The modified diagonal cycles of Hypergeometric curves
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911367382958080 |
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| author | Eskandari, Payman Nemoto, Yusuke |
| author_facet | Eskandari, Payman Nemoto, Yusuke |
| contents | For each $N\geq 2$, Asakura and Otsubo have recently introduced a smooth family of algebraic curves $\{X_{N,λ}\}_{λ\in \mathbb{P}^1\setminus \{0, 1, \infty\}}$ in characteristic 0 that is closely related to hypergeometric functions and the Fermat curve of degree $N$. In this paper, we study the Gross-Kudla-Schoen modified diagonal 1-cycles of these curves. We prove that if $p \ge 3$ is a prime, then for every $λ$ the Griffiths Abel-Jacobi image of the modified diagonal cycle of $X_{p,λ}$ is nontrivial for every cuspidal choice of a base point. On the other hand, we show that the modified diagonal cycle and hence the Ceresa cycle of $X_{3,λ}$ is torsion in the Chow group for every $λ$ and every choice of a base point. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_06008 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The modified diagonal cycles of Hypergeometric curves Eskandari, Payman Nemoto, Yusuke Algebraic Geometry Number Theory 14C25, 14F25, 14H40 For each $N\geq 2$, Asakura and Otsubo have recently introduced a smooth family of algebraic curves $\{X_{N,λ}\}_{λ\in \mathbb{P}^1\setminus \{0, 1, \infty\}}$ in characteristic 0 that is closely related to hypergeometric functions and the Fermat curve of degree $N$. In this paper, we study the Gross-Kudla-Schoen modified diagonal 1-cycles of these curves. We prove that if $p \ge 3$ is a prime, then for every $λ$ the Griffiths Abel-Jacobi image of the modified diagonal cycle of $X_{p,λ}$ is nontrivial for every cuspidal choice of a base point. On the other hand, we show that the modified diagonal cycle and hence the Ceresa cycle of $X_{3,λ}$ is torsion in the Chow group for every $λ$ and every choice of a base point. |
| title | The modified diagonal cycles of Hypergeometric curves |
| topic | Algebraic Geometry Number Theory 14C25, 14F25, 14H40 |
| url | https://arxiv.org/abs/2508.06008 |