Geodesic
On $S$-units for valuations of the second degree in hyperelliptic fields
Izvestiya. Mathematics , Tome 84 (2020) no. 2, pp. 392-435

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In this paper we propose a new effective approach to the problem of finding and constructing non-trivial $S$-units of a hyperelliptic field $L$ for a set $S=S_h$ consisting of two conjugate valuations of the second degree. The results obtained are based on a deep connection between the problem of torsion in the Jacobians of hyperelliptic curves and the quasiperiodicity of continued $h$-fractions, that is, generalized functional continued fractions of special form constructed with respect to a valuation of the second degree. We find algorithms for searching for fundamental $S_h$-units which are comparable in effectiveness with known fast algorithms for two linear valuations.
Keywords: generalized continued fractions, hyperelliptic curves, fundamental $S$-units, divisor class group, torsion group of a Jacobian variety. 
@article{IM2_2020_84_2_a7,
     author = {G. V. Fedorov},
     title = {On $S$-units for valuations of the second degree in hyperelliptic fields},
     journal = {Izvestiya. Mathematics },
     pages = {392--435},
     publisher = {mathdoc},
     volume = {84},
     number = {2},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2020_84_2_a7/}
}
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G. V. Fedorov. On $S$-units for valuations of the second degree in hyperelliptic fields. Izvestiya. Mathematics , Tome 84 (2020) no. 2, pp. 392-435. http://geodesic.mathdoc.fr/item/IM2_2020_84_2_a7/