This paper contains estimates of several exponential sums involving Fourier coefficients of certain modular forms. Although the questions which we consider date back, in some cases, to the twenties, our motivation is modern. All of these sums are connected with one approach to the problem of calculating the norm of the Poincaré 0-operator. In this section we state our results (and their antecedents) and then describe their relevance to the θ-operator problem. For simplicity we state and prove all our results for the modular cusp form of weight six, the famous discriminant function

If П is a k-dimensional vector subspace of Rn and E is a subset of Rn, let projп(E) denote the orthogonal projection of E onto П. Marstrand [8] and Kaufman [6] have developed results on the Hausdorff dimension and measure of projп(E) in terms of the dimension of E, leading to the very general theory of Mattila [11]. In particular, Mattila shows that if the Hausdorff dimension dim E of the Souslin set E is greater than k, then projп(E) has positive k-dimensional Lebesgue measure for almost all П ∈ Gn, k (in the sense of the usual normalized invariant measure on the Grassmann manifold Gn, k of k-dimensional subspaces of Rn).

Let G be a locally compact abelian group. Then there is a finitely additive regular set function m defined on an algebra A of Borel sets in G, m(G) = 1, such that m(T-1F) = m(F) for all F ∈ A and all surjective group endomorphisms T of G onto G.

Let

be a cusp form of even integral weight k > 2 for the full modular group. Then the Dirichlet series

is absolutely convergent for σ > ½(k + 1). Hecke showed that LF is an entire function of s satisfying the functional equation

The purpose of this paper is to provide additional evidence to support our view that the modules of generalized fractions introduced in [8] are worth further investigation: we show that, for a module M over a (commutative, Noetherian) local ring A (with identity) having maximal ideal m and dimension n, the n-th local cohomology module may be viewed as a module of generalized fractions of M with respect to a certain triangular subset of An + 1, and we use this work to formulate Hochster's ‘Monomial Conjecture’ [2, Conjecture 1]; in terms of modules of generalized fractions and to make a quick deduction of one of Hochster's results which supports that conjecture.

For a convex polytope P, let s(P) and σ(P) denote, respectively, the largest t such that P has a t-dimensional face (section) which is a simplex. Using Gale diagrams, lower bounds are obtained for these numbers and formulae are developed for the behaviour of s and σ over Cartesian and other products.

Let ℱ denote a set of subsets of X = {1, 2,…, n). Let deg(i) be the number of members of ℱ containing i and val(ℱ) = min {deg (i): i ∈ X). Suppose no k members of ℱ have union X. We conjecture val(ℱ) ≤ 2n-k-1 for k ≥ 3. This is known for n ≤ 2k and we prove it for k ≥ 25. For k = 2 an example has val(ℱ) > 2n-2(l−n-0·651) and we prove val(ℱ) ≤ 2n-2(1–n-1). We also prove that if the union of k sets one from each of ℱ1,…, ℱk has cardinality at most n – t then min {cardinality ℱj} < 2nαt where αk = 2α − 1 and ½ < α < 1.

The aim of this paper is to give a clear statement, and, I hope, a reasonably clear proof, of a theorem of Thorn, which occurs in his important and difficult paper “Ensembles et morphismes stratifiés” [10]. The theorem to which I refer is Théorème 1.D.1 of [10]. “Tout espace stratifié compact admet une présentation associée aux applications kYX données”. At least, I think that the theorem herein described is equivalent to the above, but I could not swear to it. The main difficulty is that, despite strenuous efforts on my part, I have always found it easier to rig up my own system of definitions than to work within the framework suggested by Thorn. However, the two accounts clearly say the same sort of thing. In particular, §1 of the present paper is closely related to, and heavily influenced by, the material on page 250 of [10].