Recently Bergelson and Leibman proved an extension to van der Waerden's theorem involving polynomials. They also generalised the Hales–Jewett theorem in a similar way. The paper presents short and purely combinatorial proofs of those results.

An independence algebra is an algebra A in which the subalgebras satisfy the exchange axiom, and any map from a basis of A into A extends to an endomorphism. Independence algebras fall into two classes; the first is specified by a set X, a group G, and a G-space C. The second is much more restricted; it is shown that the subalgebra lattice is a projective or affine geometry, and a complete classification of the finite algebras is given.

Four methods for constructing anti-Pasch Steiner triple systems are developed. The first generalises a construction of Stinson and Wei to obtain a general singular direct product construction. The second generalises the Bose construction. The third employs a construction due to Lu. The fourth employs Wilson-type inflation techniques using Latin squares having no subsquares of order 2. As a consequence of these constructions we are able to produce anti-Pasch systems of order v for v 31 or 7 (mod 18), for v ≡ 49 (mod 72), and for many other values of v.

The classical canonical Ramsey theorem of Erdős and Rado states that, for any integer q [ges ] 1, any edge colouring of a large enough complete graph contains one of three canonically coloured complete subgraphs of order q. Of these canonical subgraphs, one is coloured monochromatically while each of the other two has its edge set coloured with many different colours. The paper presents a condition on colourings that, roughly speaking, requires them to make effective use of many colours (‘essential infiniteness’); this condition is then shown to imply that the colourings in question must contain large refinements of one of two ‘unavoidable’ colourings that are rich in colours. As it turns out, one of these unavoidable colourings is one of the canonical colourings of Erdős and Rado, and the other is a ‘bipartite variant’ of this colouring.

The second Stiefel–Whitney class of the quadratic form trA/k(ax2) is computed, where A is a central simple algebra over a perfect field k of characteristic different from 2, a ∈ A is a fixed element, and trA/k is the reduced trace. This class is related on the one hand to the class of A in the Brauer group, and on the other hand to corestrictions of quaternion algebras over certain factors arising from E[otimes ]kE, where E is a commutative étale algebra over k that depends on the semisimple part of a.

Arithmetic Fuchsian groups are the most interesting and most important Fuchsian groups owing to their significance for number theory and owing to their geometric properties. However, for a fixed signature there exist only finitely many non- conjugate arithmetic Fuchsian groups; it is therefore desirable to extend this class of Fuchsian groups. This is the motivation of our definition of semi-arithmetic Fuchsian groups. Such a group may be defined as follows (for the precise formulation see Section 2). Let Γ be a cofinite Fuchsian group and let Γ2 be the subgroup generated by the squares of the elements of Γ. Then Γ is semi-arithmetic if Γ is contained in an arithmetic group Δ acting on a product Hr of upper halfplanes. Equivalently, Γ is semi-arithmetic if all traces of elements of Γ2 are algebraic integers of a totally real field. Well-known examples of semi-arithmetic Fuchsian groups are the triangle groups (and their subgroups of finite index) which are almost all non-arithmetic with the exception of 85 triangle groups listed by Takeuchi [16].

While it is still an open question as to what extent the non-arithmetic Fuchsian triangle groups share the geometric properties of arithmetic groups, it is a fact that their automorphic forms share certain arithmetic properties with modular forms for arithmetic groups. This has been clarified by Cohen and Wolfart [5] who proved that every Fuchsian triangle group Γ admits a modular embedding, meaning that there exists an arithmetic group Δ acting on Hr, a natural group inclusion

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and a compatible holomorphic embedding

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that is with

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for all γ∈Γ and all z∈H.

Suppose that G(x) is a form, or homogeneous polynomial, of odd degree d in s variables, with real coefficients. Schmidt [15] has shown that there exists a positive integer s0(d), which depends only on the degree d, so that if s [ges ] s0(d), then there is an x ∈ ℤs\{0} satisfying the inequality

formula here

In other words, if there are enough variables, in terms of the degree only, then there is a nontrivial solution to (1). Let s0(d) be the minimum integer with the above property. In the course of proving this important result, Schmidt did not explicitly give upper bounds for s0(d). His methods do indicate how to do so, although not very efficiently. However, in fact much earlier, Pitman [13] provided explicit bounds in the case when G is a cubic. We consider a general cubic form F(x) with real coefficients, in s variables, and look at the inequality

formula here

Specifically, Pitman showed that if

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then inequality (2) is non-trivially soluble in integers. We present the following improvement of this bound.

Let f(x) be an integer-valued polynomial with no fixed integer divisor [ges ] 2, that is, for no integer d [ges ] 2 does d[mid ]f(x) for all integers x. One generalization of the famous Waring problem is to determine whether, for large enough s, the equation

formula here

is solvable in positive integers x1, …, xs for sufficiently large integers n. The existence of such s for every f was established by Kamke [5] in 1921. Subsequent authors (Pillai, Hua [2–4], Vinogradov, Načaev [7], and others) have studied the problem of bounding G(f), the least s for which (1.1) is solvable for all large n.

Questions of local solubility of (1.1), that is, solubility of the congruence

formula here

play a more important and complicated role in this problem than in the classical Waring problem. Let Γ0(f) denote the least number s so that (1.2) is solvable for every pair n, q. It is well known that Γ0(xk) [les ] 4k for every k, but Hua [4] found that for every k, the polynomial

formula here

satisfies Γ0(fk) [ges ] 2k − 1 (take s = 2k−2, q = 2k and n = (−1)k in (1.2)). Clearly G(f) [ges ] Γ0(f), but one can say more by restricting the values of n under consideration, as has been done by several authors in the case f(x) = x4 (for example [1, 6]).

The singular series

formula here

where e(z) = e2πiz, encapsulates the local solubility information. In particular, [Sfr ]s, f(n) [ges ] 0 for every n and [Sfr ]s, f(n) > 0 if and only if (1.2) is soluble for every q. Define G(f) to be the least number s so that for every δ >0 and every n>n0(δ) with [Sfr ]s, f(n) [ges ] δ, (1.1) is soluble. The reason for taking [Sfr ]s, f(n) [ges ] δ instead of [Sfr ]s, f(n) > 0 is that we wish to exclude from consideration certain n lying in sparse sequences for which (1.1) is insoluble but [Sfr ]s, f(n) > 0. For example, taking f(x) = x4, s = 15 and nj = 79·16j (j = 0, 1, …), it can be shown that (1.1) is not soluble for n = nj, that [Sfr ]s, f(nj) > 0 for all j, and that [Sfr ]s, f(nj) → 0 as j → ∞. It is known that G(x4) = 16 (see [1]) and that G(x4) [ges ] 11 almost holds (see [6]).

In Merel's recent proof [7] of the uniform boundedness conjecture for the torsion of elliptic curves over number fields, a key step is to show that for sufficiently large primes N, the Hecke operators T1, T2, …, TD are linearly independent in their actions on the cycle e from 0 to i∞ in H1(X0(N) (C), Q). In particular, he shows independence when max(D8, 400D4) < N/(log N)4. In this paper we use analytic techniques to show that one can choose D considerably larger than this, provided that N is large.

In 1920 Chowla made the following conjecture. Let pn denote the nth prime; if q [ges ] 3, (q, a) = 1 then there are infinitely many pairs of consecutive primes pn and pn+1 such that

formula here

By considering the sum

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where χ is the non-principal character modulo 4 or 6, it is possible to prove the conjecture for q = 4 and q = 6 (a = ±1). In this paper we prove Chowla's conjecture for all q and a with (q, a) = 1. Moreover, we shall show that for any k there exist ‘strings’ of congruent primes such that

formula here

For each modulus q the method used applies best to the following two sets of residue classes:

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Larger values of k in terms of pn+1 can be found for residue classes belonging to these sets.

Let d be a square-free number and let CL(−d) denote the ideal class group of the imaginary quadratic number field ℚ(√−d). Further let h(−d) = #CL(−d) denote the class number. For integers g [ges ] 2, we define [Nscr ]g(X) to be the number of square-free d [ges ] X such that CL(−d) contains an element of order g. Gauss' genus theory demonstrates that if d has at least two odd prime factors (in particular, for almost all d) then CL(−d) contains ℤ2 as a subgroup. Thus N2(X) ∼ 6X/π2. The behaviour of [Nscr ]g(X) is not understood for any other value of g. It is believed that [Nscr ]g(X) ∼ CgX for some positive constant Cg. For odd primes g, H. Cohen and H. Lenstra [3] conjectured that

formula here

N. Ankeny and S. Chowla [1] first showed that [Nscr ]g(X) → ∞ as X → ∞. Although they did not point this out, their method demonstrates that [Nscr ]g(X) [Gt ] X1/2. Recently, M. R. Murty [11] improved this to [Nscr ]g(X) [Gt ] X1/2+1/g. Hitherto this represented the best known lower bounds for [Nscr ]g(X) except in the cases g = 4 and g = 8. In the cases g = 4 or 8, P. Morton [9] used class field theory techniques to show that [Nscr ]g(X) [Gt ] X1−ε. In fact, he demonstrated the elegant result that given any non-negative integers r, s and t, there are ‘many’ d with CL(−d)/CL(−d)8 = ℤr2 × ℤs4 × ℤt8 (see [9] for a precise statement). The complementary question of finding d with p [nmid ] h(−d) has also attracted a lot of attention. H. Davenport and H. Heilbronn [5] proved the striking result that the proportion of d with 3 [nmid ] h(−d) is at least 1/2. For larger primes p, recently W. Kohnen and K. Ono [7] have shown that there are [Gt ] √X/log X square- free integers d [les ] X such that p [nmid ] h(−d).

In this paper, we sharpen Murty's lower bounds on [Nscr ]g(X) for all values of g; see Theorem 1 below. We also offer a simple proof that [Nscr ]4(X) [Gt ] X/√log X; see Proposition 2 below. In §5 we express the hope that these methods may lead to [Nscr ]3(X) [Gt ] X1−ε.

A formula of Poisson summation type is established for sums involving relative norms in an extension of number fields.

It is determined when a block for the reduced enveloping algebra of a classical Lie algebra corresponding to a character in Levi form has finite representation type. These results refine earlier work of Premet.

The paper gives the best quantitative forms of Kronecker's theorem.

A certain property of some type-definable subgroups of superstable groups with finite U-rank is closely related to the Mordell–Lang conjecture. This property is discussed in the context of algebraic groups.

In [1] Brauer puts forward a series of questions on group representation theory in order to point out areas which were not well understood. One of these, which we denote by (B1), is the following: what information in addition to the character table determines a (finite) group? In previous papers [5, 7–13], the original work of Frobenius on group characters has been re-examined and has shed light on some of Brauer's questions, in particular an answer to (B1) has been given as follows.

Frobenius defined for each character χ of a group G functions χ(k)[ratio ]G(k) → [Copf ] for k = 1, …, degχ with χ(1) = χ. These functions are called the k-characters (see [10] or [11] for their definition). The 1-, 2- and 3-characters of the irreducible representations determine a group [7, 8] but the 1- and 2-characters do not [12]. Summaries of this work are given in [11] and [13].

The paper studies the permutation representations of a finite general linear group, first on finite projective space and then on the set of vectors of its standard module. In both cases the submodule lattices of the permutation modules are determined. In the case of projective space, the result leads to the solution of certain incidence problems in finite projective geometry, generalizing the rank formula of Hamada. In the other case, the results yield as a corollary the submodule structure of certain symmetric powers of the standard module for the finite general linear group, from which one obtains the submodule structure of all symmetric powers of the standard module of the ambient algebraic group.

Let q be a prime and χ be a non-principal character modulo q. Let

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where 1 [les ] t [les ] q is the character polynomial associated to χ (cyclically permuted t places). The principal result is that for any non-principal and non-real character χ modulo q and 1 [les ] t [les ] q,

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where the implicit constant is independent of t and q. Here ∥·∥4 denotes the L4 norm on the unit circle.

It follows from this that all cyclically permuted character polynomials associated with non-principal and non-real characters have merit factors that approach 3. This complements and completes results of Golay, Høholdt and Jensen, and Turyn (and others). These results show that the merit factors of cyclically permuted character polynomials associated with non-principal real characters vary asymptotically between 3/2 and 6.

The averages of the L4 norms are also computed. Let q be a prime number. Then

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where the summation is over all characters modulo q.

A finite non-trivial group G is called a Hurwitz group if it is an image of the infinite triangle group

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Thus G is a Hurwitz group if and only if it can be generated by an involution and an element of order 3 whose product has order 7. The history of Hurwitz groups dates back to 1879, when Klein [9] was studying the quartic

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of genus 3. The automorphism group of this curve has order 168 = 84(3−1), and it is isomorphic to the simple group PSL2(7), which is generated by the projective images of the matrices

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with product

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and so is a Hurwitz group. In 1893, Hurwitz [7] proved that the automorphism group of an algebraic curve of genus g (or, equivalently, of a compact Riemann surface of genus g) always has order at most 84(g−1), and that, moreover, a finite group of order 84(g−1) can act faithfully on a curve of genus g if and only if it is an image of Δ(2, 3, 7).

The problem of determining which finite simple groups are Hurwitz groups has received considerable attention. In [10], Macbeath classified the Hurwitz groups of type PSL2(q); there are infinitely many of them. In [1] Cohen proved that no group PSL3(q) is a Hurwitz group except PSL3(2), which is isomorphic to PSL2(7). Certain exceptional groups of Lie type, and some of the sporadic groups, are known to be Hurwitz groups. For discussions of the results on these groups we refer the reader to [3, 5, 11].

Let G be a group and P be a property of groups. If every proper subgroup of G satisfies P but G itself does not satisfy it, then G is called a minimal non-P group. In this work we study locally nilpotent minimal non-P groups, where P stands for ‘hypercentral’ or ‘nilpotent-by-Chernikov’. In the first case we show that if G is a minimal non-hypercentral Fitting group in which every proper subgroup is solvable, then G is solvable (see Theorem 1.1 below). This result generalizes [3, Theorem 1]. In the second case we show that if every proper subgroup of G is nilpotent-by-Chernikov, then G is nilpotent-by-Chernikov (see Theorem 1.3 below). This settles a question which was considered in [1–3, 10]. Recently in [9], the non-periodic case of the above question has been settled but the same work contains an assertion without proof about the periodic case.

The main results of this paper are given below (see also [13]).