For a hyperelliptic curve ${\hbox{\ax C}}$ of genus $g$ with a divisor class of order $n = g + 1$, we shall consider an associated covering collection of curves ${\hbox{\ax D}}_\delta$, each of genus $g^2$. We describe, up to isogeny, the Jacobian of each ${\hbox{\ax D}}_\delta$ via a map from ${\hbox{\ax D}}_\delta$ to ${\hbox{\ax C}}$, and two independent maps from ${\hbox{\ax D}}_\delta$ to a curve of genus $g(g-1)/2$. For some curves, this allows covering techniques that depend on arithmetic data of number fields of smaller degree than standard 2-coverings; we illustrate this by using 3-coverings to find all ${\Bbb Q}$-rational points on a curve of genus 2 for which 2-covering techniques would be impractical.

Let $a$ be a positive integer with $a > 1$. We show \[ [0;a,a^2,a^3,a^4,\ldots]=\frac{\sum\nolimits^{\infty}_{s=0}a^{-(s+1)^2}\prod\nolimits^s_{i=1}(a^{2i}-1)^{-1}}{\sum\nolimits^{\infty}_{s=0}a^{-s^2}\prod\nolimits^s_{i=1}(a^{2i}-1)^{-1}} \] and \[ [0;a,a,a^2,a^2,a^3,a^3,\ldots]=\frac{\sum\nolimits^{\infty}_{s=0}a^{-\frac{(s+1)(s+2)}{2}}\prod\nolimits^s_{i=1}(a^{i}-1)^{-1}}{\sum\nolimits^{\infty}_{s=0}a^{-\frac{s(s+1)}{2}}\prod\nolimits^s_{i=1}(a^{i}-1)^{-1}} \] A more general case is discussed.

By avoiding a conventional application of Bessel’s inequality in favour of explicitly controlling an exponential sum over the exceptional set, in our previous work [9, 10], we have exploited additional variables so as to enhance exceptional set estimates in various additive problems of Waring type. In this paper we turn to the problem of establishing the expected asymptotic formula in Waring’s problem. The methods introduced herein, although discussed in the context of the asymptotic formula, should nonetheless provide a useful model for future excursions involving exceptional sets in additive problems.

Let $D\,{=}\,\pi_1\cdots \pi_n$, where $\pi_1,\dots,\pi_n$ are distinct Gaussian odd primes and $n$ is any positive integer. Let $E_{4D^2}$ be the elliptic curve $y^2\,{=}\,x^3-(2D)^2x$. We prove that the value at $s\,{=}\,1$ of the complex $L$-function of $E_{4D^2}$, divided by its natural period(which is $\omega/ \surd{(2D)}$, where $\omega\,{=}\,2{\cdot}6220575\dots$), is always divisible by $2^{n-1}$, and we give a simple combinatorial criterion for it to be exactly divisible by $2^{n-1}$. As a corollary, we construct a series of even non-congruent numbers having an arbitrarily large number of prime factors. Our result is in accord with the predictions of the conjecture of Birch and Swinnerton–Dyer.

Working in characteristic two, we classify nonsmooth Enriques surfaces with normal crossing singularities. Using Kato’s theory of logarithmic structures, we show that such surfaces are smoothable and lift to characteristic zero, provided they are d-semistable.

Let $K$ be a convex body in ${\bb R}^n$ with volume $|K| = 1$. We choose $N$ points $x_1,\ldots, x_N$ independently and uniformly from $K$, and write $C(x_1,\ldots, x_N)$ for their convex hull. If $n(\log n)^2 \leqslant N \leqslant \exp(c_1n)$, we show that the expected volume radius \[ {\bb E}_{\frac{1}{n}}(K,N)=\int_K\cdots\int_K |C(x_1,\ldots, x_N)|^{\frac{1}{n}}dx_N\cdots dx_1 \] of this random $N$-tope can be estimated by \[ c_1\frac{\sqrt{\log(N/n)}}{\sqrt{n}}\leqslant{\bb E}_{1/n}(K,N)\leqslant c_2L_K\frac{\log(N/n)}{\sqrt{n}}, \] where $c_1, c_2$ are absolute positive constants and $L_K$ is the isotropic constant of $K$.

Philip Hall raised the following question which is stated in The Kourovka Notebook [12, p. 88]: is there a non-trivial group which is isomorphic with every proper extension of itself by itself? We will split the problem into two parts: we want to find non-commutative splitters, that are groups $G\ne 1$ with ${\rm Ext} (G, G) = 1$. The class of splitters fortunately is quite large so that extra properties can be added to $G$. We can consider groups $G$ with the following properties: there is a complete group $L$ with cartesian product $L^w \cong G, {\rm Hom}(L^w, S^w) = 0$($S_w$ the infinite symmetric group acting on $w$) and ${\rm End} (L, L) = {\rm Inn}\, L\cup\{0\}$. We will show that these properties ensure that $G$ is a splitter and hence obviously a Hall group in the above sense. Then we will apply a recent result from our joint paper [9] which also shows that such groups exist; in fact there is a class of Hall groups which is not a set.

Let $M$ be a pure motive over $\Bbb Q$ of weight $w=w(M)$ and of the global conductor $c(M)$. Also given Dirichlet character $\chi$ one can define the twisted motive $M_\chi$; let $L(M_\chi,s)\, (s=\sigma+i t)$ denote the corresponding $L$-function.

In a recent paper [13], Kulkarni proves that, for $i\ges 0$, one has

$\Ext^{i}_{G}(M \otimes \delta(\mu), \nabla(\lambda))\cong \Ext^{i}{G}(M, \nabla(\lambda/\mu))$

– an isomorphism of spaces of extensions of rational modules. Here G is a general linear group scheme, $\lambda, \mu$ are partitions and M is a polynomial G-module. The modules $\Delta (\mu)$, $\nabla (\lambda)$ and $\nabla (\lambda/\mu)$ are respectively the Weyl module labelled by $\mu$, the induced module labelled by $\lambda$ and the skew module labelled by $\lambda /\mu$. A similar result is given in which the roles of $\Delta$ and $\nabla$ are interchanged.

Our purpose here is to set this result in the context of the representation theory of an arbitrary reductive group. Specifically, we prove a simple result which is valid for general reductive groups and derive Kulkarni's results from this. For convenience we work over an algebraically closed field k.

This paper continues the investigation of the hit problem, started in [5], for the algebra of symmetric polynomials B(n) viewed as a left ${\cal A}$-module graded by degree, where ${\cal A}$ denotes the Steenrod algebra over the field of two elements ${\bb F}_2$. We recall that a homogeneous element f of grading d in a graded left ${\cal A}$-module M is hit if there is a hit equation in the form of a finite sum $f=\sum_{k>0}Sq^{k}(h_k)$, where the homogeneous elements hk in M have grading less than d and the Sqk are the Steenrod squares, which generate ${\cal A}$. We denote by Q = Q(M) = ${\bb F}_2\otimes_{\cal A}$M the quotient of the module M by the hit elements, where ${\bb F}_2$ is here viewed as a right ${\cal A}$-module concentrated in grading 0. Then Q is a graded vector space over ${\bb F}_2$ and a basis for Q lifts to a minimal generating set for M as a module over ${\cal A}$. The hit problem is to find minimal generating sets for M and criteria for elements to be hit. We recall that B(n) = ${\bb F}_{2}\[\sigma_1,\ldots,\sigma_n\]$ is the polynomial subalgebra of P(n) = ${\bb F}_{2}\[x_1,\ldots,x_n\]$ generated by the elementary symmetric functions $\sigma_i$ in the variables xj . In particular, $\sigma_n=x_1\cdots x_n$. The algebras P(n) and B(n) realize respectively the cohomology of the product of n copies of infinite real projective space and the cohomology of the classifying space BO(n) of the orthogonal group O(n) over ${\bb F}_2$, where the usual grading in cohomology corresponds to degree in the polynomial algebra. The ideal M(n) in B(n), generated by $\sigma_n$, can be identified with the cohomology H*(MO(n), ${\bb F}_2$) of the Thom space MO(n) in positive dimensions. It is also convenient to introduce the notation L(n) for the polynomials in P(n) divisible by $\sigma_n$. Topologically, L(n) corresponds in positive degrees to the cohomology of the n-fold smash product of infinite real projective space. From the topological point of view, ${\cal A}$ is the algebra of universal stable operations in ordinary cohomology with ${\bb F}_2$ coefficients and this explains the action of ${\cal A}$ on P(n), L(n), B(n) and M(n). However, the whole subject may be treated in a purely algebraic fashion [5, 11].

The topology of a topological group $G$ is called an ${\cal L}_{\infty}$-topology if it can be represented as the intersection of a decreasing sequence of locally compact Hausdorff group topolgies on $G$. If ${\cal L}_1 < {\cal L}_2$ are two distinct ${\cal L}_{\infty}$-topologies on an Abelian group $G$, it is shown that the quotient of the corresponding character groups has cardinality ${\geqslant} 2^{\rm c}$. A conjecture in this sense announced by J. B. Reade in his paper [6] is thereby proved.

Let $(R,M)$ be a Noetherian local domain, $F$ its quotient field and $v$ a valuation of $F$ centred at $R$. In the hope of understanding the valuation $v$, a useful tool is the so-called graded algebra relative to $v$, $gr_{v}R$. Its study can be understood as an attempt to approach the problem of classifying valuations. On the other hand, $gr_{v}R$ is also an object of interest since it can be provided with a geometrical meaning [15].

Let ${\rm q}$ be a Lie algebra and ξ a linear form on ${\rm q}$. Denote by ${\rm q}_{\xi}$ the set of all $x \in {\rm q}$ such that $\xi([{\rm q},x]) = 0$. In other words, ${\rm q}_{\xi}=\{x\in {\rm q} \mid({\rm ad}^*x)\cdot = 0\}$, where ${\rm ad}^*: {\rm q}\rightarrow {\rm g I}({\rm q}^*)$ is the coadjoint representation of ${\rm q}$. The index of ${\rm q}$, ind ${\rm q}$, is defined by \[ {\rm ind\, q} = \min_{\xi\in {\rm q}^*}\dim {\rm q}\xi. \]

Partial combinatory algebras (pcas), models for a form of Combinatory Logic with partial application, have been studied for the last thirty years because of their close connection to Intuitionistic Logic (see, for example, [11])

For the first Painlevé equation, it is proved that every meromorphic solution satisfies T(r, w) = O(r$^{\frac {5}{2}}$). In showing this estimate, we employ two types of auxiliary function, one of which is crucial in the proof of the Painlevé property. Our method is also applicable to the second (resp. the fourth) Painlevé transcendents, and we obtain T(r, w) = O(r3) (resp. O(r4)).

We study compact Riemann surfaces of genus $g\geq2$ having a dihedral group of automorphisms. We find necessary and sufficient conditions on the signature of a Fuchsian group for it to admit a surface kernel epimorphism onto the dihedral group $D_N$. The question of extendability of the action of $D_N$ is considered. We also give an explicit parametrization of the moduli space of Riemann surfaces with maximal dihedral symmetry, showing that it is a one-dimensional complex manifold. Defining equations of all such surfaces and the formulae of their automorphisms are calculated. The locus of this moduli space consisting of those surfaces admitting some real structure is determined.

Let $N$ be a prime number, and $X$ a curve that is intermediate to the cover $X_1(N)\rightarrow X_0(N)$. We study the automorphism group of $X$, and prove that in most cases it is generated by the Galois group of the cover $X\rightarrow X_0(N)$, and any lift of the Atkin–Lehner involution to $X$.

Let M be a compact, connected, orientable 3-manifold with a torus boundary component $\partial_{0}M$. A slope on $\partial_{0}M$ is the isotopy class of an unoriented essential simple loop. For a slope r, the manifold obtained from M by r-Dehn filling is M(r) = M$\cup$Vr, where Vr is a solid torus glued to M along $\partial_{0}M$ in such a way that r bounds a meridian disc in Vr. If r and s are two slopes on $\partial_{0}M$, then $\Delta$(r, s) denotes their minimal geometric intersection number.

We introduce a new approach to the local study of maximal surfaces in Lorentz–Minkowski space, based on a complex representation formula for this kind of surfaces. As an application we solve a certain Björling-type problem in Lorentz–Minkowski space and we obtain some results related to it. We also establish, springing from this complex representation, a way of introducing examples of maximal surfaces with interesting prescribed geometric properties. Further applications of the complex representation let us inspect some known results from a different perspective, and show how our approach can be used to classify certain families of maximal surfaces.

Let $\Sigma_{g,r}^n$ be a connected orientable surface of genus $g$ with $r$ boundary components and $n$ punctures and let $\Gamma_{g,r}^n$ denote the mapping class group of $\Sigma_{g,r}^n$, namely the group of isotopy classes of orientation-preserving diffeomorphisms of $\Sigma_{g,r}^n$ which are the identity on the boundary and on the punctures. Here, we see the punctures on the surface as distinguished points. The isotopies are required to be the identity on the boundary and on the punctures. If $r$ and/or $n$ is zero, then we omit it from the notation.