Let $p{>}2$ be a prime, and let $k$ be a field of characteristic zero, linearly disjoint from the $p$th cyclotomic extension of $\Q$. Given a projective Galois representation $\varrho \colon \Gal(\kbar/k) {\To} \PGL_2(\F_p)$ with cyclotomic determinant, two twists $X_\varrho(p)$ and $ X'_\varrho(p)$ of a certain rational model of the modular curve $X(p)$ can be attached to it. The $k$-rational points of these twists classify the elliptic curves $E/k$ such that ${\rhobar}_{E, p}{=}\varrho$, where ${\rhobar}_{E, p}$ denotes the projective Galois representation associated with the $p$-torsion module $E[p]$. The octahedral ($p{=}3$) and icosahedral ($p{=}5$) genus-zero cases are discussed in further detail.

Let Ω be the set {1, 2, [ctdot ], n}, and let [emptyv ] be the empty set. Let [Gscr ] be the family of all non-empty sets of subsets of Ω. For [Ascr ]∈[Gscr ] and X⊆Ω, put

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An important discovery is the following.

It is known that, if f is a hyperbolic rational function, then the Hausdorff, packing and box dimensions of the Julia set, J(f), are equal. In this paper it is shown that, for a hyperbolic transcendental meromorphic function f, the packing and upper box dimensions of J(f) are equal, but can be strictly greater than the Hausdorff dimension of J(f).

Crispin St John Alvah Nash-Williams was born in Cardiff on 19 December 1932. His father was Keeper of Archaeology at the National Museum of Wales and also Senior Lecturer in Archaeology at University College, Cardiff. War broke out when Crispin was six; his father joined the army and his consequent absence changed the family's life.