Let P(α, β)n (x) be the Jacobi polynomials which are defined by

formula here

or given as hypergeometric series by

formula here

They satisfy the orthogonality relations

formula here

Asymptotic expansions of the Barnes double zeta-function

formula here

and the double gamma-function Γ2(α, (1, w)), with respect to the parameter w, are proved. An application to Hecke L-functions of real quadratic fields is also discussed.

In this paper we establish a theory of (quasi-)Fredholm elements in the setting of Jordan pairs. As for associative algebras, Barnes idempotents are introduced and a new numerical invariant, namely the cocapacity, is defined.

In this paper we prove that if n>30 030, then the nth element of any Lucas or Lehmer sequence has a primitive divisor.

Let p be an odd prime number. Let k be a [pfr ]-number field and [ofr ] the ring of all integers of k with a prime element π. Let K/k be a cyclic extension with Galois group G, and [Afr ] the associated order of the ring [Ofr ] of all integers in K:

[Afr ]={f∈kG[mid ]f[Ofr ]⊆[Ofr ]}.

F. Bertrandias and M-J. Ferton [1] obtained necessary and sufficient conditions that [Ofr ] is [Afr ]-free in the case K/k is of degree p. The purpose of this paper is to study such conditions in case K/k is a cyclic totally ramified Kummer extension of degree n=pm.

The study of the differential geometry of surfaces in 3-space has a long and celebrated history. Over the last 20 years a new approach using techniques from singularity theory has yielded some interesting results (see, for example [3, 5, 19] for surveys).

Of course surfaces arise in a number of ways: they are often defined explicitly as the image of a mapping f: R2→R3. Since the subject is differential geometry one normally asks that these defining mappings are smooth, that is infinitely differentiable, however it is not true, in any sense, that most such parametrisations will yield manifolds. For such mappings have self-intersections, and more significantly they may have crosscaps (also known as Whitney umbrellas). Moreover if we perturb the maps these singularities will persist; that is they are stable. (See [14, 20] for details.)

Consequently when studying the differential geometry of surfaces in 3-space there are good reasons for studying surfaces with crosscaps.

In this paper we carry out a classification of mappings from 3-space to lines, up to changes of co-ordinates in the source preserving a crosscap. We can apply our results to the geometry of generic crosscap points. In [10] we computed geometric normal forms for the crosscap and used them to study the dual of the crosscap. We shall see that the approach here yields more information than that obtained in [10], although the latter has the advantage of being more explicit (in particular various aspects of the geometry can be compared using the normal forms).

We refer the reader to [16, 18] for background material concerning singularity theory.

A framework is given for defining and comparing factorizability theories (as used, for example, to relate modules over the group rings of finite groups). The strict theory of Fröhlich [4] and the Hecke theory of [7] as well as several other related theories are shown to be equivalent. Of particular interest among the several equivalent theories examined is that of normaliser factorizability. This is perhaps the most user friendly theory and follows most closely the framework of the classical theory. Among other tools, the theory of permutation projectives and their species is used.

The purpose of this note is to present examples of nilpotent group extensions which split at every p-localization but fail to split. These examples disprove the conjecture by Peter Hilton [4, 5] that any nilpotent group extension with finitely generated quotient which splits at every prime must necessarily split. In [3] Hilton proves his conjecture for extensions with abelian kernel. Moreover, some further special cases of the conjecture are established by Casacuberta and Hilton in [1].

We describe two families of counterexamples to Hilton's conjecture. In both families the groups are torsion-free and finitely generated. The first family consists of groups of class 2 and rank 6. In the second family, the groups have rank 5, which, in view of Hilton's result in [3], is the minimal possible rank for a torsion-free counterexample to the conjecture. However, this reduction in rank is obtained at the expense of increasing the class from 2 to 3. Theorem 2·3 establishes that this increase in class is unavoidable: there are no torsion-free counterexamples of rank 5 and class 2.

The proof of Theorem 2·3 is based on a fact, Lemma 2·4, about quadratic forms. The author would like to thank J. W. S. Cassels for providing him with this crucial fact together with an outline of its proof.

The arc index α(L) of a link L is shown by a direct combinatorial argument to be related to sprv(FL(v, z)), the Laurent degree of its Kauffman polynomial, by the inequality

α(L)[ges ]sprv (FL(v, z))+2.

Equality is conjectured for alternating links.

In this paper, we prove the following:

Theorem. Let C⊂ℙ3be a complex reduced irreducible space curve which is non- degenerate (i.e., C is not contained in any plane). Let p: C→ℙ2be a general projection. Then the topological fundamental group π1(ℙ2/p(C)) is abelian.

Baumslag [Bm1] and Remeslennikov [R1] independently showed that the abstract group with finite presentation

〈a, x, y[mid ]ax =aay, [x, y] =1=[a, ay]〉.

is metabelian; the derived subgroup is free abelian of countably infinite rank. Both generalized this example to show that every finitely generated metabelian group embeds in a finitely presented metabelian group. Thomson [T] generalized this result still further to show that every finitely generated group of upper triangular matrices over an abstract ring A lies inside a finitely presented group of upper triangular matrices over some extension ring of A. For a valuable survey of these results, see [St].

Baumslag proved analogous theorems for Lie algebras in [Bm2]. In this paper we shall prove some analogous results for pro-p groups.

Let E be an elliptic curve with complex multiplication by the ring of integers [Ofr ] of an imaginary quadratic field K. The purpose of this paper is to describe certain connections between the arithmetic of E on the one hand and the Galois module structure of certain arithmetic principal homogeneous spaces arising from E on the other. The present paper should be regarded as a complement to [AT]; we assume that the reader is equipped with a copy of the latter paper and that he is not averse to referring to it from time to time.

The starting point of this paper is the maximal extension Γ*t of Γt, the subgroup of Sp4(ℚ) which is conjugate to the paramodular group. Correspondingly we call the quotient [Ascr ]*t=Γ*t\ℍ2 the minimal Siegel modular threefold. The space [Ascr ]*t and the intermediate spaces between [Ascr ]t=Γt\ℍ2 which is the space of (1, t)-polarized abelian surfaces and [Ascr ]*t have not yet been studied in any detail. Using the Torelli theorem we first prove that [Ascr ]*t can be interpreted as the space of Kummer surfaces of (1, t)-polarized abelian surfaces and that a certain degree 2 quotient of [Ascr ]t which lies over [Ascr ]*t is a moduli space of lattice polarized K3 surfaces. Using the action of Γ*t on the space of Jacobi forms we show that many spaces between [Ascr ]t and [Ascr ]*t possess a non-trivial 3-form, i.e. the Kodaira dimension of these spaces is non-negative. It seems a difficult problem to compute the Kodaira dimension of the spaces [Ascr ]*t themselves. As a first necessary step in this direction we determine the divisorial part of the ramification locus of the finite map [Ascr ]t→[Ascr ]*t. This is a union of Humbert surfaces which can be interpreted as Hilbert modular surfaces.

The Bianchi groups are the groups PSL2([Oscr ]d), where [Oscr ]d denotes the ring of integers of the field ℚ (√−d) for each square-free positive integer d. These groups have long been of interest, not only because of their intrinsic interest as abstract groups, but also because they arise naturally in number theory and geometry. For a discussion of their algebraic properties we refer the reader to Fine [4]. Among the groups PSL2(R), with R the ring of integers of an algebraic number field, they are distinguished by the nature of their normal subgroup structure. It was shown by Serre [20] that if R is not isomorphic to ℤ or [Oscr ]d, then for every normal subgroup K of SL2(R) there is an ideal I of R such that the image in SL2(R)/K of the kernel of the natural map from SL2(R) to SL2(R/I) is central and isomorphic to a subgroup of the group of roots of 1 in R. On the other hand, the group PSL2(ℤ) and the Bianchi groups have many subgroups of finite index which are not of the above type: this follows easily from the fact that PSL2(ℤ) is a free product of a group of order 2 and a group of order 3, and the fact, proved by Grunewald and Schwermer [6], that each Bianchi group has a normal subgroup of finite index which can be mapped epimorphically to a non-abelian free group.

K-loops have their origin in the theory of sharply 2-transitive groups. In this paper a proof is given that K-loops and Bruck loops are the same. For the proof it is necessary to show that in a (left) Bruck loop the left inner mappings L(b)L(a) L(ab)−1 are automorphisms. This paper generalizes results of Glauberman [3], Kist [8] and Kreuzer [9].

Recall that the first homology group H1(G) of a group G is the derived quotient G/[G, G]. The first homology groups of the mapping class groups of closed orientable surfaces are well known. Let F be a closed orientable surface of genus g. Recall that the extended mapping class group [Mscr ]*F of the surface F is the group of the isotopy classes of self-homeomorphisms of F. The mapping class group [Mscr ]F of F is the subgroup of [Mscr ]*F consisting of the isotopy classes of orientation-preserving self-homeomorphisms of F. It is well known that [Mscr ]F is trivial if F is a sphere. Hence the first homology group of the mapping class group of a sphere is trivial. If the genus of F is at least three, then H1([Mscr ]F) is again trivial. This result is due to Powell [P]. The group H1([Mscr ]F) is Z10 if the genus of F is two, proved by Mumford [Mu], and Z12 if F is a torus. When a problem about orientable surfaces is solved, it is natural to ask the corresponding problem for nonorientable surfaces. This is our motivation for the present paper.

Our main purpose is to explore the relationship between normal subgroups of Hecke groups and regular maps on compact orientable surfaces. We use regular maps to find all the normal subgroups of Hecke groups of index [les ]60. In Sections 1–4 we review the basic results concerning Hecke groups and normal subgroups of Fuchsian groups and in Section 5 we outline the basic facts we need about regular maps. The regular maps of genus 0 are the Platonic solids and in Section 6 we use these to completely determine the genus zero normal subgroups of Hecke groups. The regular maps of genus 1 were determined this century by Brahana [1]. We use them in Sections 7 and 8 to determine the genus 1 normal subgroups of Hecke groups. We give alternative proofs and extend theorems of M. Newman and also Kern-Isberner and Rosenberger. Unlike the genus 0 and 1 cases there are only finitely many regular maps of genus g[ges ]2. In Section 9 we use more recent results concerning their classification to find all normal subgroups of Hecke groups of index [les ]60. This was done for the modular group in [13].

We introduce in this paper the notion of a normal system for a compact, Hausdorff, right topological group, G. This generalizes the notion of a strong normal system as introduced in [7], in which it was shown that closed subgroups of compact right topological groups with dense topological centres possess strong normal systems and have Haar measure. Their method involved the construction of a strong normal system, the Furstenberg–Namioka system. In this paper we study a variation of the Furstenberg–Namioka system which we call the N-system. The N-system for a group G is a normal system and we show that if G has dense topological centre, it possesses an N-system, though the two systems do not in general coincide. In Section 2 we give examples of groups with arbitrarily long N-systems and show that a dense topological centre is not a necessary requirement for the existence of normal systems. In Section 3 we introduce the notion of minimality of normal systems and we show that if the N-system for a group, G, is minimal then the N-system and the Furstenberg–Namioka system, if the latter exists, coincide (and therefore G has Haar measure!). Finally in our main theorem, we show that if G has a minimal N-system then the closed normal subgroups of G must lie between successive terms of the N-system.

In [11], we considered the question of existence of elliptic fibre space structures on a smooth Calabi–Yau threefold X. Necessary conditions are that there exists a nef divisor D on X with D3=0; D2[nequiv ]0 and D·c2[ges ]0. In the case when D·c2[ges ]0, it was observed that a result from [9] implies that there is an elliptic fibre space structure determined by D, and so [11] concerned itself with the case when D·c2=0. In this case, we were not able to deduce in general the existence of an elliptic fibre space structure, but only in the presence of a further condition. In particular, it follows from the Theorem in [11] that, if r denotes the number of rational surfaces E on X with D[mid ]E≡0, and if the Euler characteristic e(X)≠2r, then some positive multiple of D determines an elliptic fibre space structure on X (the theorem in fact proves a slightly stronger result, which gives the elliptic fibre space structure if there is any non-rational surface E on X with D[mid ]E≡0). The purpose of this note is to clarify the meaning of the above rather mysterious condition on the non-vanishing of e(X)/2−r.

The cabling conjecture states that a non-trivial knot K in the 3-sphere is a cable knot or a torus knot if some Dehn surgery on K yields a reducible 3-manifold. We prove that symmetric knots satisfy this conjecture. (Gordon and Luecke also prove this independently ([GLu3]), by a method different from ours.)