A relatively minimal family of curves f : S → P^1 with 2 or 3 singular fibers is called a Belyi family or fibration. In this talk, we show the structure of all Belyi families of curves of genus g ≥ 2 with two singular fibers. When the such a Belyi family is simple connected, we can define exactly it by an equation. We compute all sections of f and its Mordell-Weil group. As an application, we prove that any periodic fiber can be realized as a fiber of a Belyi fibration with two singular fibers.

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