The Hessian of elliptic curves
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913739604754432 |
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| author | Mula, Marzio Pintore, Federico Taufer, Daniele |
| author_facet | Mula, Marzio Pintore, Federico Taufer, Daniele |
| contents | We prove that the Hessian transformation of elliptic curves, both as an action on $j$-invariants and on the Hesse pencil, is a Lattès map, namely it ascends to a degree-3 endomorphism $ψ$ of a prescribed elliptic curve $E$. This result provides a powerful tool to investigate the dynamics of the Hessian transformation, which inherits its symmetries from $ψ$. In particular, we show that, over arbitrary fields of characteristic different from 2 and 3, the Hessian functional graphs can be completely determined in terms of the action of $ψ$ on the twists of $E$. When the underlying field is finite, we specialize our results to provide a complete classification of Hessian functional graphs. In such a case, we also present a practical way to compute iterated Hessians. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_17042 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Hessian of elliptic curves Mula, Marzio Pintore, Federico Taufer, Daniele Number Theory Commutative Algebra Algebraic Geometry Dynamical Systems 14H52, 37P05, 37P15, 37P25 We prove that the Hessian transformation of elliptic curves, both as an action on $j$-invariants and on the Hesse pencil, is a Lattès map, namely it ascends to a degree-3 endomorphism $ψ$ of a prescribed elliptic curve $E$. This result provides a powerful tool to investigate the dynamics of the Hessian transformation, which inherits its symmetries from $ψ$. In particular, we show that, over arbitrary fields of characteristic different from 2 and 3, the Hessian functional graphs can be completely determined in terms of the action of $ψ$ on the twists of $E$. When the underlying field is finite, we specialize our results to provide a complete classification of Hessian functional graphs. In such a case, we also present a practical way to compute iterated Hessians. |
| title | The Hessian of elliptic curves |
| topic | Number Theory Commutative Algebra Algebraic Geometry Dynamical Systems 14H52, 37P05, 37P15, 37P25 |
| url | https://arxiv.org/abs/2407.17042 |