case of cyclic branched coverings of torus knots of type $(k,hk\pm 1)$.

It is known that knot groups are right-orderable, and that many of them are not bi-orderable. Here we show that certain fibred knots in $S^3$ (or in a homology sphere) do have bi-orderable fundamental groups. In particular, this holds for fibred knots, such as $4_1$, for which the Alexander polynomial has all roots real and positive. This is an application of the construction of orderings of groups, which are moreover invariant with respect to a certain automorphism.

In this paper we consider hyperbolic manifolds which are $p$-fold cyclic branched coverings of a link where $p$ is a prime number different from two; we prove that the Sylow $p$-subgroups of the orientation-preserving isometry groups of these 3-manifolds are either abelian of rank at most three or $p$ is equal to three and a Sylow $p$-subgroup is a $\bb{Z}_3$-extension of an abelian subgroup of rank three. As a corollary of this result we obtain that there are at most three inequivalent links which have the same hyperbolic 3-manifold as a $p$-fold cyclic branched covering and we describe the relation between two such links.

In this paper several results concerning the periodic points of 1-norm non-expansive maps will be presented. In particular, we will examine the set $R(n)$, which consists of integers $p\geq 1$ such that there exist a 1-norm nonexpansive map $f{:}\ \mathbb{R}^n\rightarrow\mathbb{R}^n$ and a periodic point of $f$ of minimal period $p$. The principal problem is to find a characterization of the set $R(n)$ in terms of arithmetical and combinatorial constraints. This problem was posed in [12, section 4]. We shall present here a significant step towards such a characterization. In fact, we shall introduce for each $n\in\mathbb{N}$ a set $T(n)$ that is determined by arithmetical and combinatorial constraints only, and prove that $R(n)\subset T(n)$ for all $n\in \mathbb{N}$. Moreover, we will see that $R(n)=T(n)$ for $n=1,2,3,4,6,7$, and 10, but it remains an open problem whether the sets $R(n)$ and $T(n)$ are equal for all $n\in \mathbb{N}$.

We show that, for every structurally finite transcendental entire function, the Hausdorff dimension of its Julia set is two.