Subregular Nilpotents in Stable Gradings

• Subregular nilpotents in stable gradings are nilpotent elements in degree one of a graded Lie algebra that satisfy a specific dimension condition to produce 1-dimensional Slodowy slices.
• They arise naturally in Vinberg’s θ-group representations and are classified using invariant-theoretic methods, Weyl-group criteria, and Kac diagram techniques.
• The graded Slodowy slice construction links Lie algebra invariants directly to explicit families of algebraic curves, enabling detailed orbit parametrizations and arithmetic applications.

Searching arXiv for the cited papers and topic to ground the article in current research. Subregular nilpotents in stable gradings arise naturally in Vinberg’s $\theta$-group representations and give rise to uniform families of algebraic curves. In the setting of a $\mathbb{Z}/m\mathbb{Z}$-graded Lie algebra $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$ coming from a finite-order automorphism, the relevant degree-one representation is $V=\mathfrak h_1$ for the connected fixed-point group $G=(H^\theta)^\circ$. A nilpotent element $e\in \mathfrak h_1$ is called $\theta$-subregular when it is subregular in $\mathfrak h$ and satisfies $\dim \mathfrak h_0(e)=1$; this condition is precisely what makes the graded Slodowy slice $X_e$ into a family of curves over the Vinberg quotient $\mathbb{Z}/m\mathbb{Z}$0. Recent work classifies the subregular-adapted stable gradings for all simple $\mathbb{Z}/m\mathbb{Z}$1 and $\mathbb{Z}/m\mathbb{Z}$2, while the broader theory of positive-rank stable gradings supplies the Weyl-group criterion, little Weyl groups, and Kostant-section framework in which these constructions sit (Laga et al., 13 Aug 2025, Reeder et al., 2013).

1. Stable gradings and Vinberg representations

Let $\mathbb{Z}/m\mathbb{Z}$3 be a field of characteristic zero, let $\mathbb{Z}/m\mathbb{Z}$4 be a reductive algebraic group, and let $\mathbb{Z}/m\mathbb{Z}$5 be a $\mathbb{Z}/m\mathbb{Z}$6-action. After choosing a primitive $\mathbb{Z}/m\mathbb{Z}$7-th root of unity $\mathbb{Z}/m\mathbb{Z}$8, differentiation gives an eigenspace decomposition

$\mathbb{Z}/m\mathbb{Z}$9

The associated Vinberg representation is the action of $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$0 on $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$1 by restriction of the adjoint representation. A vector $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$2 is stable if its $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$3-orbit is closed and the stabilizer $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$4 is a finite group scheme; the grading is stable if $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$5 contains stable vectors (Laga et al., 13 Aug 2025).

For simple $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$6 over an algebraically closed field, stability is characterized by the Weyl-group condition that $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$7 be principal of order $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$8 and that the coset $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$9 contain an elliptic $V=\mathfrak h_1$0-regular element of order $V=\mathfrak h_1$1. In the language of positive-rank gradings, a Cartan subspace $V=\mathfrak h_1$2 is a maximal abelian subspace consisting of semisimple elements, its dimension is the rank of the grading, and the little Weyl group

$V=\mathfrak h_1$3

acts faithfully on $V=\mathfrak h_1$4. Chevalley restriction takes the form $V=\mathfrak h_1$5, so the invariant ring is polynomial; in the Vinberg setting this is compatible with Panyushev’s congruence rule, which selects the $V=\mathfrak h_1$6-invariants $V=\mathfrak h_1$7 whose restrictions generate $V=\mathfrak h_1$8 by the condition $V=\mathfrak h_1$9 (Reeder et al., 2013).

Stable gradings have two consequences used throughout the theory. First, in $G=(H^\theta)^\circ$0 one has “stable = regular semisimple,” equivalently $G=(H^\theta)^\circ$1 is stable if and only if the discriminant $G=(H^\theta)^\circ$2. Second, the quotient $G=(H^\theta)^\circ$3 is coregular. These properties make the degree-one piece of a stable grading simultaneously amenable to GIT, invariant theory, and singularity-theoretic slicing (Laga et al., 13 Aug 2025).

2. Subregular nilpotents in degree one

In a simple Lie algebra $G=(H^\theta)^\circ$4 over an algebraically closed field, a nilpotent element $G=(H^\theta)^\circ$5 is subregular if $G=(H^\theta)^\circ$6; equivalently, its adjoint orbit is the unique maximal non-regular nilpotent orbit. In the graded setting, the additional condition

$G=(H^\theta)^\circ$7

defines $G=(H^\theta)^\circ$8-subregularity. The second condition is not decorative: it ensures that the graded transverse slice has relative dimension one, so the quotient map produces a family of curves rather than a higher-dimensional family (Laga et al., 13 Aug 2025).

Stable gradings always contain a regular nilpotent element in degree one. For principal gradings, the sum of simple root vectors $G=(H^\theta)^\circ$9 lies in $e\in \mathfrak h_1$0 and is regular nilpotent. The position of subregular nilpotents is subtler. In the classification of stable gradings of positive rank, stable gradings with normalized Kac coordinate $e\in \mathfrak h_1$1 are attached, in exceptional types, to distinguished nilpotent classes $e\in \mathfrak h_1$2 recorded by Bala–Carter labels. Among these labels are the subregular classes $e\in \mathfrak h_1$3, $e\in \mathfrak h_1$4, $e\in \mathfrak h_1$5, $e\in \mathfrak h_1$6, and $e\in \mathfrak h_1$7, which identify stable gradings of subregular type (Reeder et al., 2013).

A frequent source of confusion is the distinction between a grading being attached to a subregular distinguished nilpotent and a paper explicitly producing a representative of the subregular orbit inside $e\in \mathfrak h_1$8. The positive-rank classification gives the former correspondence via Kac diagrams and $e\in \mathfrak h_1$9, but does not explicitly claim that a representative of the subregular orbit lies in $\theta$0. The later theory of subregular-adapted stable gradings does exactly that: it isolates the cases where a $\theta$1-subregular element exists in degree one and then uses it to construct the family $\theta$2 (Reeder et al., 2013, Laga et al., 13 Aug 2025).

3. Graded Slodowy slices and the curve construction

Given a nilpotent element $\theta$3, choose a normal $\theta$4-triple $\theta$5 with $\theta$6 and $\theta$7. The affine Slodowy slice and its graded intersection are

$\theta$8

Restricting the Vinberg quotient $\theta$9 gives

$\mathfrak h$0

The map $\mathfrak h$1 is flat, and the multiplication map $\mathfrak h$2 is smooth. If $\mathfrak h$3 is $\mathfrak h$4-subregular, then the fibers of $\mathfrak h$5 are $\mathfrak h$6-dimensional; moreover, under a mild “goodness” condition that holds for the $\mathfrak h$7-subregular elements in the classification, the central fiber $\mathfrak h$8 is reduced, connected, and has a unique singular point at $\mathfrak h$9 (Laga et al., 13 Aug 2025).

The mechanism is controlled by a compatible $\dim \mathfrak h_0(e)=1$0-action derived from the $\dim \mathfrak h_0(e)=1$1-triple and the grading:

$\dim \mathfrak h_0(e)=1$2

with $\dim \mathfrak h_0(e)=1$3. In coordinates $\dim \mathfrak h_0(e)=1$4 whose weights are determined by $\dim \mathfrak h_0(e)=1$5, the central fiber of $\dim \mathfrak h_0(e)=1$6 is given by one of the simple surface singularities

$\dim \mathfrak h_0(e)=1$7

possibly with an explicit $\dim \mathfrak h_0(e)=1$8-action in the non-simply laced cases. Passing to $\dim \mathfrak h_0(e)=1$9-fixed points sets one of the coordinates $X_e$0 to zero and yields the explicit plane or weighted-plane curve equations appearing in the classification (Laga et al., 13 Aug 2025).

The dimension formula explains why subregularity is the correct threshold. For stable $X_e$1, one has $X_e$2. The nilpotent cone in $X_e$3 has dimension $X_e$4, and nilpotent orbits have codimension equal to $X_e$5. Thus the condition $X_e$6 forces $X_e$7 to have relative dimension $X_e$8, which is the precise geometric condition for a family of curves (Laga et al., 13 Aug 2025).

4. Classical subregular-adapted stable gradings

The classification of subregular-adapted stable gradings for simple $X_e$9 and $\mathbb{Z}/m\mathbb{Z}$00 is given up to isogeny of $\mathbb{Z}/m\mathbb{Z}$01 and canonical identification of $\mathbb{Z}/m\mathbb{Z}$02. In the classical series, the resulting families already display the main geometric forms: genus-$\mathbb{Z}/m\mathbb{Z}$03 degenerations, hyperelliptic families, and mixed hyperelliptic-trigonal families (Laga et al., 13 Aug 2025).

Type $\mathbb{Z}/m\mathbb{Z}$04 occurs only in the Coxeter grading $\mathbb{Z}/m\mathbb{Z}$05. Here $\mathbb{Z}/m\mathbb{Z}$06, $\mathbb{Z}/m\mathbb{Z}$07 is a sum of characters, and the family is

$\mathbb{Z}/m\mathbb{Z}$08

This is a genus-$\mathbb{Z}/m\mathbb{Z}$09 family.

Type $\mathbb{Z}/m\mathbb{Z}$10 with $\mathbb{Z}/m\mathbb{Z}$11 has $\mathbb{Z}/m\mathbb{Z}$12 acting on $\mathbb{Z}/m\mathbb{Z}$13. The corresponding curves are hyperelliptic:

$\mathbb{Z}/m\mathbb{Z}$14

The generators $\mathbb{Z}/m\mathbb{Z}$15 have degrees $\mathbb{Z}/m\mathbb{Z}$16.

Type $\mathbb{Z}/m\mathbb{Z}$17 with $\mathbb{Z}/m\mathbb{Z}$18 has $\mathbb{Z}/m\mathbb{Z}$19 acting on $\mathbb{Z}/m\mathbb{Z}$20, and the family is

$\mathbb{Z}/m\mathbb{Z}$21

These are hyperelliptic curves, with invariant degrees $\mathbb{Z}/m\mathbb{Z}$22.

Type $\mathbb{Z}/m\mathbb{Z}$23 with $\mathbb{Z}/m\mathbb{Z}$24 has $\mathbb{Z}/m\mathbb{Z}$25 on $\mathbb{Z}/m\mathbb{Z}$26. There are two families, depending on the subregular element:

$\mathbb{Z}/m\mathbb{Z}$27

and

$\mathbb{Z}/m\mathbb{Z}$28

The classification identifies these as hyperelliptic or trigonal according to the weights entering the Slodowy construction.

Type $\mathbb{Z}/m\mathbb{Z}$29 with $\mathbb{Z}/m\mathbb{Z}$30 splits according to parity and outer twisting. For even $\mathbb{Z}/m\mathbb{Z}$31, one has $\mathbb{Z}/m\mathbb{Z}$32 acting on $\mathbb{Z}/m\mathbb{Z}$33, with family

$\mathbb{Z}/m\mathbb{Z}$34

There are two degree-$\mathbb{Z}/m\mathbb{Z}$35 invariants $\mathbb{Z}/m\mathbb{Z}$36, reflecting the spinor splitting. For $\mathbb{Z}/m\mathbb{Z}$37 with $\mathbb{Z}/m\mathbb{Z}$38 odd, the analogous family is

$\mathbb{Z}/m\mathbb{Z}$39

In all of these classical cases, the invariants $\mathbb{Z}/m\mathbb{Z}$40 generate $\mathbb{Z}/m\mathbb{Z}$41, and the discriminant $\mathbb{Z}/m\mathbb{Z}$42 controls smoothness: the fiber over $\mathbb{Z}/m\mathbb{Z}$43 is smooth when $\mathbb{Z}/m\mathbb{Z}$44 (Laga et al., 13 Aug 2025).

5. Exceptional types and explicit families

The exceptional types exhibit the full range of phenomena emphasized by the classification: multiple stable orders $\mathbb{Z}/m\mathbb{Z}$45, non-simply laced cases, and several genus-one or elliptic families that already occur in arithmetic statistics. The same construction applies uniformly, but the resulting equations depend sharply on the grading order and on the $\mathbb{Z}/m\mathbb{Z}$46-weights of the invariant generators (Laga et al., 13 Aug 2025).

For type $\mathbb{Z}/m\mathbb{Z}$47, there are two relevant orders. When $\mathbb{Z}/m\mathbb{Z}$48, $\mathbb{Z}/m\mathbb{Z}$49 with $\mathbb{Z}/m\mathbb{Z}$50, and the families are

$\mathbb{Z}/m\mathbb{Z}$51

When $\mathbb{Z}/m\mathbb{Z}$52, in the Coxeter-twisted case, $\mathbb{Z}/m\mathbb{Z}$53 with

$\mathbb{Z}/m\mathbb{Z}$54

and the family is the elliptic curve

$\mathbb{Z}/m\mathbb{Z}$55

For type $\mathbb{Z}/m\mathbb{Z}$56, the case $\mathbb{Z}/m\mathbb{Z}$57 has $\mathbb{Z}/m\mathbb{Z}$58 and $\mathbb{Z}/m\mathbb{Z}$59. Two families occur:

$\mathbb{Z}/m\mathbb{Z}$60

and

$\mathbb{Z}/m\mathbb{Z}$61

For $\mathbb{Z}/m\mathbb{Z}$62, with $\mathbb{Z}/m\mathbb{Z}$63 and $\mathbb{Z}/m\mathbb{Z}$64, the family is

$\mathbb{Z}/m\mathbb{Z}$65

For type $\mathbb{Z}/m\mathbb{Z}$66, three stable orders contribute. In the outer case $\mathbb{Z}/m\mathbb{Z}$67 with $\mathbb{Z}/m\mathbb{Z}$68, $\mathbb{Z}/m\mathbb{Z}$69 and $\mathbb{Z}/m\mathbb{Z}$70, giving

$\mathbb{Z}/m\mathbb{Z}$71

For $\mathbb{Z}/m\mathbb{Z}$72, $\mathbb{Z}/m\mathbb{Z}$73 and $\mathbb{Z}/m\mathbb{Z}$74, with family

$\mathbb{Z}/m\mathbb{Z}$75

For the outer case $\mathbb{Z}/m\mathbb{Z}$76 with $\mathbb{Z}/m\mathbb{Z}$77, $\mathbb{Z}/m\mathbb{Z}$78 and $\mathbb{Z}/m\mathbb{Z}$79, yielding

$\mathbb{Z}/m\mathbb{Z}$80

For type $\mathbb{Z}/m\mathbb{Z}$81 with $\mathbb{Z}/m\mathbb{Z}$82, $\mathbb{Z}/m\mathbb{Z}$83 acts on $\mathbb{Z}/m\mathbb{Z}$84, and the family is

$\mathbb{Z}/m\mathbb{Z}$85

For type $\mathbb{Z}/m\mathbb{Z}$86, the classification yields three orders. When $\mathbb{Z}/m\mathbb{Z}$87, $\mathbb{Z}/m\mathbb{Z}$88 on the half-spin representation, with family

$\mathbb{Z}/m\mathbb{Z}$89

When $\mathbb{Z}/m\mathbb{Z}$90, $\mathbb{Z}/m\mathbb{Z}$91 and $\mathbb{Z}/m\mathbb{Z}$92, with family

$\mathbb{Z}/m\mathbb{Z}$93

When $\mathbb{Z}/m\mathbb{Z}$94, $\mathbb{Z}/m\mathbb{Z}$95 for $\mathbb{Z}/m\mathbb{Z}$96 and $\mathbb{Z}/m\mathbb{Z}$97, with family

$\mathbb{Z}/m\mathbb{Z}$98

These exceptional families include hyperelliptic, trigonal, plane quartic, genus-one, and elliptic cases. The classification identifies them as arising from a single Lie-theoretic mechanism rather than as isolated invariant-theoretic constructions (Laga et al., 13 Aug 2025).

6. Relation to the earlier classification and conceptual caveats

The classification of families of curves associated with subregular nilpotents generalizes the earlier stable-grading framework in two directions. First, it extends Thorne’s $\mathbb{Z}/m\mathbb{Z}$99-grading constructions for simply laced types $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$00, $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$01, and $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$02 to all $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$03. Second, it includes non-simply laced types $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$04, $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$05, $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$06, and $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$07. New cases explicitly singled out by the classification include the $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$08 families for $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$09, $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$10, and $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$11, the $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$12 families for $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$13 and $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$14, and the $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$15 family for $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$16 (Laga et al., 13 Aug 2025).

The broader theory of positive-rank gradings supplies the ambient classification of stable gradings, their little Weyl groups, and their invariant degrees. It proves that stable $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$17 principal $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$18 elliptic $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$19-regular on the root system, identifies $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$20 in the stable principal cases, and establishes Kostant sections for inner exceptional types through a Levi reduction. In particular, for inner $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$21, $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$22, and $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$23, there exists a $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$24-stable Levi $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$25 such that $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$26 is principal and the graded invariant theory reduces to the principal case in the Levi (Reeder et al., 2013).

Two clarifications are essential. The first is that subregular-adapted is stricter than stable. While every stable $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$27-grading is subregular-adapted, for $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$28 there are stable gradings that are not subregular-adapted; the listed examples are $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$29- and $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$30-gradings on $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$31, and $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$32 on $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$33. The second is that non-subregular-adapted gradings may still yield curve families, but these are often genus $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$34 or transverse slices inside families already arising from subregular-adapted gradings. This suggests that $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$35-subregularity isolates the cases in which the curve family is both intrinsic to the grading and maximally uniform (Laga et al., 13 Aug 2025).

7. Orbit parametrizations and the $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$36, $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$37 case

A central consequence of the classification is that almost all classical coregular representations used in arithmetic statistics can be interpreted as Vinberg representations attached to stable gradings and subregular slices. Hyperelliptic curves with a marked Weierstrass point arise from type $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$38, $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$39, where $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$40 for $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$41 and the invariant ring is generated by $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$42. Genus-one descent examples fit the same pattern: $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$43 comes from type $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$44, $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$45, with $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$46 acting on ternary cubic forms and invariants $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$47; $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$48 comes from $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$49, $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$50, with invariants $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$51; and $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$52 comes from type $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$53, $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$54, with invariants of degrees $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$55 and $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$56. In each case, the dictionary is “invariants $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$57 coefficients in the curve equation,” while integral orbits with specified invariants correspond to arithmetic objects via

$$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$58

(Laga et al., 13 Aug 2025).

The extended example is the principal $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$59-grading of split $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$60 over $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$61. Here

$$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$62

and

$$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$63

The invariant ring is

$$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$64

where $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$65 and $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$66 have degrees $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$67 and $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$68. The discriminant satisfies

$$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$69

for some $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$70, and the subregular slice produces the elliptic family

$$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$71

This equation is obtained from the graded Slodowy surface

$$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$72

by taking $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$73-fixed points and setting $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$74, so that only $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$75 and $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$76 survive on $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$77 (Laga et al., 13 Aug 2025).

The associated orbit-parametrization theorem states that there exists $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$78 such that, for any field $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$79 and any parameters $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$80 with $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$81, the elliptic curve

$$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$82

admits a natural injection, functorial in $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$83,

$$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$84

where

$$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$85

Over $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$86, this extends to an injection

$$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$87

If $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$88, then all orbits in the image have representatives in $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$89, so they are integral up to uniformly bounded denominators (Laga et al., 13 Aug 2025).

The Lie-theoretic proof combines a Kostant section $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$90 coming from a regular nilpotent in $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$91, a $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$92-subregular slice $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$93, and an identification over $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$94 of the finite group scheme $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$95 with the $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$96-torsion $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$97 of the elliptic curve family. The comparison passes through the Picard groups of the associated elliptic surfaces, where $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$98 identifies with the $$\mathfrak h=\bigoplus_{i\in \mathbb{Z}/m\mathbb{Z}}\mathfrak h_i$$99 root lattice and $V=\mathfrak h_1$00-coinvariants yield $V=\mathfrak h_1$01. Applying the Bhargava–Gross cohomological description of orbits then produces the injection from $V=\mathfrak h_1$02 and, over $V=\mathfrak h_1$03, from $V=\mathfrak h_1$04 (Laga et al., 13 Aug 2025).