Gross lattices of supersingular elliptic curves

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

arxiv: 2503.03478 · v3 · pith:HTICPTWQnew · submitted 2025-03-05 · 🧮 math.NT

Gross lattices of supersingular elliptic curves

Chenfeng He , Gaurish Korpal , Ha T. N. Tran , Christelle Vincent This is my paper

Pith reviewed 2026-05-23 01:30 UTC · model grok-4.3

classification 🧮 math.NT

keywords supersingular elliptic curvesGross latticesuccessive minimaj-invariantendomorphism ringfinite fieldsnumber theory

0 comments

The pith

The third successive minimum D3 of the Gross lattice determines whether the j-invariant of a supersingular elliptic curve lies in F_p or in F_{p^2} excluding F_p.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends earlier results that the successive minima of the Gross lattice characterize the isomorphism class of the endomorphism ring of a supersingular elliptic curve. It proves that the third minimum D3 supplies necessary and sufficient conditions for the j-invariant to belong to F_p or to F_{p^2} but not F_p. When the j-invariant lies in F_p and p is congruent to 3 modulo 4, the same value of D3 yields additional information on the endomorphism ring. The work also examines the geometry of these lattices. A reader would care because the conditions tie a lattice invariant directly to the arithmetic field of definition of the curve.

Core claim

The value of the third successive minimum D3 of the Gross lattice gives necessary and sufficient conditions for the curve to have its j-invariant in the field F_p or in the set F_{p^2} excluding F_p, as well as finer information about the endomorphism ring of E when its j-invariant belongs to F_p and p ≡ 3 mod 4.

What carries the argument

The Gross lattice attached to the endomorphism ring O of the supersingular elliptic curve, with its third successive minimum D3 carrying the information about the field containing the j-invariant.

If this is right

D3 satisfying one set of inequalities forces the j-invariant into F_p.
D3 satisfying a different set of inequalities forces the j-invariant into F_{p^2} but not F_p.
When the j-invariant is in F_p and p ≡ 3 mod 4, D3 distinguishes additional structural features of the endomorphism ring.
The geometry of the Gross lattices admits a direct description in terms of these minima.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The criterion may simplify the search for supersingular curves whose j-invariants lie in a prescribed subfield.
The geometric investigation of the lattices could produce new distribution results for supersingular curves.
Analogous statements might hold for higher successive minima and supply further arithmetic invariants.

Load-bearing premise

Earlier results that the successive minima of the Gross lattice characterize the isomorphism class of the endomorphism ring O.

What would settle it

A supersingular elliptic curve over the algebraic closure of F_p whose third successive minimum D3 fails to indicate correctly whether its j-invariant lies in F_p or in F_{p^2} excluding F_p.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper cleanly extends prior lattice minima results by showing that D3 alone distinguishes whether the j-invariant lies in F_p or in F_{p^2} minus F_p.

read the letter

The core new piece is that the third successive minimum D3 of the Gross lattice gives necessary and sufficient conditions on the location of the j-invariant, plus some extra detail on the endomorphism ring when j is in F_p and p ≡ 3 mod 4. This sits directly on top of the Chevyrev-Galbraith and Goren-Love characterizations of the orders via successive minima, without any visible circularity or extra parameters. The abstract states the claim plainly and the stress-test note finds no internal gaps in the logic of the extension. The geometry investigation at the end is secondary but harmless. The main limitation is narrow scope: it refines an existing tool rather than opening a new direction, so the payoff stays inside the supersingular curves subfield. Proofs are not visible from the abstract, but the foundation looks independent and the claim is concrete enough to check. This is for readers who already work with Gross lattices or supersingular isogenies and know the two cited papers. They would get a usable new criterion. It deserves a serious referee because the extension is specific, builds on verified prior results, and adds a falsifiable distinction without fitting or self-reference issues.

Referee Report

0 major / 2 minor

Summary. The manuscript extends results of Chevyrev-Galbraith and Goren-Love showing that successive minima of the Gross lattice characterize the isomorphism class of the endomorphism ring O of a supersingular elliptic curve E. It proves that the third successive minimum D3 supplies necessary and sufficient conditions determining whether the j-invariant lies in F_p or in F_{p^2} excluding F_p, together with finer information on O when j lies in F_p and p ≡ 3 mod 4. The paper concludes with an investigation of the geometry of these Gross lattices.

Significance. If the stated extension holds, the work supplies a direct lattice invariant that distinguishes the field of definition of j-invariants of supersingular curves, refining the prior characterization of endomorphism rings. This strengthens the connection between quadratic forms/lattices and arithmetic properties of supersingular curves without introducing circular dependence on the cited independent results. The geometric investigation of the lattices provides additional structural insight.

minor comments (2)

[Introduction] The introduction would benefit from a short paragraph recalling the precise definition of the Gross lattice and the successive minima D1, D2, D3 before stating the new results on D3.
Notation for the set F_{p^2} excluding F_p is used without an explicit symbol; introducing a compact notation (e.g., F_{p^2}^*) early would improve readability in the statements of the main theorems.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive recommendation to accept the manuscript. The report accurately summarizes the main contributions.

Circularity Check

0 steps flagged

No significant circularity; extends independent external results

full rationale

The paper's central extension—that the third successive minimum D3 of the Gross lattice distinguishes whether j(E) lies in F_p or in F_{p^2} minus F_p, plus refinements when j(E) is in F_p and p ≡ 3 mod 4—rests on the prior characterization (by Chevyrev-Galbraith and Goren-Love) that successive minima determine the isomorphism class of the endomorphism ring O. Those cited works are by different authors, are invoked only as the foundation being extended, and are not self-citations. No equations or claims in the abstract or described derivation reduce a prediction to a fitted parameter, rename a known result, or smuggle an ansatz via self-citation; the new conditions on D3 are presented as additional consequences of the lattice geometry for the already-classified orders. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract does not introduce or rely on any free parameters, ad-hoc axioms, or invented entities beyond standard concepts from elliptic curve theory and lattice theory.

pith-pipeline@v0.9.0 · 5716 in / 1065 out tokens · 32911 ms · 2026-05-23T01:30:46.290498+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

the successive minima of the Gross lattice of O characterize the isomorphism class of O … the value of the third successive minimum D3 … j-invariant in Fp or in Fp²∖Fp
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

det(OT)=4p² … 4p²≤D1D2D3≤8p²

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.