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Explicit constructions for Ramanujan-type congruences

This paper presents explicit constructions of modular forms to establish a unified framework for deriving both known and new Ramanujan-type congruences for a broad class of generating functions, including eta-quotients, weakly holomorphic modular forms, and mock modular forms.

Original authors: Wei Wang

Published 2026-06-23

📖 5 min read🧠 Deep dive

CC0 1.0

Original authors: Wei Wang

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 you have a giant, infinite list of numbers, like the number of ways you can break a whole number into smaller pieces (a "partition"). For over a century, mathematicians have been fascinated by patterns hidden inside these lists. Specifically, they look for Ramanujan-type congruences.

Think of a congruence like a magic trick where, if you pick numbers from your list at specific intervals (like every 5th number, or every 7th), they all magically disappear when you divide them by a certain number. They become zero. Srinivasa Ramanujan, a mathematical genius, found these tricks for the partition function long ago.

This paper by Wei Wang is like a master key or a blueprint that explains how to build these magic tricks for a huge variety of number lists, not just the one Ramanujan found.

Here is the breakdown of the paper's ideas using simple analogies:

1. The Problem: Finding the "Hidden Zero"

Imagine you have a machine that spits out numbers. You want to know: "If I look at every 5th number, will they all be divisible by 5?"

Old Way: Mathematicians used to prove these patterns existed by saying, "We know a pattern must be there because of some abstract math rules," but they couldn't always show you exactly what the pattern looked like. It was like saying, "There is a ghost in the house," without showing you the ghost.
This Paper's Way: The author says, "Let's build the ghost." He provides an explicit construction. He gives a specific recipe (a formula) to build the exact mathematical object that creates these zero patterns.

2. The Tools: The "Rankin-Cohen Bracket"

To build these patterns, the author uses a special mathematical tool called the Rankin-Cohen bracket.

The Analogy: Imagine you have two different types of musical instruments (let's call them "Function A" and "Function B"). If you play them separately, they make nice sounds. But if you use this special "bracket" tool, you can mix them together to create a new sound (a new function) that has very specific, predictable properties.
The paper shows that if you mix the right ingredients using this tool, the resulting sound always has a "silent spot" (a zero) at the specific intervals we are looking for.

3. The Three Types of Recipes

The author organizes his findings into three "flavors" or types of number lists, each requiring a slightly different recipe:

Type I (The Classic Ingredients): This deals with lists built from "Eta-quotients." Think of these as the basic building blocks of number theory, like the original partition function Ramanujan studied. The paper proves that for almost all of these basic blocks, the "magic zero" trick only works for the specific primes Ramanujan found (5, 7, 11). If you try to force this trick to work on a 13th prime, it fails. The author proves this failure explicitly by showing the "ghost" doesn't exist for those numbers.
Type II (The Stronger Mix): This involves slightly more complex lists (Weight 3/2). Here, the author shows that if you mix the ingredients correctly, you can predict how the numbers behave not just once, but in a repeating cycle. It's like a drumbeat that repeats every few steps.
Type III (The "Mock" Ingredients): This is the most modern and tricky part. Some number lists are "Mock Modular Forms."
- The Analogy: Imagine a "Mock" form is like a hologram. It looks like a solid object (a modular form) from the front, but if you try to touch it, it's not quite there. It's missing a piece. To make it solid, you have to add a "non-holomorphic" part (a shadow).
- The paper shows that even though these "holograms" are tricky, if you add the right shadow and mix them with the "bracket" tool, you can still find the hidden zero patterns.

4. The Big Payoff: Proving What Doesn't Work

One of the most exciting results isn't just finding new patterns; it's proving that some patterns don't exist.

The author uses his "blueprint" to check many different number lists. He finds that for most of them, the "magic zero" trick is impossible for large primes.
The "Partition" Example: He revisits the famous partition function. He proves that for any prime number larger than 11, the pattern of "every 13th number is divisible by 13" simply does not happen. He doesn't just guess this; he calculates the exact shape of the "ghost" and shows it's not zero.

5. Real-World Examples in the Paper

The paper doesn't just stay in theory; it applies these recipes to real, famous number problems:

Smallest Parts (spt): A function that counts the "smallest piece" in a partition. The paper recovers known patterns for this and finds new ones.
Hurwitz Class Numbers: A complex number theory concept related to shapes called quadratic forms. The paper finds new congruence rules for these numbers.

Summary

In short, this paper is a construction manual.

Before: Mathematicians knew some magic tricks existed but couldn't always see the mechanism.
Now: Wei Wang has built a machine (using the Rankin-Cohen bracket) that takes a number list, mixes it with a specific partner, and outputs a clear, visible pattern.
The Result: We can now explicitly see why certain patterns happen, and more importantly, we can definitively prove why other patterns are impossible. It turns "maybe there's a pattern" into "here is the pattern, or here is the proof it doesn't exist."

Technical Summary: Explicit Constructions for Ramanujan-Type Congruences

Problem Statement
The paper addresses the problem of establishing Ramanujan-type congruences for integer-valued sequences $\{a(n)\}_{n \ge 0}$ . Specifically, it seeks relations of the form $a(\ell n - \delta_\ell) \equiv 0 \pmod \ell$ for a prime $\ell$ and a specific shift $\delta_\ell$ . While such congruences are well-known for the partition function $p(n)$ (e.g., Ramanujan's congruences modulo 5, 7, and 11), previous approaches to proving them for broader classes of generating functions often relied on existence arguments. Techniques utilizing Serre's filtration or trace operators demonstrated that the relevant $q$ -series are congruent to modular forms of a specific weight but failed to explicitly construct these modular forms. Consequently, these methods could not easily determine whether the congruence was trivial (i.e., whether the series was identically zero modulo $\ell$ ) or provide explicit expressions for the non-zero modular forms involved.

Methodology
The author, Wei Wang, proposes a constructive approach that replaces existence proofs with explicit formulas. The core methodology relies on the Rankin-Cohen bracket, a differential operator acting on pairs of modular forms.

Rankin-Cohen Brackets: For functions $f$ and $g$ with weights $a$ and $b$ , the bracket $[f, g]_v$ is defined as a specific linear combination of derivatives $D^j f D^{v-j} g$ . The paper utilizes the property that if $f$ and $g$ are modular forms, their Rankin-Cohen bracket is also a modular form of weight $a+b+2v$ .
Explicit Construction: The paper derives a congruence identity relating the $U_\ell$ $U_{ℓ}$ -operator applied to a product of generating functions to a linear combination involving the Rankin-Cohen bracket of the generating function and an auxiliary theta series (or eta-quotient).
- The central lemma (Lemma 8) establishes that for a prime $\ell \ge 3$ , a specific combination of the product $fg$ and the bracket $[f, g]_{(\ell-1)/2}$ , when acted upon by $U_\ell$ , yields a congruence modulo $\ell$ that isolates the coefficients of interest.
Holomorphic Projection: For Type III results involving mock modular forms, the methodology extends to harmonic weak Maass forms. The author employs holomorphic projection to map the Rankin-Cohen bracket of a Maass form and a modular form back to the space of holomorphic modular forms, allowing for the extraction of explicit congruences for the holomorphic part (the mock modular form).
Non-Congruence Proofs: To prove that certain series are not congruent to zero (non-congruence results), the paper utilizes the explicit construction to compute the values of the resulting modular forms at various cusps. By showing these values are non-zero modulo $\ell$ at specific cusps (using the $q$ -expansion principle), the author demonstrates that the series cannot vanish identically.

Key Contributions and Results
The paper classifies its results into three types based on the nature of the generating function $f(\tau)$ :

Type I (Eta-quotients of weight $-1/2$ ):
- Theorem 1: Provides an explicit construction for eta-quotients $f(\tau)$ where $1/f$ is a holomorphic modular form of weight $1/2$ . The congruence is expressed in terms of the Rankin-Cohen bracket $[f, g]_{(\ell-1)/2}$ .
- Application: This framework recovers Ramanujan's congruences for $p(n)$ and provides a new proof of Ahlgren and Boylan's theorem that for $\ell \ge 13$ , $p(\ell n - \delta_\ell) \not\equiv 0 \pmod \ell$ .
- Non-Congruence: The paper proves that for most eta-quotients in Mersmann's classification of weight $1/2$ (excluding $1/\eta$ and $\eta\eta(4)/\eta(2)^3$ ), no Ramanujan-type congruences exist modulo $\ell$ for $\ell \ge 5$ .
Type II (Weakly holomorphic modular forms of weight $3/2$ ):
- Theorem 2: Extends the construction to weakly holomorphic modular forms $f$ of weight $3/2$ where $fg$ is a holomorphic modular form of weight $2$.
- Result: Establishes explicit congruences involving the $U_\ell$ operator and the Rankin-Cohen bracket. Under specific conditions (genus of $X_0(N)$ is zero), the paper proves that $U_\ell^2 \equiv \text{Id} \pmod \ell$ , leading to recursive congruence relations for the coefficients.
Type III (Mock modular forms of weight $3/2$ ):
- Theorem 3: Addresses mock modular forms $f$ (holomorphic parts of harmonic weak Maass forms) where the shadow is a holomorphic modular form of weight $1/2$ .
- Result: By utilizing holomorphic projection, the paper constructs explicit modular forms congruent to the generating functions of these mock modular forms. This allows for the derivation of congruences for coefficients of mock modular forms, such as the smallest parts function $spt(n)$ and Hurwitz class numbers $H(N)$ .

Significance and Claims
The paper claims that its primary significance lies in the explicitness of the constructions. Unlike previous methods that only guaranteed the existence of a modular form congruent to the generating function, this approach:

Provides Explicit Expressions: It gives a concrete formula for the modular form (via Rankin-Cohen brackets) that the generating function is congruent to.
Enables Non-Congruence Proofs: Because the constructed modular forms are explicit, their values at cusps can be computed. This allows the author to rigorously prove that certain generating functions are not congruent to zero modulo $\ell$ , a task difficult with non-constructive existence proofs.
Unified Framework: The method unifies the treatment of eta-quotients, weakly holomorphic forms, and mock modular forms under a single theoretical umbrella using Rankin-Cohen brackets and holomorphic projection.
Recovery and Extension: The approach recovers known congruences (e.g., for $p(n)$ , $spt(n)$, and class numbers) and establishes new non-congruence results and explicit congruences for various combinatorial functions that were not covered by previous literature.

The paper concludes that while the results for Type III might appear to be existence arguments at first glance, the proof mechanism actually yields computable expansions and explicit Eisenstein parts, offering a powerful tool for determining congruence properties in arithmetic combinatorics.