Gross lattices of supersingular elliptic curves
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913146827964416 |
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| author | He, Chenfeng Korpal, Gaurish Tran, Ha T. N. Vincent, Christelle |
| author_facet | He, Chenfeng Korpal, Gaurish Tran, Ha T. N. Vincent, Christelle |
| contents | Let $p$ be a prime, $E$ be a supersingular elliptic curve defined over $\bar{\mathbb{F}}_p$, and $\mathscr{O}$ be its (geometric) endomorphism ring. Earlier results of Chevyrev-Galbraith and Goren-Love have shown that the successive minima of the Gross lattice of $\mathscr{O}$ characterize the isomorphism class of $\mathscr{O}$. In this paper, we extend this work and show that the value of the third successive minimum $D_3$ of the Gross lattice gives necessary and sufficient conditions for the curve to have its $j$-invariant in the field $\mathbb{F}_p$ or in the set $\mathbb{F}_{p^2} \setminus \mathbb{F}_p$, as well as finer information about the endomorphism ring of $E$ when its $j$-invariant belongs to $\mathbb{F}_p$ and $p \equiv 3 \pmod{4}$. We end our article with an investigation of the geometry of Gross lattices of supersingular elliptic curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_03478 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Gross lattices of supersingular elliptic curves He, Chenfeng Korpal, Gaurish Tran, Ha T. N. Vincent, Christelle Number Theory 11G20, 11R52, 14G15, 14G50 (Primary) 11H06, 11Y40, 11Y16 (Secondary) Let $p$ be a prime, $E$ be a supersingular elliptic curve defined over $\bar{\mathbb{F}}_p$, and $\mathscr{O}$ be its (geometric) endomorphism ring. Earlier results of Chevyrev-Galbraith and Goren-Love have shown that the successive minima of the Gross lattice of $\mathscr{O}$ characterize the isomorphism class of $\mathscr{O}$. In this paper, we extend this work and show that the value of the third successive minimum $D_3$ of the Gross lattice gives necessary and sufficient conditions for the curve to have its $j$-invariant in the field $\mathbb{F}_p$ or in the set $\mathbb{F}_{p^2} \setminus \mathbb{F}_p$, as well as finer information about the endomorphism ring of $E$ when its $j$-invariant belongs to $\mathbb{F}_p$ and $p \equiv 3 \pmod{4}$. We end our article with an investigation of the geometry of Gross lattices of supersingular elliptic curves. |
| title | Gross lattices of supersingular elliptic curves |
| topic | Number Theory 11G20, 11R52, 14G15, 14G50 (Primary) 11H06, 11Y40, 11Y16 (Secondary) |
| url | https://arxiv.org/abs/2503.03478 |