Geodesic
Abel pairs and modular curves
Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part V, Tome 446 (2016), pp. 165-181
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We consider rational functions on algebraic curves which have a single zero and a single pole. A pair consisting of such a function and a curve is called Abel pair; a special case of an Abel pair is a Belyi pair. In this paper, we study moduli spaces of Abel pairs for curves of genus one. In particular, we compute a number of Belyi pairs over the fields $\mathbb C$ and $\overline{\mathbb F_p}$. This approach could be fruitfully used for the study of Hurwitz spaces and modular curves for fields of finite characteristics. 
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D. Oganesyan. Abel pairs and modular curves. Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part V, Tome 446 (2016), pp. 165-181. http://geodesic.mathdoc.fr/item/ZNSL_2016_446_a8/

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