In a recent paper Ahlgren, Ono and Penniston described the L-series of K3 surfaces from a certain one-parameter family in terms of those of a particular family of elliptic curves. The Tate conjecture predicts the existence of a correspondence between these K3 surfaces and certain Kummer surfaces related to these elliptic curves. A geometric construction of this correspondence is given here, using results of D. Morrison on Nikulin involutions.

We give a theoretical lower bound for the slope of a Siegel modular cusp form that is as least as good as Eichler's lower bound. In degrees $n=5,6$ and 7 we show that our new bound is strictly better. In the process we find the forms of smallest dyadic trace on the perfect core for ranks $n \le 8$. In degrees $n=5,6$ and 7 we settle the value of the generalized Hermite constant $\gamma_n'$ introduced by Bergé and Martinet and find all dual-critical pairs.

A cyclotomic polynomial $\Phi(z)$ is said to be of order 3 if $n=pqr$ for three distinct odd primes $p$, $q$, and $r$. We establish the existence of an infinite family of such polynomials whose coefficients do not exceed 1 in modulus.

In this paper we compute the subgroup zeta functions of nilpotent groups of the form \[G_n := \langle x_1,\dots,x_{n},y_1,\dots,y_{n-1}|\;[x_i,x_n]=y_i,\;1\leq i \leq n-1\], all other [,] trivial 〉 and deduce local functional equations.

A base of an action of a group G on a set $\Omega$ is a subset $B \subseteq \Omega$ such that the pointwise stabiliser of $B$ in $G$ is the identity. We prove that if $\Omega$ is the set of partitions of $[ 1, kl ]$ into $l$ subsets of size $k$, then the action of $S_{kl}$ on $\Omega$ has a base of size two if and only if $k \geq 3$ and $l \geq \max \{ k + 3, 8 \}$. This result completes a classification of the primitive base 2 actions of the symmetric groups. During the proof we show that there exists a $k$-regular bipartite graph $\mathcal{G}$ on $2l$ vertices with no non-trivial automorphisms fixing the bipartite blocks if and only if $k \geq 3$ and $l \geq \max \{ k + 3, 8 \}$.

We show that the Weierstrass points of the generic curve of genus $g$ over an algebraically closed field of characteristic 0 generate a group of maximal rank in the Jacobian.

Let $m,\,g,\,q \in\mathbb{N}$ with $q\,{\geq}\, 2$ and $(m,q-1)=1$. For $n\in\mathbb{N}$, denote by $s_q(n)$ the sum of digits of $n$ in the $q$-ary digital expansion. Given a polynomial $f$ with integer coefficients, degree $d\ge 1$, and such that $f(\mathbb{N})\subset\mathbb{N}$, it is shown that there exists $C=C(f,m,q)>0$ such that for any $g\in\mathbb{Z}$, and all large $N$, \[|\{ 0\,{\leq}\, n\,{\leq}\, N : s_q(f(n))\md gm\}|\,{\geq}\, CN^{\min(1,2/d!)}\]. In the special case $m=q=2$ and $f(n)=n^2$, the value $C=1/20$ is admissible.

We prove that Schanuel's conjecture for the reals is equivalent to a uniform version of itself.

Let $\mathfrak{g}=\mathfrak{g}_0\oplus\mathfrak{g}_1$ be a $\mathbb Z_2$-graded simple Lie algebra. Fix a Borel subalgebra $\mathfrak b_0\subset\mathfrak{g}_0$. Let $\mathfrak a\subset\mathfrak{g}_1$ be a $\mathfrak b_0$-stable subalgebra. Then $\mathfrak a$ is automatically commutative. It is known that if $\overline{G{\cdot}\mathfrak a}$ is a spherical $G$-variety, then $G_0{\cdot}\mathfrak a$ is a spherical $G_0$-variety. We describe all $\mathbb Z_2$-gradings having the property that $\overline{G{\cdot}\mathfrak a}$ is a spherical $G$-variety for any $\mathfrak a$.

A complete structural characterization of submonoids $S$ of a polycyclic-by-finite group such that the semigroup algebra $K[S]$ over a field $K$ is right noetherian is obtained. It follows that such algebras are also left noetherian.

It is proved that a ring $R$ is right Artinian if and only if, for each countably generated right $R$-module $M$, there exists a finite subset $F$ of $M$ such that the annihilator of $M$ in $R$ equals the annihilator of $F$ in $R$.

Let $S={\mathop{\rm Sym}(\Omega)$ be the group of all permutations of an infinite set $\Omega$. Extending an argument of Macpherson and Neumann, it is shown that if $U$ is a generating set for $S$ as a group, then there exists a positive integer $n$ such that every element of $S$ may be written as a group word of length at most $n$ in the elements of $U$. Likewise, if $U$ is a generating set for $S$ as a monoid, then there exists a positive integer $n$ such that every element of $S$ may be written as a monoid word of length at most $n$ in the elements of $U$. Some related questions and recent results are noted, and a brief proof is given of a result of Ore's on commutators, which is used in the proof of the above result.

We present a simple proof of the fact that every countable group ${\Gamma}$ is weak Rohlin, that is, there is in the Polish space $mathbb{A}_{\Gamma}$ of measure preserving ${\Gamma}$-actions an action $\mathbf{T}$ whose orbit in $\mathbb{A}_{\Gamma}$ under conjugations is dense. In conjunction with earlier results this in turn yields a new characterization of non-Kazhdan groups as those groups which admit such an action $\mathbf{T}$ which is also ergodic.

We consider the sequences of fractional parts $\{\xi \alpha^n\}, n \,{=}\,1,2,3,\dots,$ and of integer parts $[\xi \alpha^n], n \,{=}\,1,2,3,\dots,$ where $\xi$ is an arbitrary positive number and $\alpha\,{>}\,1$ is an algebraic number. We obtain an inequality for the difference between the largest and the smallest limit points of the first sequence. Such an inequality was earlier known for rational $\alpha$ only. It is also shown that for roots of some irreducible trinomials the sequence of integer parts contains infinitely many numbers divisible by either 2 or 3. This is proved, for instance, for $[\xi((\sqrt {13}-1)/2)^n], n\,{=}\,1,2,3,\dots.$ The fact that there are infinitely many composite numbers in the sequence of integer parts of powers was proved earlier for Pisot numbers, Salem numbers and the three rational numbers 3/2, 4/3, 5/4, but no such algebraic number having several conjugates outside the unit circle was known.

The rank of a vector space $\mathcal{A}$ of $n\times n$-matrices is by definition the maximal rank of an element of $\mathcal{A}$. The space $\mathcal{A}$ is called rank-critical if any matrix space that properly contains $\mathcal{A}$ has a strictly higher rank. This paper exhibits a sufficient condition for rank-criticality, which is then used to prove that the images of certain Lie algebra representations are rank-critical. A rather counter-intuitive consequence, and the main novelty in this paper, is that for infinitely many $n$, there exists an eight-dimensional rank-critical space of $n \times n$-matrices having generic rank $n-1$, or, in other words: an eight-dimensional maximal space of non-invertible matrices. This settles the question, posed by Fillmore, Laurie, and Radjavi in 1985, of whether such a maximal space can have dimension smaller than $n$. Another consequence is that the image of the adjoint representation of any semisimple Lie algebra is rank-critical; in both results, the ground field is assumed to have characteristic zero.

I exhibit a pair of non-symmetric operads that, while not themselves isomorphic, induce isomorphic monads. The existence of such a pair implies that if ‘algebraic theory’ is understood as meaning ‘monad’ then operads cannot be regarded as algebraic theories of a special kind.

A result is proved which implies the following conjecture of Osgood and Yang from 1976: if $f$ and $g$ are non-constant entire functions, such that $ T(r, f) = O(T(r, g))$ as $ r \to \infty $ and such that $g(z) \in {\mathbb Z}$ implies that $f(z) \in {\mathbb Z}$, then there exists a polynomial $G$ with coefficients in ${\mathbb Q}$, such that $G({\mathbb Z}) \subseteq {\mathbb Z}$ and $f = G \circ g$.

We extend a recent result on the existence of wandering domains of polynomial functions defined over the $p$-adic field ${\mathbb{C}_p}$ to any algebraically closed complete non-archimedean field ${\mathbb{C}_K}$ with residue characteristic $p>0$. We also prove that polynomials with wandering domains form a dense subset of a certain one-dimensional family of degree $p+1$ polynomials in ${\mathbb{C}_K}[z]$.

Using properties of their Cayley graphs, specific examples of Ozawa kernels are constructed for both free and amenable groups, thus showing that these groups satisfy Property $O$. It is deduced both that these groups are exact and satisfy Yu's Property $A$.

In a recent paper, Külshammer, Olsson, and Robinson proved a deep generalization of the Nakayama conjecture for symmetric groups. We provide a similar but a shorter and relatively elementary proof of their result. Our method enables us to obtain a more general $H$-analogue of the Nakayama conjecture where $H$ is a set of positive integers.