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Exact Stratification and Affine Mass Formulas for Split Richelot Data over Finite Fields

This paper establishes the exact stratification of the input space for split Richelot isogenies over finite fields of odd characteristic, deriving closed mass formulas and a complete classification of stabilizers that enable efficient, provably redundant post-checks for determining the field of definition and Weil polynomial shape of the resulting Jacobians.

Original authors: Hung T. Dang, Diep V. Nguyen

Published 2026-08-06

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Hung T. Dang, Diep V. Nguyen

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Secret Life of Math Curves

Imagine you are a master architect designing a bridge. In the world of mathematics, specifically a field called number theory, these "bridges" are often curves drawn on a grid made of numbers. But instead of an endless, smooth grid like the one on your computer screen, these architects build on a "finite field." Think of this as a grid where the numbers wrap around like a clock; once you pass a certain point, you start over at zero. It's a tiny, self-contained universe of math.

One of the most famous tools for moving between these universes is called a Richelot isogeny. You can think of this as a special kind of elevator or a teleportation device. It takes a complex, two-humped curve (called a genus-2 curve) and transforms it into a new, equally complex curve. This isn't just a drawing exercise; these transformations are the backbone of modern cryptography, the secret codes that protect your online banking and private messages. To build these codes securely, mathematicians need to know exactly how many different ways this teleportation can happen and what the destination looks like.

However, there's a catch. Sometimes, when you try to use this elevator, it doesn't go to a new, complex curve at all. Instead, it crashes into a simpler destination: a pair of single-humped curves (elliptic curves) stuck together. For a long time, mathematicians knew this could happen, but they didn't have a precise map of when it happens, how often, or what the rules are for the crash landing. They were flying blind, hoping the elevator worked, and if it didn't, they just threw the result away and tried again. This paper is about finally drawing that map, figuring out the exact odds of the crash, and realizing that the "crash" isn't a failure at all—it's just a different, very interesting kind of destination.

The Map of the Math Universe

The authors of this paper, Hung T. Dang and Diep V. Nguyen, have done something remarkable: they have created a perfect, exact map of the "input space" for this Richelot elevator. They didn't just guess or run a computer simulation to see what might happen; they used pure logic and geometry to prove exactly how many inputs lead to which outcome for any odd-sized finite field.

To understand their discovery, imagine you have three special quadratic equations (think of them as three unique, two-piece puzzle pieces). You want to combine them to build a bridge. The authors discovered that the way these three pieces fit together depends entirely on a hidden geometric game played on a flat plane.

The Geometry Game
They realized that every one of these quadratic pieces can be represented as a single dot on a 2D grid. There is a special curved line on this grid called the "discriminant parabola."

If a dot is off the curve, the piece is "safe" and usable.
If two dots lie on a line that just touches the curve (a tangent), the pieces share a secret root and can't be used together.
If three dots lie on a vertical line, the pieces are "aligned" in a specific way.
If the three dots lie on any straight line (collinear), something magical happens: the elevator doesn't go to a new complex curve; it splits into two simpler ones.

The paper proves that the entire universe of possible inputs is divided into three distinct "strata" or layers based on these geometric rules:

The Hexagon Layer (G6): The three dots are scattered such that no special lines connect them. The elevator works perfectly, creating a new, complex six-degree curve. This is the most common outcome.
The Pentagon Layer (G5): Two dots are aligned vertically. The elevator still works, but the new curve is slightly simpler (a five-degree curve). This happens about 3 times out of every $q$ attempts (where $q$ is the size of the number field).
The Split Layer (D): The three dots lie on a straight line. The elevator "crashes" into a product of two elliptic curves. This happens about 1 time out of every $q$ attempts.

The "Crash" is a Feature, Not a Bug
The most exciting part of the paper is how they handle the "Split Layer" (where $d=0$ ). In the past, if a mathematician saw this happening, they would say, "Oh no, the input is broken," and throw it away. This paper says, "Wait! This isn't broken; it's a different kind of treasure."

The authors found that when the three dots are collinear, the resulting split isn't random. It depends on a specific "square class" (a property related to whether a number is a perfect square in that field).

Case A (Square): The two resulting curves are defined right there in the original number field. They are stable and ready to use.
Case B (Non-Square): The two resulting curves are "twins" that live in a slightly larger number field. They swap places if you look at them through the lens of the original field.

Crucially, the paper proves that in Case B, the original curve has exactly $q + 1$ points. This is a massive shortcut! Instead of doing a hard calculation to count the points on a complex curve, you can just check if the input is in this "Split Layer" with a non-square property, and you instantly know the answer is $q + 1$ .

No More Guessing or Retrying
The paper also tackles a practical problem: how do you know before you do the heavy lifting whether you are in the Hexagon, Pentagon, or Split layer?
Previously, people would run the full calculation, check the result, and if it looked weird, they would try a different coordinate system (a "retry") or check the output again. The authors prove that all of this checking is unnecessary.

They show that the data needed to classify the outcome (the "certificate") is already being calculated during the standard process. It's like checking the weather while you are already driving; you don't need to stop the car to look at the sky. They provide a simple, fast formula (costing only 5 multiplications and 6 squarings) that tells you exactly which layer you are in. If you are in the Split layer, you get a bonus: a description of the two simpler curves. If you are in the Hexagon or Pentagon layer, you know the output is valid without needing to double-check it.

The Final Count
The authors didn't stop at just saying "it happens." They gave the exact number of ways this can happen for any field size $q$ .

The total number of valid inputs is a complex polynomial: $(q-2)^2(q^2 - q + 4)$ .
The number of "Split" inputs is $(q-2)(q^2 - 3q + 3)$ .
They even counted how many of these inputs are "symmetric" (meaning they look the same if you flip the coordinates), finding that only a tiny fraction have this special symmetry, and they all belong to the Split layer.

In short, this paper turns a messy, uncertain process into a precise, predictable machine. It tells us exactly how often the Richelot elevator splits, exactly what the split looks like, and proves that we never need to throw away a result or run a second check. The "failure" of the elevator is actually a guaranteed path to a simpler, well-understood destination, and we now have the exact map to find it.

Technical Summary: Exact Stratification and Affine Mass Formulas for Split Richelot Data over Finite Fields

Problem Statement
The Richelot $(2, 2)$ -isogeny is the fundamental construction for explicit genus-2 isogeny computation. It operates on a domain curve $C: y^2 = f(x)$ , where $f$ is a square-free sextic presented as a product of three monic quadratics, $f = uvw$. The standard step computes three "brackets" $U, V, W$ derived from the derivatives of these factors and returns a quotient curve. However, the classical construction does not fully describe the distribution of inputs over the possible geometric outcomes of this step, nor does it characterize the arithmetic of the quotient when the standard genus-2 output fails.

Specifically, three outcomes are possible:

Genus-2 Output (Degree 6): The product $UVW$ is square-free of degree 6.
Genus-2 Output (Degree 5): The product $UVW$ is square-free of degree 5 (occurring when two factors are "aligned," i.e., share a middle coefficient).
Decomposable Output: The coefficient determinant $d$ vanishes, causing the three quadratics to span a pencil. In this case, the quotient is not a genus-2 curve but a product of elliptic curves.

The paper addresses the lack of precise counting formulas for these strata over finite fields $\mathbb{F}_q$ (odd characteristic) and the absence of a complete arithmetic classification for the decomposable case, particularly regarding the field of definition of the resulting elliptic factors.

Methodology
The authors employ a geometric dictionary to translate algebraic conditions of the Richelot step into incidence geometry within the affine plane $\mathbb{A}^2(\mathbb{F}_q)$ .

Geometric Identification: A monic quadratic $x^2 + ax + b$ is identified with a point $(a, b)$ . The discriminant condition (separability) corresponds to avoiding the discriminant parabola $\Pi: b = a^2/4$ .
Incidence Dictionary:
- Two quadratics share a root (non-coprime) if and only if their points lie on a common tangent to $\Pi$ .
- Two quadratics are "aligned" (bracket degree drops) if and only if their points lie on a vertical line.
- The determinant $d=0$ (decomposable case) occurs if and only if the three points are collinear.
Combinatorial Counting: The stratification problem is reduced to counting incidences of lines (tangent, vertical, secant, external) with the set of separable points. This allows for the derivation of exact closed-form formulas for the cardinality of the admissible locus and its strata.
Affine Groupoid Mass: To account for coordinate freedom, the authors quotient by the affine group $\text{Aff}(\mathbb{F}_q)$ acting on the displayed splittings. They compute the "affine groupoid mass," defined as the weighted sum of orbits $\sum 1/\#\text{Stab}(O)$ , which yields polynomials of degree four in $q$ .
Certificate Construction: The paper constructs a runtime certificate using the Bezoutian minors of the input pairs. These minors, already computed for the Richelot brackets, determine the discriminants, resultants, and the branch determinant $d$ .

Key Contributions and Results

Exact Stratification Formulas:
The paper proves exact closed formulas for the number of ordered triples $(u, v, w)$ in each stratum for any odd prime power $q$ :
- Total admissible locus $A_q$ .
- Decomposable stratum $D_q$ (where $d=0$ ).
- Genus-2 strata $G_{5,q}$ (degree 5 output) and $G_{6,q}$ (degree 6 output).
  The densities of the non-generic strata are shown to be $1/q + O(q^{-3})$ for the decomposable case and $3/q + O(q^{-2})$ for the quintic case.
Affine Mass Formulas and Stabilizer Classification:
The authors provide the affine groupoid masses for each stratum, which are degree-four polynomials in $q$ . They classify all stabilizers of the affine action, showing that non-trivial stabilizers (involutions $x \mapsto -x+b$ ) occur exclusively within the decomposable stratum $D$ , specifically on the sublocus where the three quadratics share a middle coefficient.
Runtime Certificate and Redundancy Proofs:
- Algorithm 1: A certified Richelot step is presented that classifies inputs into Invalid, Genus2, or Decomposable using only data computed during the standard bracket assembly.
- Cost: The classification adds a cost of $5M + 6S$ (multiplications and squarings) beyond the bracket computation, or just $2M$ if the input is already certified.
- Redundancy: The paper proves that an output post-check (verifying the degree and coprimality of $U, V, W$ ) and an affine retry (shifting coordinates to fix a failure) are provably redundant. The certificate guarantees the output properties if the input satisfies the discriminant and resultant checks.
Arithmetic of the Decomposable Stratum:
For the case $d=0$ , the paper provides a complete arithmetic classification:
- Splitting Class: The square class of the resultant of any pair of factors is invariant and determines the geometry of the line joining the three parameter points.
- Two Cases:
  - Square Splitting Class: The elliptic factors are defined over $\mathbb{F}_q$ .
  - Non-Square Splitting Class: The elliptic factors are defined over $\mathbb{F}_{q^2}$ and exchanged by the Galois group. In this case, the Weil polynomial of the Jacobian is even, and the point count satisfies the identity $\#C(\mathbb{F}_q) = q + 1$ .
- Involution: The paper explicitly constructs the extra involution of the curve $C$ that lifts the deck transformation of the pencil, proving that $d=0$ corresponds to the existence of this extra symmetry.

Significance and Claims
The paper claims that its results are new in the following areas:

The exact counting theorems for the stratification of Richelot inputs over finite fields.
The affine groupoid mass analysis with a complete classification of stabilizers.
The identification of the "splitting class" as a line invariant, leading to a refined count of the decomposable stratum and a determination of the field of definition for the elliptic factors.
The proof that the decomposable branch is not a failure state but a distinct arithmetic case with specific properties (e.g., the point count identity).

The authors emphasize that the classification is a by-product of the step's existing computations, requiring no additional passes over the data. The results are verified by exhaustive enumeration over small finite fields (up to $q=49$ ) and symbolic verification over the integers. The paper explicitly notes that it does not claim to classify sextic models before a splitting is supplied, nor does it resolve the stratification for Galois-permuted kernels where factors are not individually rational. It also notes that the characteristic 2 case requires a separate classification due to the vanishing of middle coefficients in the brackets.

The Secret Life of Math Curves

The Map of the Math Universe

Technical Summary: Exact Stratification and Affine Mass Formulas for Split Richelot Data over Finite Fields

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