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Divisibility of the coefficients of modular polynomials

This paper investigates the high divisibility of the coefficients of modular polynomials $\Phi_N(X+J, Y+J)$ by small primes when $J$ is an algebraic number (such as 0 or singular moduli) that has supersingular reduction at those primes.

Original authors: Florian Breuer

Published 2026-06-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Florian Breuer

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

Imagine you have a giant, incredibly complex recipe book called Modular Polynomials. This book doesn't contain recipes for cakes or soups; instead, it contains the mathematical "blueprints" that describe how different types of elliptic curves (a special kind of shape used in advanced math and cryptography) are connected to one another.

Specifically, there is a famous book in this library called $\Phi_N$ . It lists every possible pair of these curves that can be linked by a specific type of bridge called a "cyclic isogeny" of size $N$ .

The Problem: The Numbers are Messy

The numbers (coefficients) inside this book are notoriously huge. If you tried to write them out, they would fill libraries. However, the author, Florian Breuer, noticed something strange: despite being huge, these numbers are also extremely divisible by small prime numbers like 2, 3, and 5.

Think of these numbers like massive, heavy boulders. You might expect them to be solid rock, but Breuer discovered they are actually made of layers of soft clay. If you try to divide them by a small prime (like 2 or 3), they crumble apart easily, revealing many layers of that prime factor.

The Discovery: Shifting the Perspective

The paper's main trick is to look at these numbers not in their original form, but after we "shift" them. Imagine taking the blueprint and sliding the entire grid over by a certain amount (mathematically, replacing $X$ with $X+J$ ).

The author asks: If we shift the blueprint by a specific, special number $J$ (called a "singular modulus"), how many layers of divisibility do we find?

He found that if you shift the blueprint by these special numbers, the resulting coefficients become even more divisible by small primes. In fact, the amount of divisibility depends on how "special" the prime number is in relation to the curve.

The Analogy: The "Supersingular" Filter

To understand why this happens, imagine the elliptic curves as different types of gears.

Ordinary Gears: Most gears work normally.
Supersingular Gears: These are rare, special gears that behave differently under specific conditions (specifically, when viewed through the lens of a prime number $p$ ).

The paper proves that when you shift the blueprint by a special number $J$ , and you look at it through the lens of a prime $p$ where the gear becomes "supersingular," the numbers in the blueprint become incredibly "sticky" with factors of $p$ .

It's as if the blueprint has a hidden magnetic property. When you align it with a supersingular gear, the magnetic field (the prime $p$ ) pulls the numbers apart, revealing that they are actually composed of many, many smaller pieces of that same prime.

The Main Results in Plain English

The "Zero" Shift ( $J=0$ ):
If you shift the blueprint by zero (looking at the original numbers), the author proves that if a prime $p$ doesn't divide $N$ , the numbers are divisible by $p$ a certain minimum number of times.
- For the prime 2, the numbers are divisible by 2 at least 15 times more than you might expect based on their size.
- For the prime 3, they are divisible by 3 at least 3 times more.
- For larger primes, the rule changes slightly but the pattern holds: the "further" you are from the edge of the polynomial, the more divisible the numbers are.
The "Special" Shifts ( $J$ is a Singular Modulus):
The author goes further. He looks at 13 specific, famous numbers (like $J=0$ , $J=1728$ , etc.) that correspond to curves with "Complex Multiplication" (curves with extra symmetry).
- When you shift the blueprint by one of these special numbers, the divisibility rules get even stronger.
- The paper provides a "cheat sheet" (Table 1) that tells you exactly how many layers of divisibility you will find for each of these 13 special numbers and various primes.

Why Does This Matter? (According to the Paper)

The paper mentions two main practical uses for this discovery:

Saving Space: Since we now know exactly how many factors of small primes are guaranteed to be in these numbers, we don't need to store those factors in our computer files. We can just store the "leftover" part.
- Example: For a specific case ( $N=5$ ), this trick reduced the storage space needed by 43%. For larger numbers, the savings are smaller (around 12%), but it still helps.
Quality Control: When mathematicians use computers to calculate these massive polynomials, the results can be wrong due to errors. This paper provides a "sanity check." If a computer calculates a coefficient and it doesn't have the required number of factors of 2, 3, or 5, the mathematician knows immediately that the calculation is wrong.

Summary

Florian Breuer's paper is like finding a hidden pattern in a chaotic mess of giant numbers. He discovered that if you rearrange these numbers slightly (using special shifts), they reveal a deep, predictable structure: they are built from layers of small prime numbers. This discovery helps mathematicians store these numbers more efficiently and check their work more quickly, ensuring the "blueprints" of these mathematical shapes are accurate.

Technical Summary: Divisibility of the Coefficients of Modular Polynomials

Problem Statement
The paper investigates the arithmetic properties of the classical modular polynomial $\Phi_N(X, Y) \in \mathbb{Z}[X, Y]$ , which vanishes at pairs of $j$ -invariants of elliptic curves linked by a cyclic isogeny of degree $N$ . While the coefficients of $\Phi_N(X, Y)$ are known to be extremely large, they exhibit significant divisibility by small primes. The central problem addressed is establishing precise lower bounds on the $p$ -adic valuations of these coefficients, particularly when the polynomial is shifted by singular moduli (algebraic numbers $J$ corresponding to elliptic curves with Complex Multiplication). Specifically, the author studies the coefficients of $\Phi_N(X+J, Y+J)$ where $J$ is a rational singular modulus, focusing on the case $J=0$ and other values where the underlying elliptic curve has supersingular reduction at a prime $p$ .

Methodology
The author employs a combination of deformation theory, explicit isogeny formulas, and polynomial interpolation techniques:

Deformation Theory and Supersingular Reduction: The core strategy involves constructing families of elliptic curves $E_k$ over a complete valued field $K$ that reduce to a fixed curve $E$ modulo a high power of the uniformizer $\pi$ . By Proposition 3.6, utilizing the Serre-Tate lifting theorem, the author constructs infinitely many such curves $E_k$ that are pairwise non-isomorphic modulo $\pi^{M+1}$ but isomorphic modulo $\pi^M$ . This allows for the control of the valuations of differences in $j$ -invariants, $v(j(E_k) - j(E_l))$ .
Valuation Analysis of Isogenies: The paper analyzes the reduction of cyclic $N$ -isogenies. When the reduced curve is supersingular, the endomorphism ring is a maximal order in a quaternion algebra. The number of cyclic isogenies reducing to endomorphisms, denoted $C_J(N, \pi)$ , is explicitly calculated using theta series associated with these quaternion orders (Proposition 2.3).
Polynomial Interpolation: To transfer valuation bounds from specific evaluations of the polynomial to its coefficients, the author utilizes a generalized interpolation lemma (Lemma 3.3). By evaluating $\Phi_N(X+J_1, Y+J_2)$ at points $y_k = j(E_k) - J_2$ with controlled valuations, and applying the symmetry of the roots, lower bounds on the coefficients $a_{i,j}$ are derived (Proposition 3.4).
Explicit Formulas (Vélu's Formulae): For the specific cases where the residue characteristic $p$ is 2 or 3, the author uses Vélu's explicit formulas for isogenies to compute the valuation of the difference $j(E') - j(E)$ directly. This leads to the definition of a constant $n_v$ (Theorem 3.2) which depends on $N \pmod 4$ (for $p=2$ ) or $N \pmod 6$ (for $p=3$ ).

Key Contributions and Results

Theorem 1.1 (The Case $J=0$ ): The paper establishes lower bounds for the $p$ -adic valuation $v_p(a_{i,j})$ of the coefficients of $\Phi_N(X, Y)$ when $N$ is not divisible by $p$ .
- For $p=2$ (if $2 \nmid N$ ): $v_2(a_{i,j}) \ge 15(\psi(N) - i - j)$ .
- For $p=3$ (if $3 \nmid N$ ): $v_3(a_{i,j}) \ge 3(\psi(N) - i - j)$ , with a stronger bound of $\lceil \frac{9}{2}(\psi(N) - i - j) \rceil$ if $N \equiv 1 \pmod 3$ .
- For $p=5$ (if $5 \nmid N$ ): $v_5(a_{i,j}) \ge 3(\psi(N) - i - j)$ .
- For $p \ge 11$ with $p \equiv 2 \pmod 3$ : $v_p(a_{i,j}) \ge 3(C_0(N, p) - i - j)$ , where $C_0(N, p)$ is the order of vanishing of $\Phi_N(X, 0)$ modulo $p$ .
- These results confirm and generalize conjectures by Wang [26].
Theorem 1.2 (General Singular Moduli): The results are extended to $\Phi_N(X+J, Y+J)$ where $J$ is a rational singular modulus (corresponding to an elliptic curve with CM by an order of class number 1).
- The bound takes the form $v_p(a_{i,j}) \ge n_p(C_J(N, p) - i - j)$ .
- The constant $n_p$ depends on the prime $p$ and whether $p$ divides $J$ or $J-1728$ . For instance, if $p|J$ and $p=2$ , $n_p=15$ ; if $p|J$ and $p=3$ , $n_p=6$ .
- The paper provides a table of "exceptional valuations" for specific discriminants (e.g., $D=-12, -27$ ) where the general formula for $n_p$ requires adjustment based on $N \pmod 3$ or $N \pmod 6$ .
Optimality: The author notes that computations suggest the derived bounds for $n_p$ are optimal, except for the specific cases $D=-12$ and $D=-27$ , where the true values are conjectured to be slightly higher for certain congruence classes of $N$ .

Significance and Applications
The paper claims significance in the following areas:

Computational Number Theory: The results provide tighter bounds on the size of individual coefficients of modular polynomials. This is directly applicable to algorithms based on the Chinese Remainder Theorem (CRT) for computing $\Phi_N(X, Y)$ , potentially reducing the number of primes required for reconstruction.
Data Storage: By storing only the factors of the coefficients not predicted by these divisibility theorems, significant space savings can be achieved. The author cites a 43% reduction in decimal digits for $N=5$ and a 12% reduction for $N=101$ .
Verification: The divisibility conditions serve as a "quick sanity check" for computations of modular polynomials for large $N$ .
Theoretical Context: The work is framed as a variation on the theme of differences of singular moduli pioneered by Gross and Zagier, extending the understanding of how supersingular reduction influences the arithmetic of modular polynomials.

The author explicitly states that the results are derived from the study of reduction types of elliptic curves and do not propose new experimental algorithms or future implications beyond the immediate computational and theoretical applications mentioned.