We discuss some linear acceleration methods for alternating series which are in theory and in practice much better than that of Euler--Van Wijngaarden. One of the algorithms, for instance, allows one to calculate $\sum(-1)^ka_k$ with an error of about $17$.$93^{-n}$ from the first $n$ terms for a wide class of sequences $\{a_k\}$. Such methods are useful for high precision calculations frequently appearing in number theory.

We review a number of ways of "visualizing'' the elements of the Shafarevich-Tate group of an elliptic curve $E$ over a number field $K$. We are specifically interested in cases where the elliptic curves are defined over the rationals, and are subabelian varieties of the new part of the jacobian of a modular curve (specifically, of $X_0(N)$, where $N$ is the conductor of the elliptic curve). For a given such $E$ with nontrivial Shafarevich-Tate group, we pose the question:

Are all the curves of genus one representing elements of the Shafarevich-Tate group of $E$ isomorphic (over the rationals) to curves contained in a (single) abelian surface $A$, itself defined over the rationals, containing $E$ as a sub-elliptic curve, and contained in turn in the new part of the jacobian of a modular curve $X_0(N)$?

At first view, one might imagine that there are few $E$ with nontrivial Shafarevich-Tate group for which the answer is yes. Indeed we have a small number of examples where the answer is no, and it is very likely that the answer will be no if the order of the Shafarevich-Tate group is large enough. Nonetheless, among all (modular) elliptic curves $E$ as above, with conductors up to $5500$ and with no rational point of order $2$, we have found the answer to the question to be yes in the vast majority of cases. We are puzzled by this and wonder whether there is some conceptual reason for it. We present a substantial amount of data relating to the curves investigated.

We study a family of cubic branched coverings and matings of cubic polynomials of the form $g\mate f$, with $g=g_a:z\mapsto z^3+a$ and $f=P_i$ for $i=1$, 2, 3 or $4$. We give criteria for matability or not of critically finite $g_a$ with each $P_i$. The maps $g_a\mate P_1$ illustrate features that do not occur for matings of quadratic polynomials: they never have Levy cycles but do sometimes have Thurston obstructions.

I introduce the recurrence $D(n)= D(D(n{-}1))+D(n{-}1-D(n{-}2))$, $D(1)=D(2)=1$, and study it by means of computer experiments. The definition of $D(n)$ has some similarity to that of Conway's sequence defined by $a(n)= a(a(n{-}1))+a(n-a(n{-}1))$, $a(1)=a(2)=1$. However, unlike the completely regular and predictable behaviour of $a(n)$, the $D$-numbers exhibit chaotic patterns. In its statistical properties, the $D$-sequence shows striking similarities with Hofstadter's $Q(n)$-sequence, given by $Q(n)= Q(n-Q(n{-}1))+Q(n-Q(n{-}2))$, $Q(1)=Q(2)=1$. Compared to the Hofstadter sequence, $D$ shows higher structural order. It is organized in well-defined "generations'', separated by smooth and predictable regions. The article is complemented by a study of two further recurrence relations with definitions similar to those of the $Q$-numbers. There is some evidence that the different sequences studied share a universality class.

Doi, Hida, and Ishii have shown that the values of twisted adjoint $L$-functions $L(1,\Ad(f)\otimes\chi)$ attached to modular forms $f$ are closely connected with discriminants of Hecke fields. Goto has given a numerical example of this $L$-value for an elliptic cusp form $f$ of level $1$ and weight $20$. We shall show a method to calculate the $L$-values, which is more effective than Goto's, and give new numerical examples.

W. H. Meeks has asked the following question: For what $g$ does every (orientation preserving) periodic automorphism of a closed orientable surface of genus $g$ have an invariant circle? A variant of this question due to R. D. Edwards asks for the existence of invariant essential circles. Using a construction of Meeks we show that the answer to his question is negative for all but 43 values of $g\leq 10000$, all of which lie below $g=105$. We then show that the work of S. C. Wang on Edwards' question generalizes to nonorientable surfaces and automorphisms of odd order. Motivated by this, we ask for the maximal odd order of a periodic automorphism of a given nonorientable surface. We obtain a fairly complete answer to this question and also observe an amusing relation between this order and Fermat primes.

We consider a class of elliptic curves many of whose associated Shafarevich-Tate groups \sfSha/ are relatively large, and give examples of curves with $o(\sfSha/) = k^{2}$ for all $k \leq 100$.

Let $n \ge 2$ be an integer and consider the set $T_n$ of $n \times n$ permutation matrices $\pi$ for which $\pi_{ij}=0$ for $j\ge i+2$.

We study the convex hull $P_n$ of $T_n$, a polytope of dimension $\binom{n}{2}$. We provide evidence for several conjectures involving $P_n$, including Conjecture 1: Let $v_n$ denote the minimum volume of a simplex with vertices in the affine lattice spanned by $T_n$. Then the volume of $P_n$ is $v_n$ times the product $$\prod_{i=0}^{n-2} \frac{1}{i+1}\BINOM{2i}{i} $$ of the first $n-1$ Catalan numbers.

We also give a related result on the Ehrhart polynomial of $P_n$.

Editor's note: After this paper was circulated, Doron Zeilberger proved Conjecture 1, using the authors' reduction of the original problem to a conjectural combinatorial identity, and sketched the proofs of two others. The problems and methodology presented here gain even further interest thereby.

We construct complete, open, hyperbolic 4-manifolds of smallest volume by gluing together the sides of a regular ideal 24-cell in hyperbolic 4-space. We also show that the volume spectrum of hyperbolic 4-manifolds is the set of all positive integral multiples of $4\pi^2/3$.

Snap is a computer program for computing arithmetic invariants of hyperbolic 3-manifolds, built on Jeff Weeks's SnapPea and the number theory package Pari. Its approach is to compute the hyperbolic structure to very high precision, and use this to find an exact description of the structure. Then the correctness of the hyperbolic structure can be verified, and the arithmetic invariants of Neumann and Reid can be computed. Snap also computes high precision numerical invariants such as volume, Chern--Simons invariant, eta invariant, and the Borel regulator.

We show that there are at most 19 integers that are not of the form $xy+yz+xz$ with $x,y,z \ge 1$. Eighteen of them are small and easily found. The remaining possibility must be greater than $10^{11}$ and cannot occur if we assume the Generalized Riemann Hypothesis.