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.

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.

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.

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.

Given weight functions $\theta$, $w$, $\rho$ and $v$, the weighted modular inequality $$ Q^{-1}\bigg(\int_{0}^{\infty} Q(\theta(x)Tf(x)) w(x)\, dx\bigg) \leq P^{-1}\bigg(\int_{0}^{\infty} P(C \rho(x) f(x)) v(x)\, dx\bigg)$$ is characterized. Here $Q$ is a strictly increasing function with $Q(0) = 0$, $Q(\infty) = \infty$ and $2Q(x) \leq Q(C x)$, $P$ is a Young's function, and $T$ is the Hardy operator or a Hardy type operator. In particular, a characterizing condition for the Hardy type operator to map $L^{p}(w)$ to $L^{q}(v)$ when $0 < q < 1 \leq p < \infty$ is deduced. In addition, a new proof for the Maz'ja-Sinnamon theorem is given, and weighted Lorentz norm inequalities for Hardy type operators are established.

1991 Mathematics Subject Classification: primary 26D15, 42B25; secondary 26A33, 46E30.

Let $f$ be an orientation-preserving homeomorphism of a compact orientable manifold. Sufficient conditions are given for the persistence of a collection of periodic points under isotopy of $f$ relative to a compact invariant set $A$. Two main applications are described. In the first,~$A$ is the closure of a single discrete orbit of~$f$, and~$f$ has a Smale horseshoe, all of whose periodic orbits persist; in the second,~$A$ is a minimal invariant Cantor set obtained as the limit of a sequence of nested periodic orbits, all of which are shown to persist under isotopy relative to~$A$.

1991 Mathematics Subject Classification: 58F20, 58F15.

We prove global existence and uniqueness of solutions to the Rarita-Schwinger evolution equations compatible with the constraints. We use a gauge fixing for the Rarita-Schwinger equations for helicity 3/2 fields in curved space that leads to a straightforward Hilbert space framework for their study. We explain how these results might be applied to the global analysis of the full Einstein vacuum equations and provide a complete analysis as a basis for such applications. These and a programme for developing a scattering/inverse scattering transform for the full Einstein equations are discussed.

1991 Mathematics Subject Classification: 83C60, 35Q75, 83C05, 35L45.