We study the number of rational points of bounded height on a certain threefold. The accumulating subvarieties are Zariski dense in this example. The computations support an extension of a conjecture of Manin to this situation.

Bounded Apollonian circle packings (ACPs) are constructed by repeatedly inscribing circles into the triangular interstices of a Descartes configuration of four mutually tangent circles, one of which is internally tangent to the other three. If the original four circles have integer curvature, all of the circles in the packing will have integer curvature as well. In "Letter to Lagarias," Sarnak proves that there are infinitely many circles of prime curvature and infinitely many pairs of tangent circles of prime curvature in a primitive integral ACP. (A primitive integral ACP is one in which no integer greater than 1 divides the curvatures of all of the circles in the packing.) In this paper, we give a heuristic backed up by numerical data for the number of circles of prime curvature less than x and the number of "kissing primes," or pairs of circles of prime curvature less than x, in a primitive integral ACP.We also provide experimental evidence toward a local-to-global principle for the curvatures in a primitive integral ACP.

We consider statistical properties of the sequence of ordered pairs obtained by taking the sequence of prime numbers and reducing modulo $m$. Using an inclusion/exclusion argument and a cutoff of an infinite product suggested by Pólya, we obtain a heuristic formula for the "probability" that a pair of consecutive prime numbers of size approximately $x$ will be congruent to $(a, a + d)$ modulo $m$. We demonstrate some symmetries of our formula. We test our formula and some of its consequences against data for $x$ in various ranges.

Motivated in part by various questions of Serre, Labourie and Lubotzky, we consider the question of representing the fundamental group of the figure eight knot complement into ${\sf SL}(3, \mathbb{Z})$. We explore questions of faithfulness and finite index for such representations.

Let $E/ \mathbb{Q}$ be an elliptic curve and $K/ \mathbb{Q}$ a finite Galois extension with group $G$. We write $E_K$ for the base change of $E$ and consider the equivariant Tamagawa number conjecture for the pair $(h^1(E_K )(1), \mathbb{Z}[G])$. This conjecture is an equivariant refinement of the Birch and Swinnerton-Dyer conjecture for $E/K$. For almost all primes $l$, we derive an explicit formulation of the conjecture that makes it amenable to numerical verifications. We use this to provide convincing numerical evidence in favor of the conjecture.

We study equivalence classes relating to the Kazhdan–Lusztig $\mu(x,w)$ coefficients in order to help explain the scarcity of distinct values. Each class is conjectured to contain a “crosshatch” pair. We also compute the values attained by $\mu(x,w)$ for the permutation groups $S_10$ and $S_11$.

The goal of this paper is to give a conjectural census of complex hyperbolic sporadic triangle groups. We prove that only finitely many of these sporadic groups are lattices.

We also give a conjectural list of all lattices among sporadic groups, and for each group in the list we give a conjectural group presentation, as well as a list of cusps and generators for their stabilizers. We describe strong evidence for these conjectural statements, showing that their validity depends on the solution of reasonably small systems of quadratic inequalities in four variables.

Extensive and systematic machine computations are carried out to investigate the integral cohomology of the Euclidean Bianchi groups and their congruence subgroups. The collected data give insight into several aspects, including the asymptotic behavior of the torsion in the first homology. Along with the experimental work, some basic properties of the integral cohomology are recorded with an eye toward the liftability issue of Hecke eigenvalue systems.