Abstract
For a field k of characteristic 0, up to a natural equivalence relation, it is proved that the number of nontrivial elliptic fields \(k(x)(\sqrt f )\) with a periodic continued fraction expansion of \(\sqrt f \in k((x))\) for which the corresponding elliptic curve contains a k-point of even order at most 18 or a k-point of odd order at most 11 is finite. In the case when k is a quadratic extension of \(\mathbb{Q}\), all such fields are found.
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Platonov, V.P., Petrunin, M.M. & Shteinikov, Y.N. On the Finiteness of the Number of Elliptic Fields with Given Degrees of \(S\) -Units and Periodic Expansion of \(\sqrt f \) . Dokl. Math. 100, 440–444 (2019). https://doi.org/10.1134/S1064562419050119
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DOI: https://doi.org/10.1134/S1064562419050119