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PARAMETRIZATION OF ALGEBRAIC POINTS OF A FAMILY OF ARNTHJENSEN-FLYNN HYPERELLIPTIC CURVES OF A GIVEN DEGREE
Keywords:
Mordell-Weil group, Jacobian, linear system.DOI:
https://doi.org/10.17654/0972555523020Abstract
We explicitly define the set of algebraic points of a hyperelliptic curve $\mathcal{C}_q$ of any given degree having affine equation $y^2=(x-2 q)\left(x^2-2 q^2\right)\left(x^2+2^2 q^2\right)$. Such a curve is described in [4], where it is shown that the Mordell-Weil group is finite for $q \equiv 13$ [24]. Furthermore, the generators of the torsion group for these curves are explained.
Received: March 25, 2023
Accepted: July 3, 2023
References
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Anna Arnth-Jensen and E. Victor Flynn, Non-trivial III in the Jacobian of an infinite family of curves of genus 2, J. Théor. Nombres Bordeaux 21 (2009), 1-13.
Samir Siksek and Michael Stoll, Partial descent on hyperelliptic curves and the generalized Fermat equation $x^3 + y^4 + z^5 = 0$, Bull. Lond. Math. Soc. 44 (2012), 151-166.
P. A. Griffiths, Introduction to algebraic curves, Translation Mathematical Monographs, American Mathematical Society, Providence, Vol. 76, 1989.
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