Following the work of Chevalley and Warning, Ax obtained a bound on the $p$-divisibility of exponential sums involving multi-variable polynomials of fixed degree $d$ over a finite field of characteristic $p$. This bound was subsequently improved by Katz. More recently, Moreno and Moreno, and Adolphson and Sperber, derived bounds that in many instances improved upon the Ax–Katz result. Here we derive a tight bound on the $p$-divisibility of the exponential sums. While exact computation of this bound requires the solution of a system of modular equations, approximations are provided which in several classes of examples, improve on the results of Chevalley and Warning, Ax and Katz, Adolphson and Sperber, and Moreno and Moreno. All of the above results readily translate into bounds on the $p$-divisibility of the number of zeros of multi-variable polynomials.

An important consequence of one of our main results is a method to find classes of examples for which bounds on divisibility of the number of solutions of a system of polynomial equations over finite fields are tight. In particular, we give classes of examples for which the Moreno–Moreno bound is tight.

It is important to note that we have also found applications of our results to coding theory (computation of the covering radius of certain codes) and to Waring's problem over finite fields. These will be described elsewhere.

We present a new approach to the problem of computing the zeta function of a hypersurface over a finite field. For a hypersurface defined by a polynomial of degree $d$ in $n$ variables over the field of $q$ elements, one desires an algorithm whose running time is a polynomial function of $d^n \log(q)$. (Here we assume $d \geq 2$, for otherwise the problem is easy.) The case $n = 1$ is related to univariate polynomial factorisation and is comparatively straightforward. When $n = 2$ one is counting points on curves, and the method of Schoof and Pila yields a complexity of $\log(q)^{C_d}$, where the function $C_d$ depends exponentially on $d$. For arbitrary $n$, the theorem of the author and Wan gives a complexity which is a polynomial function of $(pd^n \log(q))^n$, where $p$ is the characteristic of the field. A complexity estimate of this form can also be achieved for smooth hypersurfaces using the method of Kedlaya, although this has only been worked out in full for curves. The new approach we present should yield a complexity which is a small polynomial function of $pd^n\log(q)$. In this paper, we work this out in full for Artin–Schreier hypersurfaces defined by equations of the form $Z^p - Z = f$, where the polynomial $f$ has a diagonal leading form. The method utilises a relative $p$-adic cohomology theory for families of hypersurfaces, due in essence to Dwork. As a corollary of our main theorem, we obtain the following curious result. Let $f$ be a multivariate polynomial with integer coefficients whose leading form is diagonal. There exists an explicit deterministic algorithm which takes as input a prime $p$, outputs the number of solutions to the congruence equation $f = 0 \bmod{p}$, and runs in ${\mathcal O}(p^{2 + \varepsilon})$ bit operations, for any $\varepsilon > 0$. This improves upon the elementary estimate of ${\mathcal O}(p^{n-1 + \varepsilon})$ bit operations, where $n$ is the number of variables, which can be achieved using Berlekamp's root counting algorithm.

Given a classical Weyl group $W$, that is, a Weyl group of type $A$, $B$ or $D$, one can associate with it a polynomial with integral coefficients $Z_W$ given by the ratio of the Hilbert series of the invariant algebras of the natural action of $W$ and $W^t$ on the ring of polynomials ${\bf C}[x_1, \ldots , x_n]^{\otimes t}$. We introduce and study several statistics on the classical Weyl groups of type $B$ and $D$ and show that they can be used to give an explicit formula for $Z_{D_n}$. More precisely, we define two Mahonian statistics, that is, statistics having the same distribution as the length function, $Dmaj$ and $ned$ on $D_n$. The statistic $Dmaj$, defined in a combinatorial way, has an analogous algebraic meaning to the major index for the symmetric group and the flag-major index of Adin and Roichman for $B_n$; namely, it allows us to find an explicit formula for $Z_{D_n}$. Our proof is based on the theory of $t$-partite partitions introduced by Gordon and further studied by Garsia and Gessel.

Using similar ideas, we define the Mahonian statistic $ned$ also on $B_n$ and we find a new and simpler proof of the Adin–Roichman formula for $Z_{B_n}$.

Finally, we define a new descent number $Ddes$ on $D_n$ so that the pair $(Ddes,Dmaj)$ gives a generalization to $D_n$ of the Carlitz identity on the Eulerian–Mahonian distribution of descent number and major index on the symmetric group.

The moduli space of stable vector bundles on a Riemann surface is smooth when the rank and degree are coprime, and is diffeomorphic to the space of unitary connections of central constant curvature. A classic result of Newstead and Atiyah and Bott asserts that its rational cohomology ring is generated by the universal classes, that is, by the Künneth components of the Chern classes of the universal bundle.

This paper studies the larger, non-compact moduli space of Higgs bundles, as introduced by Hitchin and Simpson, with values in the canonical bundle $K$. This is diffeomorphic to the space of all connections of central constant curvature, whether unitary or not. The main result of the paper is that, in the rank 2 case, the rational cohomology ring of this space is again generated by universal classes.

The spaces of Higgs bundles with values in $K(n)$ for $n > 0$ turn out to be essential to the story. Indeed, we show that their direct limit has the homotopy type of the classifying space of the gauge group, and hence has cohomology generated by universal classes.

The moduli space of principally polarised abelian 4-folds can be compactified in several different ways by toroidal methods. Here we consider in detail the Igusa compactification and the (second) Voronoi compactification. We describe in both cases the cone of nef Cartier divisors. The proof depends on a detailed description of the Voronoi compactification, which makes it possible to proceed by induction, using the known description of the nef cone for compactifications of ${\mathcal A}_3$. The Igusa compactification has a non-${\mathbb Q}$-factorial singularity, which is resolved by a single blow-up: this resolution is the Voronoi compactification. The exceptional divisor $E$ is a toric Fano variety (of dimension 9): the other boundary divisor, $D$, corresponds to degenerations with corank~1. After imposing a level structure in order to avoid certain technical complications, we show that the closure of $D$ in the Voronoi compactification maps to the Voronoi compactification of ${\mathcal A}_3$. The toric description of the exceptional divisor allows us to describe the map in sufficient detail to estimate the intersection numbers needed. This inductive process is only valid for the Voronoi compactification: the result for the Igusa compactification is deduced from the Voronoi compactification.

We define a support variety for a finitely generated module over an artin algebra $\Lambda$ over a commutative artinian ring $k$, with $\Lambda$ flat as a module over $k$, in terms of the maximal ideal spectrum of the algebra ${\rm HH}^*(\Lambda)$ of $\Lambda$. This is modelled on what is done in modular representation theory, and the varieties defined in this way are shown to have many of the same properties as those for group rings. In fact the notions of a variety in our sense and those for principal and non-principal blocks are related by a finite surjective map of varieties. For a finite-dimensional self-injective algebra over a field, the variety is shown to be an invariant of the stable component of the Auslander–Reiten quiver. Moreover, we give information on nilpotent elements in ${\rm HH}^*(\Lambda)$, give a thorough discussion of the ring ${\rm HH}^*(\Lambda)$ on a class of Nakayama algebras, give a brief discussion of a possible notion of complexity, and make a comparison with support varieties for complete intersections.

We study ergodic random Schrödinger operators on a covering manifold, where the randomness enters both via the potential and the metric. We prove measurability of the random operators, almost sure constancy of their spectral properties, the existence of a self-averaging integrated density of states and a Pastur–šubin type trace formula.

Let $A = (a^{ij})$ be a Borel mapping on $[0, 1] \times \mathbb{R}^d$ with values in the space of non-negative operators on $\mathbb{R}^d$ and let $b = (b^i)$ be a Borel mapping on $[0, 1] \times \mathbb{R}^d$ with values in $\mathbb{R}^d$. Let

\[Lu(t, x) = \partial_{t} u(t, x) + a^{ij}(t, x)\partial_{x_i}\partial_{x_j}u(t, x) + b^i(t, x)\partial_{x_i}u(t, x), \quad u \in C_0^\infty((0, 1)\times \mathbb{R}^d).\]

Under broad assumptions on $A$ and $b$, we construct a family $\mu = (\mu_t)_{t \in [0, 1)}$ of probability measures $\mu_t$ on $\mathbb{R}^d$ which solves the Cauchy problem $L^{*}\mu = 0$ with initial condition $\mu_0 = \nu$, where $\nu$ is a probability measure on $\mathbb{R}^d$, in the following weak sense:

$$\[ \int_0^1\int_{\mathbb{R}^d} Lu(t, x)\, \mu_t(dx)\, dt=0, \quad u \in C_0^\infty((0, 1) \times \mathbb{R}^d), \] $$

and

$$\[ \lim\limits_{t \to 0} \int_{\mathbb{R}^d} \zeta(x)\, \mu_t(dx) = \int_{\mathbb{R}^d} \zeta(x)\, \nu(dx), \quad \zeta \in C_0^\infty(\mathbb{R}^d). \] $$

Such an equation is satisfied by transition probabilities of a diffusion process associated with $A$ and $b$ provided such a process exists. However, we do not assume the existence of a process and allow quite singular coefficients, in particular, $b$ may be locally unbounded or $A$ may be degenerate. An infinite-dimensional analogue is discussed as well. Main methods are $L^p$-analysis with respect to suitably chosen measures and reduction to the elliptic case (studied previously) by piecewise constant approximations in time.

We prove that, for every bounded and measurable forcing $p(t)$, the differential equation $\ddot{u}+u^{1/3} =p(t)$ has bounded solutions with arbitrarily large amplitude. In general it is not possible to say that all solutions are bounded, as shown by an example due to Littlewood.

The proof is based on a variational method which can be seen as a dual version of Nehari's method for boundary value problems on compact intervals.

The main purpose of this paper is to study the validity of the Hausdorff–Young inequality for vector-valued functions defined on a non-commutative compact group. As we explain in the introduction, the natural context for this research is that of operator spaces. This leads us to formulate a whole new theory of Fourier type and cotype for the category of operator spaces. The present paper is the first step in this program, where the basic theory is presented, the main examples are analyzed and some important questions are posed.