We prove that there are two incomplete d.r.e.\ degrees (the Turing degrees of differences of two recursively enumerable sets) such that every non-zero recursively enumerable degree cups at least one of them to ${\bf 0}′$, the greatest recursively enumerable (Turing) degree.

This paper proves conditional existence results for non-trivial solutions of the equation

\begin{equation} \sum_{i=1}^{n}a_{i}X_{i}^{3}=0 \quad (n=4\mbox{ or }5), \tag{$*$} \end{equation}

where the coefficients $a_{i}$ and the unknowns $X_{i}$ are taken to be rational integers.

No such results were previously known for $n\leq 6$. The proofs use elementary facts about the 3-descent procedure for elliptic curves of the form $E_{A}: X^{3}+Y^{3}=AZ^{3}$.

Thus, when $n=4$, and the $a_{i}$ are each prime, and are all congruent to 2 modulo 3, it is shown that ($*$) will have non-trivial solutions, providing that the Selmer conjecture holds for the curves $E_{A}$. One may replace the Selmer conjecture by an appropriate form of the Generalized Riemann Hypothesis, when $n=5$ and the $a_{i}$ are again taken to be primes, all congruent to 8 modulo 9. Finally, when $n=5$, one may require only that the $a_{i}$ be square-free and coprime to 3, providing one assumes both the Selmer conjecture and a special case of Schinzel's conjecture (on the representation of primes by cubic polynomials).

1991 Mathematics Subject Classification: 11D25, 11G05, 14G05.

Arakelov and Faltings developed an admissible theory on {\it regular} arithmetic surfaces by using Arakelov canonical volume forms on the associated Riemann surfaces. Such volume forms are induced from the associated Kähler forms of the flat metric on the corresponding Jacobians. So this admissible theory is in the nature of Euclidean geometry, and hence is not quite compatible with the moduli theory of Riemann surfaces. In this paper, we develop a general admissible theory for arithmetic surfaces (associated with stable curves) with respect to any volume form. In particular, we have a theory of arithmetic surfaces in the nature of hyperbolic geometry by using hyperbolic volume forms on the associated Riemann surfaces. Our theory is proved to be useful as well: we have a very natural Weil function on the moduli space of Riemann surfaces, and show that in order to solve the arithmetic Bogomolov-Miyaoka-Yau inequality, it is sufficient to give an estimation for Petersson norms of some modular forms.

1991 Mathematics Subject Classification: 11G30, 11G99, 14H15, 53C07, 58A99.

We prove that the zeta function and normal zeta function of a virtually abelian group have meromorphic continuation to the whole complex plane. We do this by relating the functions to classical L-functions of arithmetic orders considered by Hey, Solomon, Bushnell and Reiner. We calculate the zeta functions and normal zeta functions of the plane crystallographic groups. As a corollary of these calculations we produce \begin{enumerate} \item[(1)] examples of two isospectral residually finite groups with non-isomorphic profinite completions and even distinct lattices of subgroups; and \item[(2)] examples of non-nilpotent residually finite groups whose zeta functions enjoy an Euler product.

1991 Mathematics Subject Classification: 11M41, 20H15.

Let $M_{g,k}$ be the moduli space of flat SU(2)-connections on a compact Riemann surface $\Sigma_g$ of genus $g \ge 2$ punctured in a point $p$ in a small open disc $D$, with holonomy $(-1)^k$ about $\partial D$. This paper gives an explicit decomposition of the rational intersection homology of $M_{g,k}$ into irreducible representations of the oriented mapping class group $\mbox{Map}^+(\Sigma_g)$ of $\Sigma_g$.

1991 Mathematics Subject Classification: primary 14D20; secondary 14H60, 55N33.

We consider the problem of minimizing the uniform norm on $[0, 1]$ over non-zero polynomials $p$ of the form

$$p(x) = \sum_{j=0}^n a_jx^j \quad\text{with } |a_j| \le 1,\, a_j \in {\Bbb C},$$

where the modulus of the first non-zero coefficient is at least $\delta > 0$. Essentially sharp bounds are given for this problem. An interesting related result states that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that

$$ \exp (-c_1 \sqrt{n} ) \le \inf_{0 \ne p \in {\cal F}_n }\|p\|_{[0, 1]} \le \exp (-c_2 \sqrt{n}),$$

for every $n \ge 2$, where ${\cal F}_n$ denotes the set of polynomials of degree at most $n$ with coefficients from $\{-1, 0, 1 \}$. This Chebyshev-type problem is closely related to the question of how many zeros a polynomial from the above classes can have at $1$. We also give essentially sharp bounds for this problem.

{\em Inter alia} we prove that there is an absolute constant $c > 0$ such that every polynomial $p$ of the form

$$p(x) = \sum_{j=0}^n a_j x^j, \quad\text{with } |a_j| \le 1,\,|a_0|=|a_n|=1,\, a_j \in {\Bbb C},$$

has at most $c\sqrt{n}$ real zeros. This improves the old bound $c\sqrt{n\log n}$ given by Schur in 1933, as well as more recent related bounds of Bombieri and Vaaler, and, up to the constant $c$, this is the best possible result.

All the analysis rests critically on the key estimate stating that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that

$$|f(0)|^{c_1/a} \le \exp (c_2/a)\|f\|_{[1-a,1]},$$

for every $f \in {\cal S}$ and $a \in (0, 1]$, where $\cal S$ denotes the collection of all analytic functions $f$ on the open unit disk $D := \{ z \in {\Bbb C}: |z| < 1 \}$ that satisfy

$$|f(z)| \le \frac{1}{1-|z|}\quad \text{for } z \in D.$$

Suppose given a finite Galois extension $K/k$ of number fields with group $G$ and a finite sufficiently large $G$-stable set $S$ of primes of $K$, write $E$ for the group of $S$-units in $K$, and $\Delta S$ for the augmentation submodule of the $\Bbb{Z}$-free permutation module $\Bbb{Z} S$ on $S$. Let also $A$, $B$ be finitely generated cohomologically trivial $\Bbb{Z} G$-modules, and $\tau$ Tate's canonical class defining a 2-extension $E\rightarrowtail A\to B\twoheadrightarrow\Delta S$. Chinburg has assigned the algebraic invariant $\Omega=[A]-[B]\in K_0(\Bbb{Z}G)$ to $\tau$ and conjectured that $\Omega$ equals the root number class $W(K/k)$, an analytic invariant defined by Cassou-Nogués and Fröhlich, and based on Artin's root numbers. We combine $\Omega$ with $G$-injections $\varphi:\Delta S\rightarrowtail E$ to obtain lifts $\Omega_\varphi\in K_0T(\Bbb{Z}G)$ of $\Omega$ in the Grothendieck group of finite $\Bbb{Z}G$-modules of finite projective dimension. Unlike $K_0(\Bbb{Z}G)$, the group $K_0T(\Bbb{Z}G)$ has a natural decomposition as a direct sum $\bigoplus K_0T(\Bbb{Z}_lG)$, over all finite rational primes $l$. This provides a local setting for the study of the $\Omega_\varphi$.

Each of the above $G$-equivariant maps $\varphi$ determines a function $A_\varphi$ on characters of $G$ via $L$-values at zero. Stark's conjecture asserts that $A_\varphi$ is Galois-equivariant. These functions can be viewed as naming elements of $K_0T(\Bbb{Z}G)$ by using Fröhlich's Hom description. The new conjecture, referred to as the Lifted Root Number Conjecture, is that $A_\varphi$ names $\Omega_\varphi$. The truth of this conjecture would imply that of Chinburg's.

The dependence of the invariant $\Omega_\varphi$ on $S$, how it varies with $\varphi$, and its behaviour under restriction to subgroups and deflation to factor groups of $G$, are studied in individual sections. Then, the Hom description is used to compare $\Omega_\varphi$ with $A_\varphi$ whenever Stark's conjecture holds true. It is shown that it suffices to give a proof of the Lifted Root Number Conjecture at the prime $l$ in the case when the Galois group $G$ is replaced by an $l$-elementary group, that is, the direct product of an $l$-group and a cyclic group of order prime to $l$.

Mumford defined a natural isomorphism between the intermediate jacobian of a conic-bundle over ${\Bbb P}^2$ and the Prym variety of a naturally defined étale double cover of the discriminant curve of the conic-bundle. Clemens and Griffiths used this isomorphism to give a proof of the irrationality of a smooth cubic threefold, and Beauville later generalized the isomorphism to intermediate jacobians of odd-dimensional quadric-bundles over ${\Bbb P}^2$. We further generalize the isomorphism to the primitive {\em cohomology} of a smooth cubic hypersurface in ${\Bbb P}^n$. We give two applications of our construction: one is a special case of the generalized Hodge conjectures and the other is an Abel-Jacobi isomorphism.

1991 Mathematics Subject Classification: primary 14J70; secondary 14J45.

Engel-4 groups of exponent 5 are now known to be locally finite. In this paper we determine the orders of the largest finite groups of this kind on $m$ generators.

In the process we show that such a group has a subdirect decomposition into a group which is centre-by-Engel-3 and one which is nilpotent of class at most 10. We work via associated Lie algebras and Lie superalgebras and make use of computer calculations.

1991 Mathematics Subject Classification: 20F45.

We construct the moduli spaces $M(n)$ of semistable parabolic sheaves of rank $n$ and fixed Euler characteristic on a disjoint union $X$ of integral projective curves with parabolic structures over Cartier divisors on $X$. In the case where $X$ is non-singular, $M$ is a normal projective variety. Suppose that $X$ is the desingularisation of a reducible reduced curve $Y$ with at most ordinary double points as singularities. We show that, for a suitable choice of parabolic structure, $M(n)$ is the normalisation of the moduli space of torsion-free sheaves of rank $n$ and fixed Euler characteristic on $Y$, and it is a desingularisation if semistability coincides with stability. We find explicit descriptions of $M(n)$ for small $n$ in some cases.

Schwarz's lemma asserts that analytic mappings from the unit disc into itself decrease hyperbolic distances. In this paper, inner functions which decrease hyperbolic distances as much as possible, when one approaches the unit circle, are constructed. Actually, it is shown that a quadratic condition governs the best decay of the hyperbolic derivative of an inner function. This is related to a result of L. Carleson on the existence of singular symmetric measures. As a consequence, some results on composition operators are obtained, bringing out the importance of the Bloch spaces in this connection. Another consequence is a uniform way of producing singular measures which are simultaneously symmetric and Kahane.

1991 Mathematics Subject Classification: primary 30D50; secondary 30D45, 26A30, 47B38.

It is known that a number of algebraic properties of the braid groups extend to arbitrary finite Coxeter-type Artin groups. Here we show how to extend the results to more general groups that we call Garside groups.

Define a Gaussian monoid to be a finitely generated cancellative monoid where the expressions of a given element have bounded lengths, and where left and right lowest common multiples exist. A Garside monoid is a Gaussian monoid in which the left and right lowest common multiples satisfy an additional symmetry condition. A Gaussian group is the group of fractions of a Gaussian monoid, and a Garside group is the group of fractions of a Garside monoid. Braid groups and, more generally, finite Coxeter-type Artin groups are Garside groups. We determine algorithmic criteria in terms of presentations for recognizing Gaussian and Garside monoids and groups, and exhibit infinite families of such groups. We describe simple algorithms that solve the word problem in a Gaussian group, show that these algorithms have a quadratic complexity if the group is a Garside group, and prove that Garside groups have quadratic isoperimetric inequalities. We construct normal forms for Gaussian groups, and prove that, in the case of a Garside group, the language of normal forms is regular, symmetric, and geodesic, has the 5-fellow traveller property, and has the uniqueness property. This shows in particular that Garside groups are geodesically fully biautomatic. Finally, we consider an automorphism of a finite Coxeter-type Artin group derived from an automorphism of its defining Coxeter graph, and prove that the subgroup of elements fixed by this automorphism is also a finite Coxeter-type Artin group that can be explicitly determined.

1991 Mathematics Subject Classification: primary 20F05, 20F36; secondary 20B40, 20M05.

This paper makes a contribution to the classification of reductive spherical subgroups of simple algebraic groups over algebraically closed fields (a subgroup is called spherical if it has finitely many orbits on the flag variety). The author produces a class of reductive subgroups, and shows that in positive characteristic any such is spherical. The class includes all centralizers of inner involutions if the characteristic is odd, for which the result was already known thanks to work of Springer; however, the sphericality of the corresponding subgroups in even characteristic was in many cases previously undecided. The methods used are character-theoretic in nature, and rely upon bounding inner products of permutation characters.

1991 Mathematics Subject Classification: primary 20G15; secondary 20C15.

A function $f$ from the symmetric group $S_n$ into $\Bbb R$ is called a class function if $f(\sigma ) = f(\tau )$ whenever $\tau$ is conjugate to $\sigma$. Let $d_f$ be thegeneralized matrix function associated with $f$, mapping the $n$-by-$n$ positive semidefinite Hermitian matricesto $\Bbb R$. For example, if $f(\sigma ) = \mbox{sgn}(\sigma )$, then $d_f(A) = \det A$.

We consider the cone $K_n$ of those $f$ for which $d_f(A) \geq 0$ for all $n$-by-$n$ positive semidefinite Hermitian matrices. For $n = 4$ we show that $K_n$ is polyhedral, and explicitly find the extreme rays. Equivalently, $f$ belongs to $K_n$ if $d_f(A) \geq 0$ for a finite minimal set of ‘test matrices’. This solves a problem posed by Gordon James. As a corollary, we characterize all inequalities involving linear combinations of immanants on the positive semidefinite Hermitian matrices of order $n = 4$.

We obtain similar results for 4-by-4 real positive semidefinite matrices.

Let $F: {\Bbb C}^n \to {\Bbb C}^n$ be a holomorphic map, $F^k$ be the $k$th iterate of $F$, and $p \in {\Bbb C}^n$ be a periodic point of $F$ of period $k$. That is, $F^k(p) = p$, but for any positive integer $j$ with $j < k$, $F^j(p) \ne p$. If $p$ is hyperbolic, namely if $DF^k(p)$ has no eigenvalue of modulus $1$, then it is well known that the dynamical behaviour of $F$ is stable near the periodic orbit $\Gamma = \{p, F(p),\ldots, F^{k-1}(p)\}$. But if $\Gamma$ is not hyperbolic, the dynamical behaviour of $F$ near $\Gamma$ may be very complicated and unstable. In this case, a very interesting bifurcational phenomenon may occur even though $\Gamma$ may be the only periodic orbit in some neighbourhood of $\Gamma$: for given $M \in {\Bbb N}\setminus \{1\}$, there may exist a $C^r$-arc $\{F_t: t\in [0,1]\}$ (where $r \in {\Bbb N}$ or $r = \infty$) in the space ${\Bbb H}({\Bbb C}^n)$ of holomorphic maps from ${\Bbb C}^n$ into ${\Bbb C}^n$, such that $F_0 =F$ and, for $t \in (0,1]$, $F_t$ has an $Mk$-periodic orbit $\Gamma_t$ with $d(\Gamma_t, \Gamma) = \sup_{p \in \Gamma_t}\inf_{q \in \Gamma} \|p - q\| \to 0$ as $t \to 0$. The period thus increases by a factor $M$ under a $C^r$-small perturbation! If such an $F_t$ does exist, then $\Gamma$, as well as $p$, is said to be {\em $M$-tupling bifurcational.} This definition is independent of~$r$.

For the above $F$, there may exist a $C^r$-arc $F^*_t$ in ${\Bbb H}({\Bbb C}^n)$, with $t \in [0,1]$, such that $F^*_0 = F$ and, for $t \in (0,1]$, $F^*_t$ has two distinct $k$-periodic orbits $\Gamma_{t,1}$ and $\Gamma_{t,2}$ with $d(\Gamma_{t,i}, \Gamma) \to 0$ as $t \to 0$ for $i = 1,2$. If such an $F^*_t$ does exist, then $\Gamma$, as well as $p$, is said to be {\em $1$-tupling bifurcational.}

In recent decades, there have been many papers and remarkable results which deal with period doubling bifurcations of periodic orbits of parametrized maps. L. Block and D. Hart pointed out that period $M$-tupling bifurcations cannot occur for $M > 2$ in the 1-dimensional case. There are examples showing that for any $M\in {\Bbb N}$, period $M$-tupling bifurcations can occur in higher-dimensional cases.

An $M$-tupling bifurcational periodic orbit as defined here acts as a critical orbit which leads to period $M$-tupling bifurcations in some parametrized maps. The main result of this paper is the following.

{\sc Theorem.} {\em Let $k \in {\Bbb N}$ and $M \in {\Bbb N}$, and let $F: {\Bbb C}^2 \to {\Bbb C}^2$ be a holomorphic map with $k$-periodic point $p$. Then $p$ is $M$-tupling bifurcational if and only if $DF^k(p)$ has a non-zero periodic point of period $M$.}

1991 Mathematics Subject Classification: 32H50, 58F14.

Consider the differential equation

$$\ddot{x} +n^2 x+h_L (x) =p(t),$$

where $n=1,2,\dots$ is an integer, $p$ is a $2\pi$-periodic function and $h_L$ is the piecewise linear function

$$ h_L (x)=\begin{cases} L & \text{if $x\geq 1$},\\

Lx & \text{if $|x|\leq 1$},\\

-L & \text{if $x\leq -1$}.\end{cases}$$

A classical result of Lazer and Leach implies that this equation has a $2\pi$-periodic solution if and only if

\begin{equation}\label{ll} |\hat{p}_n |<{2L\over \pi}, \end{equation}

where

$$\hat{p}_n :={1\over 2\pi}\int_0^{2\pi} p(t)e^{-int}\, dt.$$

In this paper I prove that if $p$ is of class $C^5$ then the condition (\ref{ll}) is also necessary and sufficient for the boundedness of all the solutions of the equation.

The proof of this theorem motivates a new variant of Moser's Small Twist Theorem. This variant guarantees the existence of invariant curves for certain mappings of the cylinder which have a twist that may depend on the angle.

1991 Mathematics Subject Classification: 34C11, 58F35.

There is a standard additive decomposition of the Hochschild cohomology ring of the group algebra of a finite group $G$ as the direct sum of the cohomology rings of the centralizers of representatives of the conjugacy classes of $G$. A special case of our main result describes the cup product in terms of this decomposition. As applications, we determine presentations for the Hochschild cohomology rings of

[(1)] the mod-$3$ group algebra of the symmetric group $S_3$,

[(2)] the mod-$2$ group algebra of the alternating group $A_4$, and

[(3)] the mod-$2$ group algebras of the dihedral $2$-groups.

We complete and complement our recent work on Drasin-Shea-Jordan theorems for Fourier and Hankel transforms. In improving on the methods of our previous work, we were led to certain ratio Mercerian theorems for general kernels; these yield definitive versions of our earlier results for Fourier and Hankel transforms.

1991 Mathematics Subject Classification: primary 44A15; secondary 47B38.

Bihermitian complex surfaces are oriented conformal four-manifolds admitting two independent compatible complex structures. Non-anti-self-dual bihermitian structures on ${\mathbb R}^4$ and the four-dimensional torus $T^4$ have recently been discovered by P. Kobak. We show that an oriented compact 4-manifold, admitting a non-anti-self-dual bihermitian structure, is a torus or K3 surface in the strongly bihermitian case (when the two complex structures are independent at each point) or, otherwise, must be obtained from the complex projective plane or a minimal ruled surface of genus less than 2 by blowing up points along some anti-canonical divisor (but the actual existence of bihermitian structures in the latter case is still an open question). The paper includes a general method for constructing non-anti-self-dual bihermitian structures on tori, K3 surfaces and $S^1\times S^3$. Further properties of compact bihermitian surfaces are also investigated.

1991 Mathematics Subject Classification: 53C12, 53C55, 32J15.

Let $V$ be a vector space and let $\{ e_1,\hdots,e_r \}$ be a basis of $V$. An algebra structure on $V$ is given by $r^3$ structure constants $c_{ij}^h$ where $e_i\cdot e_j = \sum_h c_{ij}^h e_h$. We require this algebra structure to be associative with unit element $e_1$. This limits the sets of structure constants $(c_{ij}^h)$ to a subvariety of $k^{r^3}$, which we denote by $\mbox{Alg}_r$. Base changes in $V$ (leaving $e_1$ fixed) give rise to the natural transport of structure action on $\mbox{Alg}_r$; isomorphism classes of $r$-dimensional algebras are in one-to-one correspondence with the orbits under this action.

In this paper we classify the smooth closed subvarieties of $\mbox{Alg}_r$ which are invariant under the transport of structure action and study the singularities which may occur. In particular, we show that if $r=n^2$ then the closure of the locus corresponding to the matrix algebra $M_n(k)$ is not smooth for $n \geq 3$. This gives a negative answer to a question of Seshadri on the desingularization of moduli spaces of vector bundles over curves.