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Generalized Kummer theory and its applications

Toru Komatsu (2009)

Annales mathématiques Blaise Pascal

In this report we study the arithmetic of Rikuna’s generic polynomial for the cyclic group of order n and obtain a generalized Kummer theory. It is useful under the condition that ζ ∉ k and ω ∈ k where ζ is a primitive n -th root of unity and ω = ζ + ζ - 1 . In particular, this result with ζ ∈ k implies the classical Kummer theory. We also present a method for calculating not only the conductor but also the Artin symbols of the cyclic extension which is defined by the Rikuna polynomial.

Generalized Staircases: Recurrence and Symmetry

W. Patrick Hooper, Barak Weiss (2012)

Annales de l’institut Fourier

We study infinite translation surfaces which are ℤ -covers of compact translation surfaces. We obtain conditions ensuring that such surfaces have Veech groups which are Fuchsian of the first kind and give a necessary and sufficient condition for recurrence of their straight-line flows. Extending results of Hubert and Schmithüsen, we provide examples of infinite non-arithmetic lattice surfaces, as well as surfaces with infinitely generated Veech groups.

Currently displaying 661 – 680 of 2022