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POINTS ON $x^4+y^4=z^4$ OVER QUADRATIC EXTENSIONS OF ${\mathbb {Q}}(\zeta _8)(T_1,T_2,\ldots ,T_n)$

Part of: Diophantine equations Arithmetic algebraic geometry

Published online by Cambridge University Press:  07 November 2024

NGUYEN XUAN THO Open the ORCID record for NGUYEN XUAN THO [Opens in a new window]
NGUYEN XUAN THO*
Affiliation:
School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, Hanoi, Vietnam
*
e-mail: tho.nguyenxuan1@hust.edu.vn
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Abstract

Ishitsuka et al. [‘Explicit calculation of the mod 4 Galois representation associated with the Fermat quartic’, Int. J. Number Theory 16(4) (2020), 881–905] found all points on the Fermat quartic ${F_4\colon x^4+y^4=z^4}$ over quadratic extensions of ${\mathbb {Q}}(\zeta _8)$, where $\zeta _8$ is the eighth primitive root of unity $e^{i\pi /4}$. Using Mordell’s technique, we give an alternative proof for the result of Ishitsuka et al. and extend it to the rational function field ${\mathbb {Q}}({\zeta _8})(T_1,T_2,\ldots ,T_n)$.

Keywords

Diophantine geometryDiophantine equationsFermat quarticsquadratic points

MSC classification

Primary: 11D25: Cubic and quartic equations
Secondary: 11G05: Elliptic curves over global fields

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 111 , Issue 1 , February 2025 , pp. 19 - 31
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The author is supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) (grant number 101.04-2023.21).

Dedicated to Professor Andrew Bremner

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