Quadratic Congruences for half-integral weight cusp forms with the eta multiplier

This paper establishes that quadratic congruences modulo $\ell$ hold for a wide range of half-integral weight cusp forms with the eta multiplier and an arbitrary Dirichlet character, extending previous results limited to real characters through the study of modular Galois representations and a new theorem on the existence of specific conjugacy classes in their images.

The Gist

Imagine you are a detective trying to find hidden patterns in a massive, chaotic library of numbers. This library is filled with "partition numbers"—a famous sequence that counts the different ways you can break a number down into smaller pieces (like breaking the number 4 into 4, 3+1, 2+2, 2+1+1, or 1+1+1+1).

For over a century, mathematicians have known that if you look at these numbers through a special "mathematical lens" (called a modular form), they start to whisper secrets. Specifically, they sometimes follow strict rules where certain numbers in the sequence are divisible by a prime number (like 5, 7, or 11). This is called a congruence.

The Problem: The "Real-Only" Rule

In recent years, a team of mathematicians (including the author, Robert Dicks) discovered a powerful new way to find these secrets. They found that if you look at these numbers through a specific type of lens, you can predict when the numbers will be zero (divisible by a prime) based on a simple "yes or no" test involving squares.

However, there was a catch. Their method only worked if the lens had a very specific, "real" setting. It was like having a camera that only worked in black and white. If you tried to use it in color (with more complex mathematical settings), the picture would blur, and the secrets would disappear.

The Breakthrough: The "Universal Lens"

In this paper, Robert Dicks (building on previous work) says: "We don't need to be limited to black and white anymore."

He proves that these hidden patterns exist even when the lens is set to "color" (using what mathematicians call an arbitrary Dirichlet character). He shows that the rules for these number patterns are much more universal than we thought.

How Did He Do It? The "Shadow Puppet" Analogy

To solve this, Dicks didn't just look at the numbers directly. Instead, he used a technique involving Galois representations.

Think of a Galois representation as a shadow puppet show.

• The numbers are the puppets on the screen.
• The Galois representation is the light source and the hand moving the puppets.
• The congruences (the patterns) are the shapes the shadows make.

Previously, mathematicians could only predict the shapes of the shadows if the light source was very simple (the "black and white" setting). Dicks realized that even with a complex, multi-colored light source, the shadows still form predictable shapes, provided the light source is strong enough and the puppets are moving in a specific way.

His key insight was a new mathematical trick:

1. He looked at a group of different "shadow shows" (different modular forms) happening at the same time.
2. He proved that there is a specific "magic moment" (a specific element in the symmetry group of numbers) where all these different shows align perfectly.
3. At this moment, the shadows of all the different shows look like the same shape (specifically, a square of a specific rotation).
4. Because they all align, the hidden rules (congruences) that apply to one apply to all, even in the complex "color" settings.

Why Does This Matter?

This is like discovering that a secret code used by a spy agency isn't just one language, but a universal language that works in every dialect.

• The "Partition Function" Connection: The original discovery was about the partition function (how many ways to break down numbers). This paper generalizes that discovery to a huge family of similar mathematical objects.
• The "Eta Multiplier": This is a fancy name for a specific type of "glue" that holds these number patterns together. Dicks shows that this glue works even when the patterns get very complicated.
• The Result: We now know that for a vast range of these number patterns, there are infinitely many prime numbers where the pattern "breaks" in a predictable way (becomes divisible by that prime).

The Takeaway

Imagine you have a giant, complex machine with thousands of gears. For a long time, we only knew how to predict the machine's behavior when the gears were made of a specific metal. Robert Dicks has shown that the machine works the exact same way, even if you swap the gears for any other type of metal, as long as the machine is running fast enough.

He didn't just find one new pattern; he built a universal key that unlocks the secrets of a whole new class of mathematical objects, proving that the beautiful, hidden order of numbers is far more robust and widespread than we ever imagined.

Technical Summary

Here is a detailed technical summary of the paper "Quadratic Congruences for Half-Integral Weight Cusp Forms with the Eta Multiplier" by Robert Dicks.

1. Problem Statement

The paper addresses the existence of quadratic congruences for the Fourier coefficients of half-integral weight cusp forms. Specifically, it investigates congruences of the form:
$$a(p^2 n) \equiv 0 \pmod{\ell^m}$$
under specific conditions on the prime $p$ and the Legendre symbol $\left(\frac{n}{p}\right)$.

Context and Motivation:

• Historical Background: The study is motivated by Ramanujan's congruences for the partition function $p(n)$ and Atkin's subsequent generalizations involving quadratic conditions on primes.
• Previous Work: Recent work by Ahlgren, Andersen, and the author (AAD24) established such congruences for forms with real Dirichlet characters ($\psi$) and the Dedekind eta multiplier ($\nu_\eta^r$).
• The Gap: The previous results relied on the character $\psi$ being real. This paper aims to generalize these results to arbitrary Dirichlet characters $\psi$, which introduces significant complexity in the associated Galois representations.

2. Methodology

The author employs a Galois-theoretic approach centered on modular Galois representations modulo $\ell$. The core strategy involves translating congruences for half-integral weight forms into congruences for integral weight forms via the Shimura correspondence, and then analyzing the images of these representations.

Key Technical Components:

1. Shimura Correspondence:

   • The paper utilizes a precise Shimura lift developed in [AAD24] that maps half-integral weight forms $F \in S_{\lambda+1/2}(N, \psi\nu_\eta^r)$ to integral weight newforms $St(F)$ in spaces like $S_{2\lambda}^{new}(6N, \psi^2, \dots)$.
   • A crucial property is that $F \equiv 0 \pmod{\ell} \iff St(F) \equiv 0 \pmod{\ell}$.

2. Modular Galois Representations:

   • For an integral weight newform $f$, there is an associated continuous irreducible representation $\rho_f: G_\mathbb{Q} \to GL_2(\mathbb{Q}_\ell)$.
   • The paper studies the reduction of these representations modulo $\ell$, denoted $\bar{\rho}_f$.
   • The condition of "suitability" is introduced: A pair $(k, \ell)$ is suitable if the image of $\bar{\rho}_f$ for every newform in the relevant space contains a conjugate of $SL_2(\mathbb{F}_\ell)$. This ensures the image is "large" (not solvable or exceptional).

3. The Core Obstacle and Insight:

   • In previous work with trivial or real characters, the Galois characters arising in the analysis were easily identified (e.g., trivial or powers of the cyclotomic character).
   • With arbitrary characters $\psi$, identifying the specific Galois characters is difficult.
   • Key Insight: The author proves that one can establish the existence of a Galois element $\sigma$ such that the images $\bar{\rho}_f(\sigma)$ lie in a specific conjugacy class (specifically $\gamma^2$) "up to a constant factor" (sign). The theory of modular Galois representations is then used to show these constant factors are well-behaved, allowing the recovery of the result for $\gamma^2$.

4. Chebotarev Density Theorem:

   • Once the existence of a specific $\sigma$ in the Galois group is established (whose images under all relevant representations are conjugate to a target matrix), the Chebotarev Density Theorem is used to find a set of primes $p$ (of positive density) whose Frobenius elements correspond to this $\sigma$.

3. Key Contributions and Results

A. Generalization to Arbitrary Characters (Theorem 1.1)

The main result extends the congruence theorems to arbitrary Dirichlet characters $\psi$.

• Conditions: Let $\ell \ge 5$, $r$ be an odd integer, and $(k, \ell)$ be "suitable" for $(N, \psi, r)$.
• Result: There exists a set of primes $S$ of positive density such that for $p \in S$ (with $p \equiv 1 \pmod{\ell^m}$):
  $$a(p^2 n) \equiv 0 \pmod{\ell^m} \quad \text{if} \quad \left(\frac{n}{p}\right) = \epsilon_p$$
  where $\epsilon_p$ is explicitly determined by $\psi(p)$, $r$, and Legendre symbols involving $-1$ and $-3$.
• Significance: This vastly generalizes previous results by removing the restriction that $\psi$ must be real.

B. Congruences without Suitability (Theorem 1.2)

The paper provides a result that does not require the "suitability" condition (which can be restrictive on the weight $k$).

• Condition: Requires the existence of an integer $a$ such that $2^a \equiv -2 \pmod{\ell}$.
• Result: There exists a set of primes $S$ (density $>0$) such that for $p \in S$ (with $p \equiv -2 \pmod{\ell^m}$), the congruence holds for some $\epsilon_p \in \{\pm 1\}$.
• Note: By Hasse's result, the condition $2^a \equiv -2 \pmod{\ell}$holds for a proportion of$17/24$ of all primes.

C. New Technical Proposition (Proposition 1.3 / 3.5)

A critical new lemma is established regarding the simultaneous behavior of Galois representations:

• Statement: Given a finite set of newforms $f_1, \dots, f_s$ whose associated representations $\bar{\rho}_{f_i}$ have large images ($SL_2(\mathbb{F}_\ell)$), and given any $\gamma \in SL_2(\mathbb{F}_\ell)$, there exists a $\sigma \in \text{Gal}(\mathbb{Q}/\mathbb{Q}(\zeta_\ell))$ such that $\bar{\rho}_{f_i}(\sigma^2)$ is conjugate to $\gamma^2$ for all $i$.
• Importance: This generalizes a result by Ahlgren, Allen, and Tang (which was limited to trivial characters) to the case of arbitrary characters, overcoming the difficulty of identifying specific Galois characters by working "up to sign."

4. Significance and Impact

1. Broadening the Scope of Modular Congruences: The paper removes a major technical barrier (the requirement of real characters) in the theory of quadratic congruences for half-integral weight forms. This allows the application of these congruences to a much wider class of modular forms, including those with complex Nebentypus.
2. Advancement in Galois Representation Theory: The proof of Proposition 3.5 represents a significant technical achievement in handling modular Galois representations with arbitrary characters. It demonstrates that deep arithmetic properties (like the existence of specific Frobenius elements) can be derived even when the precise nature of the twisting characters is unknown, provided the images are large.
3. Connection to Partition Functions: Since the partition function and its generalizations are central examples of such forms, these results deepen the understanding of the arithmetic structure underlying partition congruences, extending Atkin's legacy.
4. Methodological Robustness: The paper successfully adapts the "Galois theoretic" machinery to more complex multiplier systems (eta multipliers with arbitrary characters), providing a template for future research in congruences for forms with non-trivial multipliers.

In summary, Robert Dicks successfully generalizes the theory of quadratic congruences for half-integral weight cusp forms to arbitrary Dirichlet characters by developing new tools in the theory of modular Galois representations, specifically addressing the challenge of identifying Galois characters in the presence of arbitrary Nebentypus.