Asymptotic expansions of the Barnes double zeta-function

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

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.

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.

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.

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.)

Let [Ascr ]* be the mod-2 Steenrod algebra of cohomology operations and χ its canonical antiautomorphism. For all positive integers f and k, we show that the excess of the element χ[Sq (2k−1f)· Sq (2k−2f)… Sq (2f)·Sq (f)] is (2k−1)μ(f), where μ(f) denotes the minimal number of summands in any representation of f as a sum of numbers of the form 2i−1. We also interpret this result in purely combinatorial terms. In so doing, we express the Milnor basis representation of the products Sq (a1)…Sq (an) and χ[Sq (a1)…Sq (an)] in terms of the cardinalities of certain sets of matrices.

For s[ges ]1, let ℙs=[ ]2= [x1, …, xs] be the mod-2 cohomology of the s-fold product of ℝP∞ with itself, with its usual structure as an [Ascr ]*-module. A polynomial P∈ℙs is hit if it is in the image of the action [Ascr ]*×ℙs→ℙs, where [Ascr ]* is the augmentation ideal of [Ascr ]*. We prove that if the integers e, f, and k satisfy e<(2k−1)μ(f), then for any polynomials E and F of degrees e and f respectively, the product E·F2k is hit. This generalizes a result of Wood conjectured by Peterson, and proves a conjecture of Singer and Silverman.

Let Γ be a connected G-symmetric graph of valency r, whose vertex set V admits a non-trivial G-partition [Bscr ], with blocks B∈[Bscr ] of size v and with k[les ]v independent edges joining each pair of adjacent blocks. In a previous paper we introduced a framework for analysing such graphs Γ in terms of (a) the natural quotient graph Γ[Bscr ] of valency b=vr/k, and (b) the 1-design [Dscr ](B) induced on each block. Here we examine the case where k=v and Γ[Bscr ]=Kb+1 is a complete graph. The 1-design [Dscr ](B) is then degenerate, so gives no information: we therefore make the additional assumption that the stabilizer G(B) of the block B acts 2-transitively on B. We prove that there is then a unique exceptional graph for which [mid ]B[mid ]=v>b+1.

Analysis of projections of a convex body is a familiar topic in tomography. However, instead of considering standard projection bodies, this work investigates a convex body introduced by Schneider [8] which is a Minkowski average of projections. The question addressed here is similar to that posed by Goodey and Weil [4] with respect to Minkowski averages of sections, as opposed to projections, that is, can the shape of a convex body be determined from random sections? Their main result shows that a body K is determined by the average of its two-dimensional sections, but not by the average of its one-dimensional sections. The goal of this study is to uncover the extent to which a convex body is determined by the average of its projections.