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We construct a new kind of elliptic bration in terms of group action and group representation: a dihedral group acts on a regular polygon – thickening its edges yields a torus on which the dihedral group acts. We give a complex structure to the torus such that this action is holomorphic. Then from representations of the dihedral group, we construct fibrations of elliptic curves (dihedral quotient families). We classify them – their singular loci/fibers, as well as the dynamic motion of the singular fibers, are completely described. Moreover, we construct binary dihedral quotient families and classify them. Although the representations of the dihedral and binary dihedral groups are analogous, it turns out that their associated quotient families are totally different.
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Ryota Hirakawa, Shigeru Takamura, Quotient Families of Elliptic Curves Associated with Representations of Dihedral Groups. Publ. Res. Inst. Math. Sci. 55 (2019), no. 2, pp. 319–367DOI 10.4171/PRIMS/55-2-4