For a wide class of one-parameter families of Thue equations of arbitrary degree n[ges ]3 all solutions are determined if the parameter is sufficiently large. The result is based on the Lang–Waldschmidt conjecture, on the primitivity of the associated number fields and on an index bound, which does not depend on the coefficients. By applying the theory of Hilbertian fields and results on thin sets, primitivity is proved for almost all choices (in the sense of density) of the parameters.

It is proved that the number of countable models of a countable supersimple theory is either 1 or infinite. This result is an extension of Lachlan‘s theorem on a superstable theory.

New lower bounds are given for the size of a point set in a Desarguesian projective plane over a finite field that contains at least a prescribed number s of points on every line. These bounds are best possible when q is square and s is small compared with q. In this case the smallest set is shown to be the union of disjoint Baer subplanes. The results are based on new results on the structure of certain lacunary polynomials, which can be regarded as a generalization of Rédei's results in the case when the derivative of the polynomial vanishes.

One of the most famous theorems in number theory states that there are infinitely many positive prime numbers (namely p = 2 and the primes p ≡ 1 mod4) that can be represented in the form x21+x22, where x1 and x2 are positive integers. In a recent paper, Fouvry and Iwaniec [2] have shown that this statement remains valid even if one of the variables, say x2, is restricted to prime values only. In the sequel, the letter p, possibly with an index, is reserved to denote a positive prime number. As p21+p22 = p is even for p1, p2 > 2, it is reasonable to conjecture that the equation p21+p22 = 2p has an infinity of solutions. However, a proof of this statement currently seems far beyond reach. As an intermediate step in this direction, one may quantify the problem by asking what can be said about lower bounds for the greatest prime divisor, say P(N), of the numbers p21+p22, where p1, p2 [les ] N, as a function of the real parameter N [ges ] 1. The well-known Chebychev–Hooley method combined with the Barban–Davenport–Halberstam theorem almost immediately leads to the bound P(N) [ges ] N1−ε, if N [ges ] No(ε); here, ε denotes some arbitrarily small fixed positive real number. The first estimate going beyond the exponent 1 has been achieved recently by Dartyge [1, Théorème 1], who showed that P(N) [ges ] N10/9−ε. Note that Dartyge's proof provides the more general result that for any irreducible binary form f of degree d [ges ] 2 with integer coefficients the greatest prime divisor of the numbers [mid ]f(p1, p2)[mid ], p1, p2 [les ] N, exceeds Nγd−ε, where γd = 2 − 8/(d + 7). We in particular want to point out that Dartyge does not make use of the specific features provided by the form x21+x22. By taking advantage of some special properties of this binary form, we are able to improve upon the exponent γ2 = 10/9 considerably.

Let E2(T) denote the error term in the asymptotic formula for ∫T0[mid ]ζ(½+it)[mid ]4dt. It is proved that there exist constants A>0, B>1 such that for T[ges ]T0>0 every interval [T, BT] contains points T1, T2 for which

formula here

and that ∫T0[mid ]E2(t) [mid ]adt[Gt ]T1+(a/2) for any fixed a[ges ]1. These results complement earlier results of Motohashi and Ivić that ∫T0E2(t) dt[Lt ]T3/2 and that ∫T0E22(t) dt[Lt ]T2logCT for some C>0. Omega-results analogous to the above ones are obtained also for the error term in the asymptotic formula for the Laplace transform of [mid ]ζ(½+it)[mid ]4.

The paper shows that any graph G with the maximum degree Δ(G) [ges ] 8, which is embeddable in a surface Σ of Euler characteristic χ(Σ) [ges ] 0, is totally (Δ(G)+2)-colorable. In general, it is shown that any graph G which is embeddable in a surface Σ and satisfies the maximum degree Δ(G) [ges ] (20/9) (3−χ(Σ))+1 is totally (Δ(G)+2)-colorable.

A proper vertex-colouring of a graph is acyclic if there are no 2-coloured cycles. It is known that every planar graph is acyclically 5-colourable, and that there are planar graphs with acyclic chromatic number χa = 5 and girth g = 4. It is proved here that a planar graph satisfies χa [les ] 4 if g [ges ] 5 and χa [les ] 3 if g [ges ] 7.

An integral version of a classical embedding theorem concerning quaternion algebras B over a number field k is proved. Assume that B satisfies the Eichler condition, that is, some infinite place of k is not ramified in B, and let Ω be an order in a quadratic extension of k. The maximal orders of B which admit an embedding of Ω are determined. Although most Ω embed into either all or none of the maximal orders of B, it turns out that some Ω are ‘selective’, in the sense that they embed into exactly half of the isomorphism types of maximal orders of B. As an application, the maximal arithmetic subgroups of B*/k* which contain a given element of B*/k* are determined.

Let F1, …, Ft be diagonal forms of degree k with real coefficients in s variables, and let τ be a positive real number. The solubility of the system of inequalities

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in integers x1, …, xs has been considered by a number of authors over the last quarter-century, starting with the work of Cook [9] and Pitman [13] on the case t = 2. More recently, Brüdern and Cook [8] have shown that the above system is soluble provided that s is sufficiently large in terms of k and t and that the forms F1, …, Ft satisfy certain additional conditions. What has not yet been considered is the possibility of allowing the forms F1, …, Ft to have different degrees. However, with the recent work of Wooley [18,20] on the corresponding problem for equations, the study of such systems has become a feasible prospect. In this paper we take a first step in that direction by studying the analogue of the system considered in [18] and [20]. Let λ1, …, λs and μ1, …, μs be real numbers such that for each i either λi or μi is nonzero. We define the forms

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and consider the solubility of the system of inequalities

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in rational integers x1, …, xs. Although the methods developed by Wooley [19] hold some promise for studying more general systems, we do not pursue this in the present paper. We devote most of our effort to proving the following theorem.

A metric space X has the unique midset property if there is a topology-preserving metric d on X such that for every pair of distinct points x, y there is one and only one point p such that d(x, p) = d(y, p). The following are proved. (1) The discrete space with cardinality [nfr ] has the unique midset property if and only if [nfr ] ≠ 2, 4 and [nfr ] [les ] [cfr ], where [cfr ] is the cardinality of the continuum. (2) If D is a discrete space with cardinality not greater than [cfr ], then the countable power DN of D has the unique midset property. In particular, the Cantor set and the space of irrational numbers have the unique midset property.

A finite discrete space with n points has the unique midset property if and only if there is an edge colouring ϕ of the complete graph Kn such that for every pair of distinct vertices x, y there is one and only one vertex p such that ϕ(xp) = ϕ(yp). Let ump(Kn) be the smallest number of colours necessary for such a colouring of Kn. The following are proved. (3) For each k [ges ] 0, ump(K2k+1) = k. (4) For each k [ges ] 3, k [les ] ump(K2k) [les ] 2k−1.

Let G be a finite group and let K be a hilbertian field. Many finite groups have been shown to satisfy the arithmetic lifting property over K, that is, every G-Galois extension of K arises as a specialization of a geometric branched covering of the projective line defined over K. The paper explores the situation when a semidirect product of two groups has this property. In particular, it shows that if H is a group that satisfies the arithmetic lifting property over K and A is a finite cyclic group then G = A [rtimes ] H also satisfies the arithmetic lifting property assuming the orders of H and A are relatively prime and the characteristic of K does not divide the order of A. In this case, an arithmetic lifting for any A[wreath ]H-Galois extension of K is explicitly constructed and the existence of the arithmetic lifting for any G-Galois extension is deduced. It is also shown that if A is any abelian group, and H is the group with the arithmetic lifting property then A[wreath ]H satisfies the property as well, with some assumptions on the ground field K. In the construction properties of Hilbert sets in hilbertian fields and spectral sequences in étale cohomology are used.

In this paper we shall prove the main part ((3) implies (1)) of the following theorem. The proof of the theorem will be completed in a forthcoming paper [6].

The moduli theory of Calabi–Yau threefolds is investigated, and using Griffiths' work on the period map, some finiteness results are derived. In particular, a prediction of Morrison's cone conjecture is confirmed.

The paper describes the cell structure of the affine Temperley–Lieb algebra with respect to a monomial basis. A diagram calculus is constructed for this algebra.

The determination of the modular character tables of the sporadic O'Nan group, its automorphism group and its covering group is completed by the calculation of the 7-modular decomposition numbers. The results are obtained with the assistance of the systems GAP, MOC, and MeatAxe, and by applying new condensation methods.

A central issue in finite group modular representation theory is the relationship between the p-local structure and the p-modular representation theory of a given finite group. In [5], Broué poses some startling conjectures. For example, he conjectures that if e is a p-block of a finite group G with abelian defect group D and if f is the Brauer correspondent block of e of the normalizer, NG(D), of D then e and f have equivalent derived categories over a complete discrete valuation ring with residue field of characteristic p. Some evidence for this conjecture has been obtained using an important Morita analog for derived categories of Rickard [11]. This result states that the existence of a tilting complex is a necessary and sufficient condition for the equivalence of two derived categories. In [5], Broué also defines an equivalence on the character level between p-blocks e and f of finite groups G and H that he calls a ‘perfect isometry’ and he demonstrates that it is a consequence of a derived category equivalence between e and f. In [5], Broué also poses a corresponding perfect isometry conjecture between a p-block e of a finite group G with an abelian defect group D and its Brauer correspondent p-block f of NG(D) and presents several examples of this phenomena. Subsequent research has provided much more evidence for this character-level conjecture.

In many known examples of a perfect isometry between p-blocks e, f of finite groups G, H there are also perfect isometries between p-blocks of p-local subgroups corresponding to e and f and these isometries are compatible in a precise sense. In [5], Broué calls such a family of compatible perfect isometries an ‘isotypy’.

In [11], Rickard addresses the analogous question of defining a p-locally compatible family of derived equivalences. In this important paper, he defines a ‘splendid tilting complex’ for p-blocks e and f of finite groups G and H with a common p-subgroup P. Then he demonstrates that if X is such a splendid tilting complex, if P is a Sylow p-subgroup of G and H and if G and H have the same ‘p- local structure’, then p-local splendid tilting complexes are obtained from X via the Brauer functor and ‘lifting’. Consequently, in this situation, we obtain an isotypy when e and f are the principal blocks of G and H.

Linckelmann [9] and Puig [10] have also obtained important results in this area.

In this paper, we refine the methods and program of [11] to obtain variants of some of the results of [11] that have wider applicability. Indeed, suppose that the blocks e and f of G and H have a common defect group D. Suppose also that X is a splendid tilting complex for e and f and that the p-local structure of (say) H with respect to D is contained in that of G, then the Brauer functor, lifting and ‘cutting’ by block indempotents applied to X yield local block tilting complexes and consequently an isotypy on the character level. Since the p-local structure containment hypothesis is satisfied, for example, when H is a subgroup of G (as is the case in Broué's conjectures) our results extend the applicability of these ideas and methods.

Cellular algebras have recently been introduced by Graham and Lehrer [5, 6] as a convenient axiomatization of all of the following algebras, each of them containing information on certain classical algebraic or finite groups: group algebras of symmetric groups in any characteristic, Hecke algebras of type A or B (or more generally, Ariki Koike algebras), Brauer algebras, Temperley–Lieb algebras, (q-)Schur algebras, and so on. The problem of determining a parameter set for, or even constructing bases of simple modules, is in this way reduced (but of course not solved in general) to questions of linear algebra.

The present paper has two aims. First, we make explicit an inductive construction of cellular algebras which has as input data of linear algebra, and which in fact produces all cellular algebras (but no other ones). This is what we call ‘inflation’. This construction also exhibits close relations between several of the above algebras, as can be seen from the computations in [6]. Among the consequences of the construction is a natural way of generalizing Hochschild cohomology. Another consequence is the construction of certain idempotents which is used in the second part of the paper.

The second aim is to study Morita equivalences of cellular algebras. Since the input of many of the constructions of representation theory of finite-dimensional algebras is a basic algebra, it is useful to know whether any finite-dimensional cellular algebra is Morita equivalent to a basic one by a Morita equivalence that preserves the cellular structure. It turns out that the answer is ‘yes’ if the underlying field has characteristic other than 2. However, there are counterexamples in the case of characteristic 2, or more generally for any ring in which 2 is not invertible. This also tells us that the notion of ‘cellular’ cannot be defined only in terms of the module category. However, in any characteristic we find some useful Morita equivalences which are compatible with cellular structures.

Let T be a d×d matrix with integral coefficients. Then T determines a self-map of the d-dimensional torus X = ℝd/ℤd. Choose for each natural number n a ball B(n) in X and suppose that B(n+1) has smaller radius than B(n) for all n. Now let W be the set of points x ∈ X such that Tn(x) ∈ B(n) for infinitely many n ∈ ℕ. The Hausdorff dimension of W is studied by analogy with the Jarník–Besicovitch theorem on the dimension of the set of well-approximable real numbers. The dimension depends on the quantity

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A complete description is given only when the matrix is diagonalizable over ℚ. In other cases a result is obtained for sufficiently large τ. The results, in as far as they go, show that the Hausdorff dimension of W is a strictly decreasing, continuous function of τ which is piecewise of the form (Aτ+B)/(Cτ+D). The numbers A, B, C and D which arise in this way are typically sums of logarithms of the absolute values of eigenvalues of T.

In [2] Bieri and Strebel introduced a geometric invariant for finitely generated abstract metabelian groups that determines which groups are finitely presented. For a valuable survey of their results, see [6]; we recall the definition briefly in Section 4. We shall introduce a similar invariant for pro-p groups.

Let [ ] be the algebraic closure of [ ]p and U be the formal power series algebra [ ][lobrk ]T[robrk ], with group of units U×. Let Q be a finitely generated abelian pro-p group. We write ℤp[lobrk ]Q[robrk ] for the completed group algebra of Q over ℤp. Let T(Q) be the abelian group Hom(Q, U×) of continuous homomorphisms from Q to U×. We write 1 for the trivial homomorphism. Each v∈T(Q) extends to a unique continuous algebra homomorphism vmacr; from ℤp[lobrk ]Q[robrk ] to U.

Two rings A and B are said to be derived Morita equivalent if the derived categories Db(Mod A) and Db(Mod B) are equivalent. If A and B are derived Morita equivalent algebras over a field k, then there is a complex of bimodules T such that the functor T[otimes ]LA−[ratio ]Db(Mod A) → Db(Mod B) is an equivalence. The complex T is called a tilting complex.

When B = A the isomorphism classes of tilting complexes T form the derived Picard group DPic(A). This group acts naturally on the Grothendieck group Ko(A).

It is proved that when the algebra A is either local or commutative, then any derived Morita equivalent algebra B is actually Morita equivalent. This enables one to compute DPic(A) in these cases.

Assume that A is noetherian. Dualizing complexes over A are complexes of bimodules which generalize the commutative definition. It is proved that the group DPic(A) classifies the set of isomorphism classes of dualizing complexes. This classification is used to deduce properties of rigid dualizing complexes.

Finally finite k-algebras are considered. For the algebra A of upper triangular 2×2 matrices over k, it is proved that t3 = s, where t, s ∈ DPic(A) are the classes of A*:= Homk(A, k) and A[1] respectively. In the appendix, by Elena Kreines, this result is generalized to upper triangular n×n matrices, and it is shown that the relation tn+1 = sn−1 holds.