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Explicit valuation of elliptic nets for elliptic curves with complex multiplication

This paper establishes an explicit valuation formula for net polynomials associated with multiples of points on elliptic curves with complex multiplication, and applies this result to demonstrate that the corresponding elliptic divisibility sequences satisfy a recurrence relation when indexed by elements of the complex multiplication order.

Original authors: Edison H L Au-Yeung

Published 2026-06-29

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Edison H L Au-Yeung

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 Big Picture: A New Way to Count Points

Imagine you have a magical, curved surface called an Elliptic Curve. On this surface, you can pick a starting point, let's call it P. You can then perform a special kind of "jumping" operation: you can jump to a new point by adding P to itself (2P), then to that point again (3P), and so on. This creates a sequence of points: P, 2P, 3P, 4P...

Mathematicians have long been interested in the "coordinates" of these points. Specifically, they look at the denominators (the bottom numbers) of the fractions that describe where these points land. These denominators form a pattern called an Elliptic Divisibility Sequence (EDS).

Usually, we count these jumps using whole numbers (1, 2, 3...). However, this paper deals with a special type of curve that has a hidden symmetry called Complex Multiplication (CM). Because of this symmetry, you don't just jump by whole numbers; you can jump using "complex" steps (like $1 + \omega$ , where $\omega$ is a special imaginary number). This turns the sequence from a simple line into a multi-dimensional grid.

The author, Edison Au-Yeung, wants to answer a very specific question: When we take these complex jumps, how do the denominators behave? Specifically, he wants to calculate exactly how many "zeros" appear in the bottom of the fraction when we look at them through a specific mathematical lens (called a "valuation").

The Problem: The "Cancellation" Mystery

When you calculate the coordinates of a point like $nP$, the top number (numerator) and the bottom number (denominator) often share common factors. Think of it like a fraction $\frac{6}{8}$ . Before you look at it, it's messy. But if you simplify it to $\frac{3}{4}$ , you've "cancelled" the common factor of 2.

In the world of elliptic curves, this cancellation is tricky. Sometimes the numerator and denominator share a huge common factor that disappears, making the point look much "simpler" than it seemed. Mathematicians call this the cancellation exponent.

For a long time, mathematicians knew how to calculate this cancellation for simple, whole-number jumps (like 5P or 10P). But for these special "Complex Multiplication" curves, where the jumps are complex numbers (like $3 + 2\omega$ ), no one had a clear formula. It was like trying to predict the weather in a new dimension without a map.

The Solution: A New Map (Elliptic Nets)

The author introduces a tool called an Elliptic Net.

The Analogy: Imagine a standard EDS is a single string of beads. An Elliptic Net is like a fishing net made of many strings woven together. Each intersection of the net represents a point on the curve created by a combination of different jumps.
The Innovation: Instead of trying to force the old formulas to work, the author uses these "nets" to map out the behavior of the points. He treats the problem not just as a sequence of numbers, but as a geometric shape in a higher dimension.

The Main Discovery: The "Height" Formula

The core of the paper is a new formula that predicts the "cancellation exponent" (how much simplification happens) for these complex jumps.

The author uses a concept called Local Height, which is like measuring the "altitude" or "energy" of a point on the curve.

Good Reduction (Smooth Terrain): If the point lands on a smooth part of the curve, the formula is straightforward. The cancellation depends on the "altitude" of the starting point and its neighbors.
Bad Reduction (Rough Terrain): If the point lands on a "singular" or rough spot (where the curve has a sharp corner or cusp), the math gets messy. The author proves that even here, the cancellation follows a predictable pattern, but it depends on a specific "annihilator" (a special number that, when used to jump, brings you back to a safe, smooth spot).

The Result: The paper provides a precise recipe. If you know the starting point and the type of curve, you can plug in your complex jump number ( $z$ ) and instantly calculate exactly how much the denominator will simplify. The formula looks like a quadratic equation (a parabola), meaning the cancellation grows in a predictable, curved way as your jumps get larger.

The Application: A New Recurrence Rule

Once the author has this formula, he applies it to see if the sequence of denominators still follows a "rule of the road" (a recurrence relation).

The Old Rule: In simple cases, if you have three points in a sequence, their denominators follow a specific multiplication rule (like a family tree where the children are determined by the parents).
The New Rule: The author shows that for these complex curves, the rule mostly still holds, but with a catch.
- If you pick your jumps carefully (specifically, if they belong to a certain "ideal" or group of numbers related to the curve's rough spots), the rule works perfectly.
- If you pick random jumps, the rule breaks down slightly.

Why this matters (according to the paper): This allows mathematicians to count how many points on the curve have "nice" coordinates (integral points). By understanding the cancellation, they can set better limits on how many such points exist. The author suggests this could help solve problems about counting points on curves that are currently too difficult to tackle.

Summary in a Nutshell

The Setup: We have special curves with extra symmetry (Complex Multiplication).
The Challenge: We need to know how the "bottom numbers" of these points simplify when we make complex jumps.
The Tool: The author uses "Elliptic Nets" (multi-dimensional grids) instead of simple lists.
The Breakthrough: He derives a precise formula that predicts exactly how much simplification happens, whether the curve is smooth or rough at that point.
The Payoff: This formula allows mathematicians to create a new set of rules for these sequences, which can be used to count and understand the points on these complex curves more accurately.

The paper is essentially a manual for navigating the "rough terrain" of these special mathematical curves, providing a compass (the formula) that tells you exactly how the landscape changes as you travel further out.

Technical Summary: Explicit Valuation of Elliptic Nets for Elliptic Curves with Complex Multiplication

Problem Statement
The paper addresses the explicit valuation of "division polynomials" associated with elliptic curves $E/K$ that possess Complex Multiplication (CM). In the classical setting (without CM), the coordinates of multiples $nP$ of a point $P$ are expressed via division polynomials $\phi_n$ and $\psi_n$ . The "cancellation exponent," defined as $g_{n,\nu}(P) = \min(2\nu(\psi_n(P)), \nu(\phi_n(P)))$ for a discrete valuation $\nu$ , is crucial for understanding the arithmetic of these points, particularly in relation to Siegel's Theorem on integral points and the study of Elliptic Divisibility Sequences (EDS).

Existing results, such as those by Cheon and Hahn, provide formulas for this exponent when the index $n$ is an integer. However, when $E$ has CM by an order $\mathcal{O} = \mathbb{Z}[\omega]$ in a quadratic imaginary field, the natural indices for multiples are elements of $\mathcal{O}$ rather than just $\mathbb{Z}$ . The paper seeks to derive an explicit formula for the common valuation $g_{z,\nu}(P)$ where $z \in \mathbb{Z}[\omega]$ , extending the integer-indexed theory to the CM setting. Furthermore, the paper investigates whether the resulting sequences of denominator ideals satisfy recurrence relations analogous to the classical EDS recurrence.

Methodology
The author departs from the polynomial-based strategies used in previous works (e.g., Cheon and Hahn), which rely heavily on the degrees and recursive structures of division polynomials. Instead, the methodology employs elliptic nets and the Néron local height function.

Elliptic Nets: The paper utilizes elliptic nets, which are higher-dimensional analogues of EDS, to define the CM division polynomials $\psi_z$ and $\phi_z$ . These are rational functions associated with the linear combination of points $[z]P$ .
Néron Local Height: The core of the proof relies on the normalized Néron local height function $\tilde{\lambda}$ . The author establishes that for a point $P$ and an index $\alpha = a + b\omega$ , the local height of $[\alpha]P$ is a quadratic function in $a$ and $b$ .
Filtration and Reduction: The analysis distinguishes between points with good reduction (non-singular modulo $\nu$ $ν$ ) and bad reduction (singular modulo $\nu$ $ν$ ).
- For good reduction, the valuation is derived directly from the properties of the local height and the structure of the denominator ideals.
- For bad reduction, the paper utilizes a subgroup filtration $E_n(K_\nu)$ of points with non-singular reduction. By analyzing the annihilator of $P$ in the quotient module $E(K)/E_0(K)$ , the author derives explicit formulas for the valuation based on the behavior of the point under the action of the order $\mathbb{Z}[\omega]$ .
Inductive Proofs: The derivation of the main valuation formula involves inductive arguments on the coefficients of the index $z = a + b\omega$ , utilizing the quasi-parallelogram law of the local height function and specific recurrence relations of elliptic nets.

Key Contributions and Results

Explicit Valuation Formula (Theorem 1.5): The paper provides a closed-form formula for the common valuation $g_{z,\nu}(P)$ for $z \in \mathbb{Z}[\omega]$ .
- Case 1 (Non-singular reduction): If $P$ is non-singular modulo $\nu$ , the valuation is given by a quadratic form involving the valuations of the $x$ -coordinates of $P$ , $[\omega]P$ , and $[1+\omega]P$ .
- Case 2 (Singular reduction): If $P$ is singular modulo $\nu$ , the formula depends on the choice of an annihilator element $r$ in the module $E(K)/E_0(K)$ . The valuation is expressed as a linear combination of three base values: $\mu(r)$ , $\mu_\omega(r)$ , and $\mu_{1+\omega}(r)$ , which correspond to the valuations at $[r]P$ , $[r\omega]P$ , and $[r(1+\omega)]P$ . The coefficients of this linear combination are quadratic polynomials in the components of $z$ .
Recurrence Relations for Denominator Ideals (Theorem 1.8 and 1.9): The paper extends results by Verzobio regarding recurrence relations for EDS to the CM setting.
- It is shown that the sequence of denominator ideals $\{D_{[\alpha]P}\}_{\alpha \in \mathbb{Z}[\omega]}$ does not generally satisfy the standard EDS recurrence relation (1.3) when $P$ has bad reduction.
- However, by introducing a sequence of fractional ideals $\{g_\alpha\}$ derived from the cancellation exponents, the paper proves that a modified recurrence relation holds. Specifically, if at least two of the indices $\alpha, \beta, \gamma$ belong to a specific ideal $M(P)$ (related to the annihilators of $P$ at primes of bad reduction), then the sequence of generators $B_\alpha$ satisfies a recurrence relation analogous to (1.3).
Correction of Previous Work: The author notes that while the final result of Cheon and Hahn [4] regarding the integer case is correct, their proof contained a gap regarding the valuation of points in the filtration $E_0(K)$ . This paper's approach using the Néron local height function avoids this gap and provides a more flexible framework.

Significance and Claims
The paper claims to provide the first explicit valuation formula for CM division polynomials indexed by elements of an order in a quadratic imaginary field. By shifting the focus from polynomial properties to the Néron local height and elliptic nets, the argument is presented as more flexible and applicable to problems involving linear combinations of points on elliptic curves.

The primary significance lies in the potential application to counting problems. The author suggests that these results could be used to bound the number of integral points (or points with specific denominator properties) in linear combinations of the form $[n]P + [m]\omega P$ . While the paper does not prove a new bound on integral points itself, it establishes the necessary valuation machinery to generalize Ingram's and Stange's work on bounding integral points to the CM setting and to linear combinations of points. The recurrence relations derived for the denominator ideals are presented as a prototype for a "linear combination version" of the integral point problem.