Computing periodic points on Veech surfaces

Periodic points of Prym eigenforms

“…Overview of the proof. Because we can use the algorithm of Chowdhury-Everett-Freedman-Lee [5] to compute the periodic points of any prototypical eigenform M (w, e), it suffices to assume that D is sufficiently large. (Our argument is effective in that it computes the specific lower bounds on D.) For such a prototype M (w, e), we first classify the periodic points that lie in the interior of a horizontal or vertical cylinder.…”

“…He [15] The above families all yield good prototypes when D ∉ {5, 8, 12, 17}. In these remaining cases of low discriminant, we use the algorithm of Chowdhury-Everett-Freedman-Lee [5] to compute the periodic points. Applying Lemma 3.1 with the long horizontal cylinder and the long vertical cylinder, we find that the potential interior periodic points are the solid dots in Figure 11.…”

“…Examples of periodic points are the singular points of M and the fixed points of the hyperelliptic involution. Chowdhury-Everett-Freedman-Lee [5] produced an algorithm that inputs a (non-square-tiled) Veech surface and outputs its set of periodic points. They used this algorithm to conjecture that the periodic points of certain Veech surfaces, Prym eigenforms in the minimal stratum in genus 3, are the fixed points of an involution.…”

“…Previous work. Gutkin-Hubert-Schmidt [8] first showed that (nonarithmetic) closed GL(2, R)-orbits have a finite number of periodic points (see Chowdhury-Everett-Freedman-Lee [5] for another proof). Eskin-Filip-Wright [6] then showed finiteness for all nonarithmetic affine invariant subvarieties.…”

“…The philosophy of the algorithm in Chowdhury-Everett-Freedman-Lee [5] is that the Rational Height Lemma (see Lemma 2.1) applied with three "independent" parabolic directions is enough to determine the periodic points of the surface. Our argument in genus two uses three such directions, yet our arguments in genera four and five use slightly more.…”