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arXiv · 2503.03478

Gross lattices of supersingular elliptic curves

Chenfeng He·Gaurish Korpal·Ha T. N. Tran·Christelle Vincent
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Abstract

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.

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Chenfeng He, Gaurish Korpal, Ha T. N. Tran, Christelle Vincent. 2025-03-05. Gross lattices of supersingular elliptic curves. https://arxiv.org/abs/2503.03478

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