Periodic points on Veech surfaces and the Mordell-Weil group over a Teichmüller curve

By Martin Möller. 2004

99 citations recorded by the source corpus.

Periodic points are points on Veech surfaces, whose orbit under the group of affine diffeomorphisms is finite. We characterize those points as being torsion points if the Veech surfaces is suitably mapped to its Jacobian or an appropriate factor thereof. For a primitive Veech surface in genus two we show that the only periodic points are the Weierstraß points and the singularities.Our main tool is the Hodge-theoretic characterization of Teichmüller curves. We deduce from it a finiteness result for the Mordell-Weil group of the family of Jacobians over a Teichmüller curve. The link to the classification of periodic points is provided by interpreting them as sections of the family of curves over a covering of the Teichmüller curve.

Subfields: Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/fb7a9dccc90d86b5b6e96499d1d975a47e69fb63

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

Citations in the corpus

Prym Varieties and Teichmüller Curves (2006, background): “Classification. The primitive Teichm¨uller curves in genus 2 are classified in [Mc2] and [Mc4]; in particular, it is shown (using [ Mo ]) that there is only one such curve lying outside S WD ⊂ M2. It would be interesting to have”
Euler characteristics of Teichmüller curves in genus two (2006, background): “Tabachnikov [51], Veech [76; 77; 78; 79], Kontsevich [42], Zorich [84], Möller [60], or Bouw and Möller [16].”
Teichmüller curves in genus two: Torsion divisors and ratios of sines (2006, background): “2 See [ Mo , Cor 3.4]. The proof also uses an elementary fact about cyclotomic fields (§2):”
The Deligne–Mumford compactification of the real multiplication locus and Teichmüller curves in genus 3 (2012, background): “2) together with a torsion condition from [Möl06a] which gives strong restrictions on Teichmüller curves generated by forms with more than one zero.”
A geometric criterion for the non-uniform hyperbolicity of the Kontsevich--Zorich cocycle (2010, background): “ by all the traces of elements of the Veech group. By a theorem of R. Kenyon and J. Smillie [KS], the trace ﬁeld of a translation surface of genus g ≥ 1 has degree at most g over Q. Deﬁnition 5. (see [Mö1], §2) A Veech surface (S,ω) of genus g ≥ 1 is called algebraically primitive if the trace ﬁeld ”