A Van der Corput exponential sum is $S = \Sigma \exp (2 \pi i f(m))$, where $m$ has size $M$, the function $f(x)$ has size $T$ and $\alpha = (\log M) / \log T < 1$. There are different bounds for $S$ in different ranges for $\alpha $. In the middle range where $\alpha $ is near ${1\over 2}$, $S = O(\sqrt{M} T^{\theta + \epsilon })$. This $\theta $ bounds the exponent of growth of the Riemann zeta function on its critical line ${\rm Re} s = {1\over 2}$. Van der Corput used an iteration which changed $\alpha$ at each step. The Bombieri–Iwaniec method, whilst still based on mean squares, introduces number-theoretic ideas and problems. The Second Spacing Problem is to count the number of resonances between short intervals of the sum, when two arcs of the graph of $y = f'(x)$ coincide approximately after an automorphism of the integer lattice. In the previous paper in this series [Proc. London Math. Soc. (3) 66 (1993) 1–40] and the monograph Area, lattice points, and exponential sums we saw that coincidence implies that there is an integer point close to some ‘resonance curve’, one of a family of curves in some dual space, now calculated accurately in the paper ‘Resonance curves in the Bombieri–Iwaniec method’, which is to appear in Funct. Approx. Comment. Math.

We turn the whole Bombieri–Iwaniec method into an axiomatised step: an upper bound for the number of integer points close to a plane curve gives a bound in the Second Spacing Problem, and a small improvement in the bound for $S$. Ends and cusps of resonance curves are treated separately. Bounds for sums of type $S$ lead to bounds for integer points close to curves, and another branching iteration. Luckily Swinnerton-Dyer's method is stronger. We improve $\theta $ from 0.156140... in the previous paper and monograph to 0.156098.... In fact $(32/205 + \epsilon , 269/410 + \epsilon)$ is an exponent pair for every $\epsilon > 0$.

When $G$ is abelian and $l$ is a prime we show how elements of the relative K-group $K_{0}({\bf Z}_{l}[G], {\bf Q}_{l})$ give rise to annihilator/Fitting ideal relations of certain associated ${\bf Z}[G]$-modules. Examples of this phenomenon are ubiquitous. Particularly, we give examples in which $G$ is the Galois group of an extension of global fields and the resulting annihilator/Fitting ideal relation is closely connected to Stickelberger's Theorem and to the conjectures of Coates and Sinnott, and Brumer. Higher Stickelberger ideals are defined in terms of special values of L-functions; when these vanish we show how to define fractional ideals, generalising the Stickelberger ideals, with similar annihilator properties. The fractional ideal is constructed from the Borel regulator and the leading term in the Taylor series for the L-function. En route, our methods yield new proofs, in the case of abelian number fields, of formulae predicted by Lichtenbaum for the orders of K-groups and étale cohomology groups of rings of algebraic integers.

In this paper we obtain optimal bounds for the length of the longest arithmetic progression in various kinds of sum-sets. As an application, we derive a sharp estimate for the number of sets $A$ of residues modulo a prime $n$ such that no subsum of $A$ equals $x$ modulo $n$, where $x$ is a fixed residue modulo $n$.

Classification of real K3 surfaces $X$ with non-symplectic involution $\tau$ is considered. We define a natural notion of degeneration for them. We show that the connected component of moduli of non-degenerate surfaces of this type is defined by the isomorphism class of the action of $\tau$ and the anti-holomorphic involution $\varphi$ in the homology lattice. (There are very few similar results known.) For their classification we apply invariants of integral lattice involutions with conditions that were developed by the first author in 1983. As a particular case, we describe connected components of moduli of real non-singular curves $A \in | -2 K_V|$ for the classical real surfaces: $V = P^2$, hyperboloid, ellipsoid, $F_1$, $F_4$.

As an application, we describe all real polarized K3 surfaces that are deformations of general real K3 double rational scrolls (the surfaces $V$ above). There are very few exceptions. For example, any non-singular real quartic in $P^3$ can be constructed in this way.

A general mean value theorem for Dirichlet series, with a sharp error estimate near the boundary of the critical strip, is proved. Applications of this theorem to various automorphic $L$-functions are discussed. Moreover, sharp upper bounds of mean square values of $L$-functions are obtained when they are attached to lifted forms, such as Doi–Naganuma and Ikeda lifts in the theory of Siegel modular forms.

The structure of the point-line collinearity graph of the maximal 2-local geometry for the Monster simple group is investigated. The main results describe the first two discs around an arbitrary vertex.

We study the relationship between the cohomology of $q$-Schur algebras and Hecke algebras in type $A$. A key tool, which has independent interest, involves a specific Hecke algebra resolution, introduced in a special case by Deodhar. This resolution is somewhat akin to the Koszul resolution for Lie algebra cohomology, in that it has an explicit description yet applies for all modules. An application of the properties of this resolution establishes a ‘quantum Weyl reciprocity’ connecting, in a range of degrees, the cohomology of the $q$-Schur algebra with that of the Hecke algebra. This work also provides, independently of the resolution, cohomological inequalities which apply in all degrees and which sometimes can be related to Kazhdan–Lusztig polynomials. Finally, we extend to cohomology a decomposition number method of Erdmann, and illustrate its usefulness in computations. This method enables us to exhibit some properties of symmetric group cohomology which, if true, would imply the $SL_n$-version of Lusztig's characteristic $p$ character conjecture.

For a finite group $H$, let $Irr(H)$ denote the set of irreducible characters of $H$, and define the ‘zeta function’ $\zeta^H(t) = \sum_{\chi \in Irr(H)} \chi(1)^{-t}$ for real $t > 0$. We study the asymptotic behaviour of $\zeta^H(t)$ for finite simple groups $H$ of Lie type, and also of a corresponding zeta function defined in terms of conjugacy classes. Applications are given to the study of random walks on simple groups, and on base sizes of primitive permutation groups.

The Chowla–Selberg formula is applied in approximating a given Epstein zeta function. Partial sums of the series derive from the Chowla–Selberg formula, and although these partial sums satisfy a functional equation, as does an Epstein zeta function, they do not possess an Euler product. What we call partial sums throughout this paper may be considered as special cases concerning a more general function satisfying a functional equation only. In this article we study the distribution of zeros of the function. We show that in any strip containing the critical line, all but finitely many zeros of the function are simple and on the critical line. For many Epstein zeta functions we show that all but finitely many non-trivial zeros of partial sums in the Chowla–Selberg formula are simple and on the critical line.

Let $X / T$ be a one parameter family of canonical 3-folds and let $D$ be a Weil divisor on it flat over $T$. We study the problem of when the $D_t$-minimal models of $X_t$ form a family and we obtain conditions for this to happen. As an application of this we classify terminal divisorial contractions $E \subset Y \leftarrow C \subset X$ contracting an irreducible surface $E$ onto the smooth curve $C$, in the case when the general section of $X$ through $C$ is a $D_5$ DuVal singularity.

Let $F$ be a non-Archimedean local field. In an earlier paper, we constructed certain types (in the sense of Bushnell and Kutzko) in $SL_n (F)$, in fact enough to describe the non-supercuspidal components of the Bernstein decomposition of this group. We now give an almost fully explicit presentation of the Hecke algebras of these types.

Let $G$ be a semisimple Lie group of $\mathbb{R}$-rank at least 2 and $\varGamma$ a discrete subgroup of $G$. We consider the limit set of $\varGamma$ in the geometric boundary of the symmetric space associated with $G$. We define the notion of conical and horospherical limit points. In the case of irreducible non-uniform lattices, by using the two Tits building structures, we distinguish the location of their conical limit points. The limit sets of generalized Schottky groups contained in Hilbert modular groups are studied.

The alternating group of degree 6 is located at the junction of three series of simple non-commutative groups: simple sporadic groups, alternating groups and simple groups of Lie type. It plays a very special role in the theory of finite groups. We shall study its new roles both in a finite geometry of a certain pentagon in the Leech lattice and also in the complex algebraic geometry of K3 surfaces.

The center symmetry set (CSS) of a smooth hypersurface $S$ in an affine space $\mathbf{R}^n$ is the envelope of lines joining pairs of points where $S$ has parallel tangent hyperplanes. The idea stems from a definition of Janeczko, in an alternative version due to Giblin and Holtom. For $n = 2$ the envelope is always real, while for $n \ge 3$ the existence of a real envelope depends on the geometry of the hypersurface. In this paper we make a local study of the CSS, some results applying to $n \le 5$ and others to the cases $n = 2,3$. The method is to construct a generating function whose bifurcation set contains the CSS and possibly some other redundant components. Focal sets of smooth hypersurfaces are a special case of the construction, but the CSS is an affine and not a euclidean invariant. Besides the familiar local forms of focal sets there are other local forms corresponding to boundary singularities, and yet others which do not appear to have arisen elsewhere in a geometrical context. There are connections with Finsler geometry. This paper concentrates on the theory and the proof of the local normal forms for the CSS.

We introduce a notion of a category with full projective functors. It encodes certain common properties of categories appearing in representation theory of Lie groups, Lie algebras and quantum groups. We describe the left or right exact functors which naturally commute with projective functors and provide a unified approach to the verification of relations between such functors.

In this paper, we develop the crystal basis theory for quantum generalized Kac–Moody algebras. For a quantum generalized Kac–Moody algebra $U_q(\mathfrak{g})$, we first introduce the category $\mathcal{O}_{int}$ of $U_q(\mathfrak{g})$-modules and prove its semisimplicity. Next, we define the notion of crystal bases for $U_q(\mathfrak{g})$-modules in the category $\mathcal{O}_{int}$ and for the subalgebra $U_q^-(\mathfrak{g})$. We then prove the tensor product rule and the existence theorem for crystal bases. Finally, we construct the global bases for $U_q(\mathfrak{g})$-modules in the category $\mathcal{O}_{int}$ and for the subalgebra $U_q^-(\mathfrak{g})$.

Two locally generic maps $f, g \colon M^n \to \mathbb{R}^{2n - 1}$ are regularly homotopic if they lie in the same path-component of the space of locally generic maps. Our main result is that if $n \neq 3$ and $M^n$ is a closed $n$-manifold then the regular homotopy class of every locally generic map $f \colon M^n \to \mathbb{R}^{2n - 1}$ is completely determined by the number of its singular points provided that $f$ is singular (that is, $f$ is not an immersion).

We describe a method for constructing explicit algebraic solutions to the sixth Painlevé equation, generalising that of Dubrovin and Mazzocco. There are basically two steps. First we explain how to construct finite braid group orbits of triples of elements of {\rm SL}_2({\rm C}) out of triples of generators of three-dimensional complex reflection groups. (This involves the Fourier–Laplace transform for certain irregular connections.) Then we adapt a result of Jimbo to produce the Painlevé VI solutions. (In particular, this solves a Riemann–Hilbert problem explicitly.)

Each step is illustrated using the complex reflection group associated to Klein's simple group of order 168. This leads to a new algebraic solution with seven branches. We also prove that, unlike the algebraic solutions of Dubrovin and Mazzocco and Hitchin, this solution is not equivalent to any solution coming from a finite subgroup of {\rm SL}_2({\rm C}).

The results of this paper also yield a simple proof of a recent theorem of Inaba, Iwasaki and Saito on the action of Okamoto's affine {\rm D}_4 symmetry group as well as the correct connection formulae for generic Painlevé VI equations.

For systems of elliptic equations of Fitzhugh–Nagumo type on bounded domains and with small diffusion in one equation, we construct solutions with multiple sharp peaks close to each other and close to, but not on, the boundary. This is a striking contrast to results for scalar equations.

For some symmetric domains, we also construct similar multipeak solutions except that here the peaks are not close to each other.

We construct an orientable ribbon surface $F \subset B^4$ which is universal in the following sense: any orientable 4-manifold $M \cong B^4 \cup \text{1-handles} \cup \text{2-handles}$ can be represented as a cover of $B^4$ branched over $F$.