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Modular forms for chromatic homotopy: Supersingular congruences

This paper proves Larson's conjecture for all primes $p \ge 5$ by establishing a sharp criterion for when modular forms associated with the divided beta family in the Adams-Novikov spectral sequence are pure powers of the discriminant, a result derived from a geometric property of supersingular points on modular curves.

Original authors: Ken Ono

Published 2026-07-07

📖 5 min read🧠 Deep dive

CC0 1.0

Original authors: Ken Ono

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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 the universe of mathematics as a giant, intricate city. In one district, you have Topology, the study of shapes and how they can be stretched or twisted without tearing. In another district, you have Number Theory, the study of whole numbers and their hidden patterns. For a long time, these two districts seemed very far apart.

This paper, written by mathematician Ken Ono, is like a new bridge connecting these two districts. Specifically, it solves a puzzle about how certain "shapes" (from topology) can be described using "numbers" (from modular forms).

Here is the story of the paper, broken down into simple concepts and analogies.

1. The Big Picture: The "Chromatic" City

Think of the "Stable Homotopy Groups of Spheres" (a very complex topological object) as a massive, multi-layered skyscraper. Mathematicians use a tool called the Adams-Novikov Spectral Sequence (ANSS) to try to map out the floors of this building.

The Problem: Some floors of this building are hard to see. They are made of "divided beta elements."
The Solution: A mathematician named Mark Behrens figured out that you can describe these hidden floors using Modular Forms. Think of modular forms as a special kind of "blueprint" or "recipe" that generates numbers.
The Goal: We want to know: Can we describe these specific blueprints using a very simple, pure ingredient? The "pure ingredient" in this story is a famous formula called $\Delta$ (Delta).

2. The Conjecture: The "Pure Power" Rule

Mathematician Donald Larson made a bold guess (a conjecture) about when these complex blueprints can be replaced by a simple, pure power of the Delta formula.

Imagine you are baking a cake. You have a complex, fancy recipe (the Behrens form). Larson guessed:

Rule: You can swap the fancy recipe for a simple "Delta Cake" only if you are baking a specific size of cake (defined by a number $j$ ).
The Limit: If the cake size is small enough ( $1 \le j \le p^n$ ), the simple Delta Cake works perfectly.
The Boundary: If the cake is too big ( $j > p^n$ ), the simple Delta Cake fails. You must use the complex recipe.

For a long time, this rule was only proven for a few specific numbers (primes like 5, 7, 11). Ken Ono's paper proves that this rule works for every single prime number 5 or larger.

3. The Detective Work: Supersingular Points

How did Ono prove this? He didn't just crunch numbers; he went on a geometric treasure hunt.

To understand the "Delta Cake," Ono had to look at Modular Curves. Imagine these curves as a landscape with hills and valleys.

The Special Spots: On this landscape, there are special "Supersingular" spots. These are like unique, magical islands where the rules of geometry behave differently.
The Test: Ono needed to check what happens to the Delta formula when you stand on these magical islands.

He discovered a beautiful symmetry:

If you stand on any supersingular island and look at the ratio of two specific Delta values, the result is a "root of unity."

The Analogy: Imagine standing on a magical island and spinning a compass. No matter which island you are on, the compass always points to a direction that, if you spin it a specific number of times (determined by the prime number), brings you exactly back to "North" (1).

This geometric fact (Theorem 1.2) is the key. It proves that the "Delta Cake" works perfectly up to the limit Larson predicted, and fails exactly when you cross that limit.

4. The "Deuring" Models: The Perfect Mirror

To prove the compass always points North, Ono used a clever trick involving Deuring Models.

Imagine you have a weird, twisted shape (an elliptic curve) that is hard to study.
Ono showed that for every one of these weird shapes, there is a "perfect mirror" version (a Deuring model) defined over a specific number field.
In this mirror world, the math becomes incredibly clean. The "discriminant" (a value that measures the shape's complexity) behaves in a very predictable way: it becomes a perfect power of a simple number.

This allowed him to calculate the "compass direction" on the magical islands with absolute certainty.

5. The Conclusion: The Bridge is Built

The paper concludes by tying the geometry back to the topology:

The Geometry: The "compass" (the value of the modular function) always points to 1 when raised to the right power on supersingular islands.
The Arithmetic: This means the simple "Delta Cake" satisfies all the necessary conditions to represent the complex topological object, but only up to the specific size limit Larson guessed.
The Result: The boundary is sharp. If you try to use the simple Delta formula for a "cake" that is too big, the math breaks down (the conditions fail).

Summary in One Sentence

Ken Ono proved that a specific, simple mathematical formula (a power of Delta) can perfectly represent complex topological shapes, but only if those shapes are within a precise size limit; crossing that limit requires a more complex formula, a fact confirmed by studying the geometry of "supersingular" points on modular curves.

What this paper does NOT do:

It does not apply this to physics, engineering, or medicine.
It does not predict future discoveries in other fields.
It is purely a proof within the abstract world of pure mathematics, connecting two deep areas of the field.

Technical Summary: Modular Forms for Chromatic Homotopy: Supersingular Congruences

Problem Statement
This paper addresses a conjecture by Donald Larson within the framework of Behrens' program, which seeks to describe the divided beta family in the Adams–Novikov spectral sequence (ANSS) for the stable homotopy groups of spheres using modular forms modulo $p$ . Specifically, the problem concerns the representation of the Behrens forms $f_{i/j}$ , which are modular forms of weight $12t$ (where $t = i(p^2-1)/12$ ) attached to order $p$ elements $\beta_{i/j}$ in the ANSS.

Larson conjectured that for a prime $p \ge 5$ , writing $i = r p^n$ with $(r, p) = 1$ , the Behrens form $f_{i/j}$ can be represented by the pure power of the discriminant modular form, $\Delta^t$ , if and only if the index $j$ satisfies $1 \le j \le p^n$ . For admissible indices $j > p^n$ , no representative of $f_{i/j}$ equals $\Delta^t$ . The paper aims to prove this conjecture for all primes $p \ge 5$ , establishing a sharp criterion for when these modular forms are pure powers of $\Delta$ .

Methodology
The proof reduces the topological problem of identifying divided beta elements to a geometric problem concerning the behavior of modular forms on modular curves $X_0(\ell)$ and their values at supersingular points. The methodology proceeds through the following steps:

Reduction to Congruence Conditions: The paper utilizes Behrens' four defining conditions (C1–C4) for the forms $f_{i/j}$ . Conditions (C1)–(C3) relate to the non-vanishing and $q$ -order of the form, while condition (C4) imposes a divisibility constraint involving the Hasse invariant $A_p$ and the operator $L_\ell = V_\ell - \iota$ (where $V_\ell$ is the $U_\ell$ -operator acting on $q$ -expansions and $\iota$ is the inclusion of level one forms). Specifically, (C4) requires $A_p^j \mid L_\ell(\bar{f}_{i/j})$ in the ring of modular forms modulo $p$ on $\Gamma_0(\ell)$ .
Level 2 Analysis: The author performs an explicit calculation at level $\ell=2$ . Using Larson's generators for the ring of modular forms on $\Gamma_0(2)$ , the operator $L_2(\Delta^t)$ is factored. The divisibility of this expression by powers of the Hasse invariant $A_p$ is analyzed by comparing the zero loci of $L_2(\Delta^t)$ with the supersingular points on $X_0(2)$ . This analysis reveals that the divisibility holds if and only if $j \le p^n$ , provided the supersingular points satisfy a specific root-of-unity condition.
Supersingular Containment Theorem: To extend the result from level 2 to all levels $\ell \neq p$ , the paper proves a geometric theorem regarding the values of the modular function $R_\ell = V_\ell(\Delta)/\Delta$ at supersingular points. The core claim is that for any supersingular point $\tilde{P}$ on $X_0(\ell)$ over $\mathbb{F}_p$ , the value $R_\ell(\tilde{P})$ is an $m$ -th root of unity, where $m = (p^2-1)/12$ .
Rationality of Supersingular Discriminants: The proof of the containment theorem relies on a new result concerning the rationality of discriminants of supersingular elliptic curves. By utilizing Deuring models (supersingular curves defined over $\mathbb{F}_{p^2}$ with Frobenius endomorphism $[-p]$ ) and eta powers at levels 3 and 4, the author demonstrates that the quantity $\kappa(E) = \Delta(E, \omega)^{(p^2-1)/12}$ is independent of the curve $E$ and the differential $\omega$ , taking the value $(-1)^{(p+1)/2}$ . This allows the ratio of discriminants along an isogeny to be shown as 1 when raised to the power $m$ .

Key Contributions and Results

Proof of Larson's Conjecture: The paper proves Larson's conjecture in full for all primes $p \ge 5$ . It establishes that the pure power $\Delta^t$ satisfies Behrens' conditions (C1)–(C4) precisely when $1 \le j \le p^n$ .
Sharpness of the Bound: The paper proves that for admissible indices $j > p^n$ (which occur when $r > 1$ and $n \ge 2$ ), the form $\Delta^t$ fails condition (C4) at level $\ell=2$ . Consequently, no representative of $f_{i/j}$ can be $\Delta^t$ in this range.
Supersingular Containment Theorem (Theorem 1.2): It is proven that for any prime $\ell \neq p$ , the value of the modular function $R_\ell$ at every supersingular point of $X_0(\ell)$ raised to the power $m = (p^2-1)/12$ equals 1.
Rationality of Supersingular Discriminants (Theorem 1.3): The paper establishes that for a supersingular elliptic curve $E$ over $\mathbb{F}_{p^2}$ with Frobenius $[-p]$ , the value $\Delta(E, \omega)^{(p^2-1)/12}$ is equal to $(-1)^{(p+1)/2}$ . This value is independent of the choice of the curve or the invariant differential.
Level 2 Factorization: The paper provides a detailed factorization of $L_2(\Delta^t)$ in terms of the linear factors corresponding to supersingular points, showing how the multiplicity of these factors relates to the $p$ -adic valuation of the exponent $t$ .

Significance
The paper completes the calculation of the range in which Behrens forms attached to the divided beta family are represented by pure powers of the discriminant. By reducing the topological congruence conditions to geometric statements about supersingular points on modular curves, the work bridges stable homotopy theory and the arithmetic geometry of elliptic curves. The proof relies entirely on arithmetic calculations on modular curves and properties of supersingular elliptic curves, with the only topological input being the existence and defining conditions of the Behrens forms. The results confirm the sharpness of the boundary $j = p^n$ for the representability of these forms, resolving a specific prediction made within the chromatic homotopy program.