We prove that a non-solvable Demushkin group satisfies the Greenberg–Stallings property, i.e., if H and K are finitely generated subgroups of a non-solvable Demushkin group G with the property that H ∩ K has finite index in both H and K, then H ∩ K has finite index in 〈H, K〉. Moreover, we prove that every finitely generated subgroup H of G has a ‘root’, that is a subgroup K of G that contains H with |K : H| finite and which contains every subgroup U of G that contains H with |U : H| finite. This allows us to show that every non-trivial finitely generated subgroup of a non-solvable Demushkin group has finite index in its commensurator.

Let $E_{/{\mathbb{Q}}}$ be a semistable elliptic curve, and p ≠ 2 a prime of bad multiplicative reduction. For each Lie extension $\mathbb{Q}$FT/$\mathbb{Q}$ with Galois group G∞ ≅ $\mathbb{Z}$p ⋊ $\mathbb{Z}$p×, we construct p-adic L-functions interpolating Artin twists of the Hasse–Weil L-series of the curve E. Through the use of congruences, we next prove a formula for the analytic λ-invariant over the false Tate tower, analogous to Chern–Yang Lee's results on its algebraic counterpart. If one assumes the Pontryagin dual of the Selmer group belongs to the $\mathfrak{M}_{\mathcal{H}}$(G∞)-category, the leading terms of its associated Akashi series can then be computed, allowing us to formulate a non-commutative Iwasawa Main Conjecture in the multiplicative setting.

Let 0 < a ⩽ 1, s, z ∈ ${\mathbb{C}}$ and 0 < |z| ⩽ 1. Then the Hurwitz–Lerch zeta function is defined by Φ(s, a, z) ≔ ∑∞n = 0zn(n + a)− s when σ ≔ ℜ(s) > 1. In this paper, we show that the Hurwitz zeta function ζ(σ, a) ≔ Φ(σ, a, 1) does not vanish for all 0 < σ < 1 if and only if a ⩾ 1/2. Moreover, we prove that Φ(σ, a, z) ≠ 0 for all 0 < σ < 1 and 0 < a ⩽ 1 when z ≠ 1. Real zeros of Hurwitz–Lerch type of Euler–Zagier double zeta functions are studied as well.

In this paper we consider the relationship between the Assouad and box-counting dimension and how both behave under the operation of taking products. We introduce the notion of ‘equi-homogeneity’ of a set, which requires a uniformity in the cardinality of local covers at all length-scales and at all points, and we show that a large class of homogeneous Moran sets have this property. We prove that the Assouad and box-counting dimensions coincide for sets that have equal upper and lower box-counting dimensions provided that the set ‘attains’ these dimensions (analogous to ‘s-sets’ when considering the Hausdorff dimension), and the set is equi-homogeneous. Using this fact we show that for any α ∈ (0, 1) and any β, γ ∈ (0, 1) such that β + γ ⩾ 1 we can construct two generalised Cantor sets C and D such that dimBC = αβ, dimBD = α γ, and dimAC = dimAD = dimA (C × D) = dimB (C × D) = α.

The study of existence of a universal C*-completion of the *-algebra canonically associated to a Hecke pair was initiated by Hall, who proved that the Hecke algebra associated to (SL2($\mathbb{Q}$p), SL2($\mathbb{Z}$p)) does not admit a universal C*-completion. Kaliszewski, Landstad and Quigg studied the problem by placing it in the framework of Fell–Rieffel equivalence, and highlighted the role of other C*-completions. In the case of the pair (SLn($\mathbb{Q}$p), SLn($\mathbb{Z}$p)) for n ⩾ 3 we show, invoking property (T) of SLn($\mathbb{Q}$p), that the C*-completion of the L1-Banach algebra and the corner of C*(SLn($\mathbb{Q}$p)) determined by the subgroup are distinct. In fact, we prove a more general result valid for a simple algebraic group of rank at least 2 over a $\mathfrak{p}$-adic field with a good choice of a maximal compact open subgroup.

Let f be a transcendental entire function. The fast escaping set, A(f), plays a key role in transcendental dynamics. The quite fast escaping set, Q(f), defined by an apparently weaker condition is equal to A(f) under certain conditions. Here we introduce Q2(f) defined by what appears to be an even weaker condition. Using a new regularity condition we show that functions of finite order and positive lower order satisfy Q2(f) = A(f). We also show that the finite composition of such functions satisfies Q2(f) = A(f). Finally, we construct a function for which Q2(f) ≠ Q(f) = A(f).

Given a number field k and a positive integer d, in this paper we consider the following question: does there exist a smooth diagonal surface of degree d in $\mathbb{P}$3 over k which contains a line over every completion of k, yet no line over k? We answer the problem using Galois cohomology, and count the number of counter-examples using a result of Erdős.

For any smooth compact manifold W with boundary of dimension of at least two we prove that the classifying spaces of its group of diffeomorphisms which fix a set of k points or k embedded disks (up to permutation) satisfy homology stability. The same is true for so-called symmetric diffeomorphisms of W connected sum with k copies of an arbitrary compact smooth manifold Q of the same dimension. The analogues for mapping class groups as well as other generalisations will also be proved.

We give an expression of the motivic zeta function for a real polynomial function in terms of the Newton polyhedron of the function. As a consequence, we show that the weights are determined by the motivic zeta function for convenient weighted homogeneous polynomials in three variables. We apply this result to the blow-Nash equivalence.

Let E/Q be an elliptic curve with complex multiplication. We study the average size of τ(#E(Fp)) as p varies over primes of good ordinary reduction. We work out in detail the case of E: y2 = x3 − x, where we prove that

$$\begin{equation} \sum_{\substack{p \leq x \\p \equiv 1\pmod{4}}} \tau(\#E({\bf{F}}_p)) \sim \left(\frac{5\pi}{16} \prod_{p > 2} \frac{p^4-\chi(p)}{p^2(p^2-1)}\right)x, \quad\text{as $x\to\infty$}. \end{equation}$$

Now suppose that E/Q is a non-CM elliptic curve. We conjecture that the sum of τ(#E(Fp)), taken over p ⩽ x of good reduction, is ~cEx for some cE > 0, and we give a heuristic argument suggesting the precise value of cE. Assuming the Generalized Riemann Hypothesis for Dedekind zeta functions, we prove that this sum is ≍Ex. The proof uses combinatorial ideas of Erdős.