By Christian Weiß

These notes introduce a brand new category of algebraic curves on Hilbert modular surfaces. those curves are known as twisted Teichmüller curves, simply because their building is particularly resembling Hirzebruch-Zagier cycles. those new items are analyzed intimately and their major houses are defined. particularly, the quantity of twisted Teichmüller curves is calculated and their parts are partly categorised. The examine of algebraic curves on Hilbert modular surfaces has been broadly coated within the literature because of their mathematics value. between those, twisted diagonals (Hirzebruch-Zagier cycles) are one of the most very important examples.

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Leads to a singularity of this metric. The total angle around a singularity, called the cone angle, is an integer multiple of 2 . If the form ! g. [Zor06], Sect. 3). A geodesic segment connecting two singular points is called saddle connections. Equivalently, flat surfaces arise from gluing rational angled planar polygons by parallel translations along their faces. This implies in particular that there exists an atlas on X such that all transition maps of X away from the zeroes are given by translations.

Denote the zeroes of !. / a flat (Riemannian) metric is given by pulling back the Euclidean metric of C via the charts. / leads to a singularity of this metric. The total angle around a singularity, called the cone angle, is an integer multiple of 2 . If the form ! g. [Zor06], Sect. 3). A geodesic segment connecting two singular points is called saddle connections. Equivalently, flat surfaces arise from gluing rational angled planar polygons by parallel translations along their faces. This implies in particular that there exists an atlas on X such that all transition maps of X away from the zeroes are given by translations.

113–115. There it is assumed that the ring is Euclidean. However, the proof also works verbatim if only hD D 1. Z/-case. There are even some more special congruence subgroups, where one can calculate the index. g. g. [Wei08]). These groups will also be important for this work. 13. n/: Accordingly, we define all these groups for proper ideals m; n. 14. n// Proof. g. in [Wei08], Chap. 1. If hD > 1 it is immediately clear that the claimed equality hold with Ä instead of D. As is commonly known in the literature, equality is also true in the latter case (weak approximation).