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Orbitwise countings in \({\mathcal H}(2)\) and quasimodular forms. (English) Zbl 1142.11025

In this very well written work, the authors apply the theory of quasimodular forms to confirm a conjecture of Hubert and Lelièvre, giving the exact number of primitive square-tiled surfaces of genus two and with a single singularity that are tiled by an odd number \(n\) of squares. A square-tiled surface is a translation surface that is a ramified cover of the flat square torus, with all ramification occurring over a single point; the surfaces considered are such that there is a single cone point of angle \(6 \pi\). A square-tiled surface naturally defines a holomorphic 1-form on the underlying Riemann surfaces; primitive here refers to the period lattice of the form being all of \(\mathbb Z^2\). The proof is given by expressing the appropriate counting functions in terms of explicit quasimodular forms. The background discussion, both of the flat surfaces and their orbit types, and of the theory of quasimodular forms is exceptionally clear.

MSC:

11F06 Structure of modular groups and generalizations; arithmetic groups
11F11 Holomorphic modular forms of integral weight
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
30F30 Differentials on Riemann surfaces
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