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We parametrize the commensurability classes of curves on Shimura surfacesthat are totally geodesic, i.e., the commensurability classes of so-called$\mathbb{C}$-Fuchsian subgroups. In particular, if a Shimura surface containsone commensurability class of totally geodesic curves, it contains infinitelymany.

Geodesic curves on Shimura surfaces