Let $C$ be the image of a canonical embedding $\phi$ of the Atkin--Lehner quotient $X_0^+(N)$ associated with the Fricke involution $w_N$. In this note we exhibit some relations among the rational points of $C$. For each $g=3$ (respectively the first $g=4$) curve $C$ we found that there are one or more lines (respectively planes) in $\PP^{g-1}$ whose intersection with $C$ consists entirely of rational Heegner points or the cusp point, where $N$ is prime. We also discuss an explanation of the first nonhyperelliptic exceptional rational point.

We give a rigorous computer-assisted proof that a triangle has a periodic billiard path when all its angles are at most one hundred degrees.

The model of a bicycle is a unit segment $AB$ that can move in the plane so that it remains tangent to the trajectory of the point $A$ (the rear wheel is fixed to the bicycle frame). The same model describes the hatchet planimeter. The trajectory of the front wheel and the initial position of the bicycle uniquely determine its motion and its terminal position; the monodromy map sending the initial position to the final position arises in this context.

According to a theorem of R. Foote, this mapping of a circle to a circle is a Möbius transformation. We extend this result to the multidimensional setting. Möbius transformations belong to one of three types: elliptic, parabolic, and hyperbolic. We prove the century-old Menzin conjecture: if the front wheel track is an oval with area at least $\pi$, then the respective monodromy is hyperbolic. We also study bicycle motions introduced by D. Finn in which the rear wheel follows the track of the front wheel. Such a ``unicycle'' track becomes more and more oscillatory in the forward direction. We prove that it cannot be infinitely extended backward and relate the problem to the geometry of the space of forward semi-infinite equilateral linkages.

We present an intriguing question about lattice points in triangles where Pick's formula is ``almost correct.'' The question has its origin in knot theory, but its statement is purely combinatorial. After more than 30 years, the topological question was recently solved, but the lattice-point problem is still open.

A fast new algorithm is used compute the zeros of $10^6$ quadratic character $L$-functions for negative fundamental discriminants with absolute value $d>10^{12}$. These are compared to the $1$-level density, including various lower-order terms. These terms come from, on the one hand, the explicit formula, and on the other, the $L$-functions ratios conjecture. The latter give a much better fit to the data, providing numerical evidence for the conjecture.

We give an example of a Bieberbach group $\Gamma$ for which $\Out(\Gamma)$ is a cyclic group of order $3$. We also calculate the outer automorphism group of a direct product of $n$ copies of a Bieberbach group with trivial center, for $n \in \mathbb{N}$. As a corollary we get that every symmetric group can be realized as an outer automorphism group of some Bieberbach group.

A pro-$p$-group is a CF group if all (except possibly the first) of its lower central series factors have order $p$. We describe a construction for a class of just-infinite CF pro-$p$-groups with fixed coclass. Based on that, we observe that the class of infinite CF pro-$p$-groups is comparatively large within the class of all just-infinite pro-$p$-groups of fixed coclass.

Let $h_n$ denote the class number of $\Q(2\cos(2\pi/2^{n+2}))$. Weber proved that $h_n$ is odd for all $n\geq 1$. We claim that if $\ell$ is a prime number less than $10^7$, then for all $n\geq 1$, $\ell$ does not divide $h_n$.

R. Hirota and K. Kimura discovered integrable discretizations of the Euler and the Lagrange tops, given by birational maps. Their method is a specialization to the integrable context of a general discretization scheme introduced by W. Kahan and applicable to any vector field with a quadratic dependence on phase variables. According to a proposal by T. Ratiu, discretizations of Hirota--Kimura type can be considered for numerous integrable systems of classical mechanics. Due to a remarkable and not well understood mechanism, such discretizations seem to inherit the integrability for all algebraically completely integrable systems. We introduce an experimental method for a rigorous study of integrability of such discretizations.

Application of this method to the Hirota--Kimura-type discretization of the Clebsch system leads to the discovery of four functionally independent integrals of motion of this discrete-time system, which turn out to be much more complicated than the integrals of the continuous-time system. Further, we prove that every orbit of the discrete-time Clebsch system lies in an intersection of four quadrics in the six-dimensional phase space. Analogous results hold for the Hirota--Kimura-type discretizations for all commuting flows of the Clebsch system, as well as for the so(4) Euler top.

In this paper we present the results of computer searches using a variation of an energy-minimization algorithm used by Kottwitz for finding good spherical codes. We prove that exact codes exist by representing the inner products between the vectors as algebraic numbers. For selected interesting cases, we include detailed discussion of the configurations. Of particular interest are the $20$-point code in $\mathbb{R}^6$ and the $24$-point code in $\mathbb{R}^7$, each of which is the union of two cross-polytopes in parallel hyperplanes. Finally, we catalogue all of the codes we have found.