Thomas A. Schmidt

Chapter 6 - An Introduction to Veech Surfaces

This chapter describes the basics of Veech surfaces with an emphasis on the Veech dichotomy. A Veech surface has dynamical properties similar to the touchstone surface, the square torus. Veech proved an analogous result for a class of particularly nice surfaces. Separatrix is a geodesic line emanating from singularity; a saddle connection is a separatrix, connecting singularities (with no singularities on its interior). To each saddle connection, one can associate a holonomy vector: the saddle connection to the plane by using local coordinates is “developed,” the difference vector defined by the planar line segment is the holonomy vector. Any square-tiled surface satisfies the Veech alternative. The Gutkin-judge result implies that any surface of arithmetic Veech group is a branched cover of the torus, with branching above one sole point. Generally, there are surfaces that have the same (or commensurate) Veech group, but are not related by any tree of finite covers that are “balanced.”

Natural extensions for the Rosen fractions

The Rosen fractions form an infinite family which generalizes the nearest-integer continued fractions. We find planar natural extensions for the associated interval maps. This allows us to easily prove that the interval maps are weak Bernoulli, as well as to unify and generalize results of Diophantine approximation from the literature.

Veech groups and polygonal coverings

Abstract We discuss branch points of affine coverings and their effects on Veech groups. In particular, this allows us to show that even if one polygon tiles another, the respective Veech groups are not necessarily commensurable. We also show that there is no universal bound on the number of Teichmuller disks passing through the same point of Teichmuller space and having incommensurable lattice Veech groups.

Generating Elliptic Curves of Prime Order

Av ariation of the Complex Multiplication (CM) method for generating elliptic curves of known order over finite fields is proposed. We give heuristics and timing statistics in the mildly restricted setting of prime curve order. These may be seen to corroborate earlier work of Koblitz in the class number one setting. Our heuristics are based upon a recent conjecture by R. Gross and J. Smith on numbers of twin primes in algebraic number fields. Our variation precalculates class polynomials as a separate off-line process. Unlike the standard approach, which begins with a prime p and searches for an appropriate discriminant D, we choose a discriminant and then search for appropriate primes. Our on-line process is quick and can be compactly coded. In practice, elliptic curves with near prime order are used. Thus, our timing estimates and data can be regarded as upper estimates for practical purposes.

Natural extensions and entropy of α-continued fractions

We construct a natural extension for each of Nakadas α-continued fraction transformations and show the continuity as a function of α of both the entropy and the measure of the natural extension domain with respect to the density function (1 + xy)−2. For 0 < α ≤ 1, we show that the product of the entropy with the measure of the domain equals π2/6. We show that the interval is a maximal interval upon which the entropy is constant. As a key step for all this, we give the explicit relationship between the α-expansion of α − 1 and of α.

Cross sections for geodesic flows and \alpha-continued fractions

We adjust Arnouxs coding, in terms of regular continued fractions, of the geodesic flow on the modular surface to give a cross section on which the return map is a double cover of the natural extension for the \alpha-continued fractions, for each

CONTINUED FRACTIONS FOR A CLASS OF TRIANGLE GROUPS

\alpha

Parametrizing simple closed geodesy on Γ 3

in (0,1]. The argument is sufficiently robust to apply to the Rosen continued fractions and their recently introduced \alpha-variants.

GALOIS ORBITS OF PRINCIPAL CONGRUENCE HECKE CURVES

We give continued fraction algorithms for each conjugacy class of triangle Fuchsian group of signature (3; n;1), with n 4. In particular, we give an explicit form of the group that is a subgroup of the Hilbert modular group of its trace field and provide an interval map that is piecewise linear fractional, given in terms of group elements. Using natural extensions, we find an ergodic invariant measure for the interval map. We also study Diophantine properties of approximation in terms of the continued fractions and show that these continued fractions are appropriate to obtain transcendence results.

Transcendence with Rosen continued fractions

We exhibit a canonical geometric pairing of the simple closed curves of the degree three cover of the modular surface, 0 3 n , with the proper single self-intersecting geodesics of Crisp and Moran. This leads to a pairing of fundamental domains for0 3 with Markoff triples. The routes of the simple closed geodesics are directly related to the above. We give two parametrizations of these. Combining with work of Cohn, we achieve a listing of all simple closed geodesics of length within any bounded interval. Our method is direct, avoiding the determination of geodesic lengths below the chosen lower bound.