We present a group-theoretic criterion under which one may verify the Artin conjecture for some (nonmonomial) Galois representations, up to finite height in the complex plane. In particular, the criterion applies to $S_5$ and $A_5$ representations. Under more general conditions, the technique allows for the possibility of verifying the Riemann hypothesis for Dedekind zeta functions of nonabelian extensions of $\Q$.

In addition, we discuss two methods for locating zeros of arbitrary $L$-functions. The first uses the explicit formula and techniques developed in A. R. Booker and A. Strömbergsson,“Numerical Computations with the Trace Formula and the Selberg Eigenvalue Conjecture,” for computing with trace formulas. The second method generalizes that of Turing for verifying the Riemann hypothesis. In order to apply it we develop a rigorous algorithm for computing general $L$-functions on the critical line via the fast Fourier transform.

Finally, we present some numerical results testing Artin's conjecture for $S_5$ representations, and the Riemann hypothesis for Dedekind zeta functions of $S_5$ and $A_5$ fields.

Using the second conjecture in the paper J. Guillera, “A New Method to Obtain Series for 1/π and 1/π2,” and inspired by the theory of modular functions, we find a method that allows us to obtain explicit formulas, involving eta or theta functions, for the parameters of a class of series for $1/ \pi$. As in J. Guillera, “A New Method to Obtain Series for 1/π and 1/π2,” the series considered in this paper include Ramanujan's series as well as those associated with the Domb numbers and Apéry numbers.

We conjecture that if $C$ is a curve of genus $>1$ over a number field $k$ such that $C(k)=\emptyset$, then a method of Scharaschkin (essentially equivalent to the Brauer--Manin obstruction in the context of curves) supplies a proof that $C(k)=\emptyset$. As evidence, we prove a corresponding statement in which $C(\F_v)$ is replaced by a random subset of the same size in $J(\F_v)$ for each residue field $\F_v$ at a place $v$ of good reduction for $C$, and the orders of Jacobians over finite fields are assumed to be smooth (in the sense of having only small prime divisors) as often as random integers of the same size. If our conjecture holds, and if Tate--Shafarevich groups are finite, then there exists an algorithm to decide whether a curve over $k$ has a $k$-point, and the Brauer--Manin obstruction to the Hasse principle for curves over the number fields is the only one.

Digital snowflakes are solidifying cellular automata on the triangular lattice with the property that a site having exactly one occupied neighbor always becomes occupied at the next time step. We demonstrate that each such rule fills the lattice with an asymptotic density that is independent of the initial finite set. There are some cases in which this density can be computed exactly, and others in which it can only be approximated. We also characterize when the final occupied set comes within a uniformly bounded distance of every lattice point. Other issues addressed include macroscopic dynamics and exact solvability.

We use a computer procedure to determine a basis of the elements of degree 5 in the nucleus of the free alternative algebra. In order to save computer memory, we do our calculations over the field $Z_{103}$. All calculations are made with multilinear identities. Our procedure is also valid for other characteristics and for determining nuclear elements of higher degree.

We give an efficient and stable algorithm for computing highest weights in a large class of prehomogeneous spaces associated with the nilpotent orbits of the real Lie algebras $\E_{6(6)}$ and $\E_{6(-26)}$. This paper concludes our classification of such prehomogeneous spaces for all complex and real reductive Lie algebras. For classical algebras using the fact that the nilpotent orbits are parameterized by partitions of integers we have given general formulas in Steven Glenn Jackson and Alfred G. Noël, “Prehomogeneous Spaces Associated with Complex Nilpotent Orbits,” and “Prehomogeneous Spaces Associated with Real Nilpotent Orbits.” For complex or inner-type real exceptional algebras we have given general algorithms and tables in Steven Glenn Jackson and Alfred G. Noël, “A LiE Subroutine for Computing Prehomogeneous Spaces Associated with Complex Nilpotent Orbits,” and “A LiE Subroutine for Computing Prehomogeneous Spaces Associated with Real Nilpotent Orbits.” The present paper considers the case of real exceptional algebras that are not of inner type.

In this paper we study elliptic curves that have a number of points whose coordinates are in arithmetic progression. We first motivate this Diophantine problem; then we prove some results, provide a number of interesting examples, and finally point out open questions that focus on the most interesting aspects of the problem for us.

In earlier papers, P. M. H. Wilson, “Sectional Curvatures of Kähler Moduli,” and B. Totaro, “The Curvature of a Hessian Metric,” the $S$-invariant of a ternary cubic $f$ was interpreted in terms of the curvature of related Riemannian and pseudo-Riemannian metrics. This is clarified further in Section 3 of this paper. In the case that $f$ arises from the cubic form on the second cohomology of a smooth projective threefold with second Betti number three, the value of the $S$-invariant is closely linked to the behavior of this curvature on the open cone consisting of Kähler classes. In this paper, we concentrate on the cubic forms arising from complete intersection threefolds in the product of three projective spaces, and investigate various conjectures of a combinatorial nature arising from their invariants.

Any stable map from a surface to the plane has an associated graph. In the case of the sphere, such graphs are of tree type. We characterize the trees that can occur as graphs of fold maps from the sphere to the plane. In order to do so, we first determine the sets of integers that may occur as winding numbers for the branch sets of these maps.

We consider the set $\Om_2$ of double zeros in $(0,1)$ for power series with coefficients in $\{-1,0,1\}$. We prove that $\Om_2$ is disconnected, and estimate $\min \Om_2$ with high accuracy. We also show that $[2^{-1/2}-\eta,1)\subset \Om_2$ for some small, but explicit, $\eta>0$ (this was known only for $\eta=0$). These results have applications in the study of infinite Bernoulli convolutions and connectedness properties of self-affine fractals.