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On the singular points of the orbifolds arising from integral, ternary quadratic forms

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Abstract

Mennicke (Proc R Soc Edinb Sect A 67:309–352, 1963; Proc R Soc Edinb Sect A 88:151–157, 1981) investigated the number of conic points with isotropies of orders 3,  4 and 6 that can appear in the orientable twofold orbifold covering of the orbifold \(Q_{f}\) associated with an integral ternary quadratic form. Mennicke made essential use of a theorem of Jones (The Arithmetic Theory of Quadratic Forms. The Carus Mathematical Monographs, vol. 10. MAA, Baltimore, 1950, Theorem 86). In this paper we revisit Mennicke’s results and we extend them to \(Q_{f},\) that is, even when \(Q_{f}\) is non-orientable. Our method is new and independent of Jones’ Theorem. We also study the possible cusp points.

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Authors and Affiliations

  1. Facultad de Matematicas, Universidad Complutense, 28040, Madrid, Spain

    José María Montesinos-Amilibia

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  1. José María Montesinos-Amilibia
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Correspondence to José María Montesinos-Amilibia.

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Dedicado con admiración y cariño a mi querido amigo Fico González-Acuña en su 75 cumpleaños.

Supported by MEC-MTM 2012-30719.

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Montesinos-Amilibia, J.M. On the singular points of the orbifolds arising from integral, ternary quadratic forms. Bol. Soc. Mat. Mex. 25, 267–332 (2019). https://doi.org/10.1007/s40590-018-0199-5

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