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Gross lattices of supersingular elliptic curves

Chenfeng He, Gaurish Korpal, Ha T. N. Tran, Christelle Vincent

TL;DR

This work links the geometry of Gross lattices to the endomorphism structure of supersingular elliptic curves over finite fields. By analyzing the third successive minimum \(D_3\) of the Gross lattice \(\mathcal{O}^T\) attached to a maximal order in the quaternion algebra \(B_p\), the authors derive precise criteria for whether the curve’s \(j\)-invariant lies in \(\mathbb{F}_p\) or in \(\mathbb{F}_{p^2}\setminus\mathbb{F}_p\); in the special case \(j(E)\in\mathbb{F}_p\) and \(p\equiv3\pmod{4}\), \(D_3\) detects whether the arithmetic endomorphism ring embeds \(\mathbb{Z}[\tfrac{1+\sqrt{-p}}{2}]\) and, when \(j(E)=1728\), yields a sharp bound. The paper also introduces a Gram-matrix invariant arising from a normalized successive minimal basis, proving its uniqueness for \(j(E)\in\mathbb{F}_p\) (except small primes) and providing an explicit algorithm to compute it from \(D_1\). Furthermore, a detailed algorithm \(\mathsf{GramGross}(p,D_1)\) computes all possible Gram matrices and is applied to the 13 CM j-invariants with class number 1, illustrating the invariant in concrete cases. Together, these results deepen the understanding of how lattice-theoretic data encode endomorphism and isogeny information of supersingular curves, with potential implications for endomorphism-ring recognition and isogeny-graph analyses.

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 refine and 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 $p \equiv 3 \pmod{4}$. Finally, in the case where $j(E)$ belongs to $\mathbb{F}_p$, we define a new invariant of the Gross lattice, the Gram matrix of a normalized successive minimal basis, and develop an algorithm to compute it explicitly given the value of the first successive minimum of the Gross lattice.

Gross lattices of supersingular elliptic curves

TL;DR

This work links the geometry of Gross lattices to the endomorphism structure of supersingular elliptic curves over finite fields. By analyzing the third successive minimum of the Gross lattice attached to a maximal order in the quaternion algebra , the authors derive precise criteria for whether the curve’s -invariant lies in or in ; in the special case \(j(E)\in\mathbb{F}_p\) and , detects whether the arithmetic endomorphism ring embeds and, when \(j(E)=1728\), yields a sharp bound. The paper also introduces a Gram-matrix invariant arising from a normalized successive minimal basis, proving its uniqueness for \(j(E)\in\mathbb{F}_p\) (except small primes) and providing an explicit algorithm to compute it from . Furthermore, a detailed algorithm \(\mathsf{GramGross}(p,D_1)\) computes all possible Gram matrices and is applied to the 13 CM j-invariants with class number 1, illustrating the invariant in concrete cases. Together, these results deepen the understanding of how lattice-theoretic data encode endomorphism and isogeny information of supersingular curves, with potential implications for endomorphism-ring recognition and isogeny-graph analyses.

Abstract

Let be a prime, be a supersingular elliptic curve defined over , and be its (geometric) endomorphism ring. Earlier results of Chevyrev-Galbraith and Goren-Love have shown that the successive minima of the Gross lattice of characterize the isomorphism class of . In this paper, we refine and extend this work and show that the value of the third successive minimum of the Gross lattice gives necessary and sufficient conditions for the curve to have its -invariant in the field or in the set , as well as finer information about the endomorphism ring of when its -invariant belongs to and . Finally, in the case where belongs to , we define a new invariant of the Gross lattice, the Gram matrix of a normalized successive minimal basis, and develop an algorithm to compute it explicitly given the value of the first successive minimum of the Gross lattice.

Paper Structure

This paper contains 18 sections, 31 theorems, 92 equations, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main results of this article
  3. Computational tools and structure of the article
  4. Background
  5. Notation and definitions
  6. Supersingular elliptic curves
  7. Lattice definitions and notations
  8. Preliminary results
  9. Successive minima of Gross lattices
  10. Rank-$2$ sublattices of determinant $4p$
  11. Successive minimal bases for $j$-invariants $0$ and $1728$
  12. The third successive minimum of Gross lattices when $j(E)\notin \mathbb{F}_p$
  13. The third successive minimum of Gross lattices when $j(E)\in \mathbb{F}_p$
  14. Inequality of successive minima when $j \in \mathbb{F}_p$
  15. Geometry of Gross lattices
  16. ...and 3 more sections

Key Result

Theorem 1.1.1

Let $E$ be a supersingular elliptic curve defined over $\overline{\mathbb{F}}_p$, $\mathcal{O}^T$ be its Gross lattice, and $D_3$ be the third successive minimum of $\mathcal{O}^T$. If $p \geq 7$, then $j(E)$, the $j$-invariant of $E$, belongs to $\mathbb{F}_p\setminus\{0\}$ if and only if and if $p \neq 3$, $j(E) = 0$ if and only if $D_3= \frac{4p+1}{3}$. Otherwise, $j(E)$ belongs to $\mathbb{F}

Theorems & Definitions (70)

  • Theorem 1.1.1
  • Theorem 1.1.2
  • Definition 2.1.1
  • Definition 2.1.2
  • Definition 2.1.3
  • Definition 2.1.4
  • Definition 2.1.5
  • Proposition 2.2.1: Proposition 3.11 of Goren-Love
  • Lemma 2.2.2
  • proof
  • ...and 60 more