Let C be a smooth proper curve of genus 2 over an algebraically closed field k. Fix a Weierstrass point ∞in C(k) and identify C with its image in its Jacobian J under the Albanese embedding that uses ∞ as base point. For any integer N[ges ]1, we write JN for the group of points in J(k) of order dividing N and J*N for the subset of JN of points of order N. It follows from the Riemann–Roch theorem that C(k)∩J2 consists of the Weierstrass points of C and that C(k)∩J*3 and C(k)∩J* are empty (see [3]). The purpose of this paper is to study curves C with C(k)∩J*5 non-empty.

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.

We are concerned in this paper with the ideal structure of group rings of infinite simple locally finite groups over fields of characteristic zero, and its relation with certain subgroups of the groups, called confined subgroups. The systematic study of the ideals in these group rings was initiated by the second author in [15], although some results had been obtained previously (see [3, 1]). Let G be an infinite simple locally finite group and K a field of characteristic zero. It is expected that in most cases, the group ring KG will have the smallest possible number of ideals, namely three, (KG itself, {0} and the augmentation ideal), and this has been verified in some cases. In some interesting cases, however, the situation is different, and there are more ideals. We mention in particular the infinite alternating groups [3] and the stable special linear groups [9], in which the ideal lattice has been completely determined. The second author has conjectured that the presence of ideals in KG, other than the three unavoidable ones, is synonymous with the presence in the group of proper confined subgroups. Here a subgroup H of a locally finite group G is called confined, if there exists a finite subgroup F of G such that Hg∩F≠1 for all g∈G. This amounts to saying that F has no regular orbit in the permutation representation of G on the cosets of H.

Let a=(a1, a2, a3, …) be an arbitrary infinite sequence in U=[0, 1). Let

formula here

Van der Corput [5] conjectured that d(a, n) (n=1, 2, …) is unbounded, and this was proved in 1945 by van Aardenne-Ehrenfest [1]. Later she refined this [2], obtaining

formula here

for infinitely many n. Here and later c1, c2, … denote positive absolute constants.

In 1954, Roth [8] showed that the quantity

formula here

is closely related to the discrepancy of a suitable point set in U2.

A linear operator between two Banach spaces X and Y is strictly-singular (or Kato) if it fails to be an isomorphism on any infinite dimensional subspace. A weaker notion for Banach lattices introduced in [8] is the following one: an operator T from a Banach lattice X to a Banach space Y is said to be disjointly strictly-singular if there is no disjoint sequence of non-null vectors (xn)n∈ℕ in X such that the restriction of T to the subspace [(xn)∞n=1] spanned by the vectors (xn)n∈ℕ is an isomorphism. Clearly every strictly-singular operator is disjointly strictly-singular but the converse is not true in general (consider for example the canonic inclusion Lq[0, 1][rarrhk ]Lp[0, 1] for 1[les ]p<q<∞). In the special case of considering Banach lattices X with a Schauder basis of disjoint vectors both concepts coincide. The notion of disjointly strictly-singular has turned out to be a useful tool in the study of lattice structure of function spaces (cf. [7–9]). In general the class of all disjointly strictly-singular operators is not an operator ideal since it fails to be stable with respect to the composition on the right.

The aim of this paper is to study when the inclusion operators between arbitrary rearrangement invariant function spaces E[0, 1] ≡ E on the probability space [0, 1] are disjointly strictly-singular operators.

A central limit theorem for a generalized Harper operator on a crystal lattice is obtained. As the limit, the continuous semigroup of a uniform magnetic Schrödinger operator is captured on a vector space equipped with a special Euclidean structure. The standard realization of the crystal lattice is a key to the Euclidean structure and a linear vector potential on the Euclidean space from combinatorial data of the generalized Harper operator.

We prove some general results about quasi-actions on trees and define Property (QFA), which is analogous to Serre's Property (FA), but in the coarse setting. This property is shown to hold for a class of groups, including SL$(n,\mathbb{Z})$ for $n\geq 3$. We also give a way of thinking about Property (QFA) by breaking it down into statements about particular classes of trees.

Let $1 < p <\infty, 0 < v < p^\prime$ , let $\Omega$ be a bounded domain in ${\bb R}^n$ , and denote by ${\rm id}_{\omega}$ the limiting compact embedding of the Besov space $B^{n/p}_{pp}({\bb R}^n)$ into the exponential Orlicz space $L_{\exp (t^v)}(\Omega)$ , mapping a function $f$ onto its restriction $f\vert_{\Omega}$ . In 1993 Triebel established, among others, two-sided estimates for the entropy numbers of ${\rm id}_{\omega}$ , which are even asymptotically optimal for ‘small’ $\nu$ . The aim of the paper is to improve the upper bounds in the case of ‘large’ $\nu$ , where Triebel's estimates are not yet sharp, thus making a further step towards the conjectured correct asymptotic behaviour.

Let $T$ be a sublinear operator such that $(T\chi_E)^*(t)\,{\le}\, h(t, |E|)$ for some positive function $h(t,s)$ and every measurable set $E$. Then it is shown that under some conditions on the operator $T$, this restricted weak type estimate can be extended to the set of all functions $f\,{\in}\, L^1$ such that $\n f.\infty\,{\le}\, 1$, in the sense that $(Tf)^*(t)\,{\le}\, h(t, \n f.1)$. This inequality allows strong type estimates for $T$ to be obtained on several classes of spaces, such as logarithmic spaces and Lorentz spaces. A similar problem for operators $T$ acting on radial functions is also studied.

Conjecture: Let [Mscr ] and [Nscr ] be two matroids (possibly of infinite ranks) on the same set S. Then there exists a set I independent in both [Mscr ] and [Nscr ], which can be partitioned as I=H∪K, where sp[Mscr ](H)∪sp[Nscr ](K)=S. This conjecture is an extension of Edmonds' matroid intersection theorem to the infinite case. We prove the conjecture when one of the matroids (say [Mscr ]) is the sum of countably many matroids of finite rank (the other matroid being general). For the proof we have also to answer the following question: when does there exist a subset of S which is spanning for [Mscr ] and independent in [Nscr ]?

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].

In this paper we are concerned with abstract homomorphisms from one algebraic group to another. That is, group homomorphisms where there is no assumption of rationality. Let K be an infinite perfect field and L an algebraically closed field of the same characteristic. Throughout G(K) will denote a Chevalley group of universal type over K.

It is proved that a cosine operator function C(·), with generator A, is locally of bounded semivariation if and only if u″(t) = Au(t)+f(t), t>0, u(0), u′(0)∈D(A), has a strong solution for every continuous function f, if and only if the function ∫t0∫t−s0C(τ) f(s)dτds, t>0, is twice continuously differentiable for every continuous function f, that is, C(·) has the maximal regularity property if and only if A is a bounded operator. Some other characterisations of bounded generators of cosine operator functions are also established in terms of their local semivariations.

A connection and a curving on a bundle gerbe associated with lifting a structure group of a principal bundle to a central extension are constructed. The construction is based on certain structures on the bundle, that is, connections and bundle splittings. The Deligne cohomology class of the lifting bundle gerbe with the connection and with the curving coincides with the obstruction class of the lifting problem with these structures.

The depth of the associated graded ring of the powers of an ideal $I$ of a local ring $R$ is studied. It is proved that the depth of the associated graded ring of $I^n$ is asymptotically constant when $n$ tends to infinity, and this value is characterized in terms of Valabrega–Valla conditions of $I^m$ for some large integer $m\ge 0$. As a corollary, a generalization is obtained of the $2$-dimensional algebraic version of the Grauert–Riemenschneider vanishing theorem (due to Huckaba and Huneke) to ideals satisfying the second Valabrega–Valla condition. The positiveness of Hilbert coefficients is also studied, and Valabrega–Valla conditions are linked to the vanishing of the cohomology groups of the closed fiber of the blowing up of ${\bf Spec}(R)$ along the closed sub-scheme defined by $I$.

We consider aperiodic shifts of finite type σ with an equilibrium state m and associated skew-products σf where f: X→G is Hölder and G is a compact Lie group. We show that generically σf is weak-mixing and give constructive methods for achieving weak-mixing by perturbing an arbitrary f at a finite number of small neighbourhoods. When σf is not ergodic we describe the (closed) ergodic decomposition precisely. The key result shows that certain measurable eigenfunctions are essentially Hölder continuous. This leads to conditions for weak-mixing or ergodicity in terms of a functional equation which involves Hölder rather than measurable functions.

The paper describes the Garnier systems as isomonodromic deformation equations of a linear system with a simple pole at 0 and a Poincaré rank 1 singularity at infinity. The extension of Okamoto's birational canonical transformations to the Garnier systems in more than one variable and to the Schlesinger systems is discussed.

We prove Landesman–Lazer type existence conditions for the solutions bounded on the real line, together with their first derivatives, for some second order nonlinear differential equations of the form x″+cx′+f(t, x)=0. The proofs use upper and lower solutions techniques together with some limiting arguments.

It is determined which Bloch-type conditions on a function $f\in \bigcap_{0 < p <\infty}H^p$ ensure that $f\in {\rm BMOA}$ .

Estimates for the Dirichlet eigenfunctions near the boundary of an open, bounded set in euclidean space are obtained. It is assumed that the boundary satisfies a uniform capacitary density condition.