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arXiv · 2107.05902

The arithmetic of a twist of the Fermat quartic

Yasuhiro Ishitsuka·Tetsushi Ito·Tatsuya Ohshita
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Abstract

We study the arithmetic of the twist of the Fermat quartic defined by $X^4 + Y^4 + Z^4 = 0$ which has no $\mathbb{Q}$-rational point. We calculate the Mordell--Weil group of the Jacobian variety explicilty. We show that the degree $0$ part of the Picard group is a free $\mathbb{Z}/2\mathbb{Z}$-module of rank $2$, whereas the Mordell--Weil group is a free $\mathbb{Z}/2\mathbb{Z}$-module of rank $3$. Thus the relative Brauer group is non-trivial. We also show that this quartic violates the local-global property for linear determinantal representations.

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Yasuhiro Ishitsuka, Tetsushi Ito, Tatsuya Ohshita. 2021-07-14. The arithmetic of a twist of the Fermat quartic. https://arxiv.org/abs/2107.05902

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