We propose a very precise conjecture on the asymptotics of the counting function for extensions of number fields with fixed Galois group and bounded norm of the discriminant. This sharpens a previous conjecture of the author. The conjecture is known to hold for abelian groups and a few nonabelian ones. We give a heuristic argument why the conjecture should be true. We also present some computational data for the nonsolvable groups of degree 5.

We present an algorithm to compute values L(s) and derivatives {\small $L^{(k)}(s)$} of L-functions of motivic origin numerically to required accuracy. Specifically, the method applies to any L-series whose {\small $\Gamma$}-factor is of the form {\small $A^s\prod_{i=1}^d \Gamma(\frac{s+\lambda_j}{2})$} with d arbitrary and complex {\small $\lambda_j$}, not necessarily distinct. The algorithm relies on the known (or conjectural) functional equation for L(s).

In this paper we present a new, practical algorithm for solving the subgroup conjugacy problem in the general linear group.

In a previous article (see Dieulefait and Manoharmayum), the modularity of a large class of rigid Calabi-Yau threefolds was established. To make that result more explicit, we recall (and reprove) a result of Serre giving a bound for the conductor of "integral" two-dimensional compatible families of Galois representations and apply this result to give an algorithm that determines the level of a modular rigid Calabi-Yau threefold. We apply the algorithm to three examples.

We classify the orientable finite-volume hyperbolic 3-manifolds having nonempty compact totally geodesic boundary and admitting an ideal triangulation with at most four tetrahedra. We also compute the volume of all such manifolds, describe their canonical Kojima decomposition, and discuss manifolds having cusps.

The eight manifolds built from one or two tetrahedra were previously known. There are 151 different manifolds built from three tetrahedra, realizing 18 different volumes. Their Kojima decomposition always consists of tetrahedra (but occasionally requires four of them). There is a single cusped manifold, which we can show to be a knot complement in a genus-2 handlebody. Concerning manifolds built from four tetrahedra, we show that there are 5,033 different ones, with 262 different volumes. The Kojima decomposition consists either of tetrahedra (as many as eight of them in some cases), of two pyramids, or of a single octahedron. There are 30 manifolds having a single cusp and one having two cusps.

Our results were obtained with the aid of a computer. The complete list of manifolds (in SnapPea format) and full details on their invariants are available on the world wide web.

Let E be an elliptic curve over q with L-function {\small $L_E(s)$}. We use the random matrix model of Katz and Sarnak to develop a heuristic for the frequency of vanishing of the twisted L-functions {\small $L_E(1, \chi)$}, as {\small $\chi$} runs over the Dirichlet characters of order 3 (cubic twists). The heuristic suggests that the number of cubic twists of conductor less than X for which {\small $L_E(1, \chi)$} vanishes is asymptotic to {\small $b_E X^{1/2} \log^{e_E}{X}$} for some constants {\small $b_E, e_E$} depending only on E. We also compute explicitly the conjecture of Keating and Snaith about the moments of the special values {\small $L_E(1, \chi)$} in the family of cubic twists. Finally, we present experimental data which is consistent with the conjectures for the moments and for the vanishing in the family of cubic twists of {\small $L_E(s)$}.

A method to generate periodic geodesics in arbitrary level surfaces is presented. The underlying algorithm resolves several technical complications posed by the constraints to stay in the surface and retain periodicity. The method exploits the "inverse'' of the parallel transport equation and its "derivative.'' This approach avoids most of the complications due to the intricate form of the geodesic curvature. The process flows any periodic curve in the surface along the negative gradient trajectory of the total squared geodesic curvature. The mathematical framework is that of an infinite-dimensional Riemannian manifold representing periodic curves of arbitrary length. The method is illustrated by an example in a sphere-like surface that is neither an ellipsoid nor a surface of revolution.

This paper deals with the spectrum of an operator associated with a special kind of random walk. The operator is related to the Metropolis algorithm, an important tool of large-scale scientific computing. The spectrum of this operator has both discrete and continuous parts. There is an interesting challenge due to the fact that any finite-dimensional approximation has only eigenvalues. Patterns are presented which give an idea of the full spectrum of this operator.

A finite {\small $\pm 1$} sequence X yields a binary triangle {\small $\Delta X$} whose first row is X, and whose {\small $(k+1)$}th row is the sequence of pairwise products of consecutive entries of its kth row, for all {\small $k \geq 1$}. We say that X is balanced if its derived triangle {\small $\Delta X$} contains as many +1s as {\small $-$}1s. In 1963, Steinhaus asked whether there exist balanced binary sequences of every length {\small $n \equiv$} 0 or 3 mod 4. While this problem has been solved in the affirmative by Harborth in 1972, we present here a different solution. We do so by constructing strongly balanced binary sequences, i.e., binary sequences of length n all of whose initial segments of length {\small $n-4t$} are balanced, for {\small $0 \leq t \leq n/4$}. Our strongly balanced sequences do occur in every length {\small $n \equiv$} 0 or 3 mod 4. Moreover, we provide a complete classification of sufficiently long strongly balanced binary sequences.

An irreducible representation of a simple Lie algebra can be a direct summand of its own tensor square. In this case, the representation admits a nonassociative algebra structure which is invariant in the sense that the Lie algebra acts as derivations. We study this situation for the Lie algebra {\small $sl(2)$}.