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Quadratic forms of modular forms

This paper conditionally establishes that the inner product of two quadratic forms in Hecke eigenforms converges to the sum of their common off-diagonal coefficients when diagonal terms are absent, while also defining $\ell^p$ -norms for holomorphic cusp forms and proving an upper bound that reveals their coefficients are dominated by small-amplitude components rather than being uniformly small.

Original authors: Shenghao Hua

Published 2026-05-28

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Shenghao Hua

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 are in a vast, magical library where every book is a "Modular Form." These aren't ordinary books; they are complex mathematical objects that act like bridges connecting different worlds of mathematics (like number theory and geometry).

This paper, written by Shenghao Hua, is about what happens when you take two of these special books and multiply them together.

The Main Characters: The "Eigenforms"

Think of the library's most famous, unique books as "Hecke Eigenforms." These are the "purest" books in the collection. They have a special property: if you look at their "table of contents" (their coefficients), they follow a very strict, predictable rhythm.

The author is interested in what happens when you take two of these pure books, say Book A and Book B, and multiply them to create a new, combined story (a "quadratic form").

The Big Question: Do They Mix?

When you mix two pure stories, you get a new, messy story. The author wants to know: If you compare two different mixed stories, how much do they resemble each other?

To measure this, mathematicians use a tool called an "Inner Product." Think of this as a "similarity score."

If the score is high, the stories are very similar.
If the score is zero, they are completely unrelated.

The Paper's Discovery:
The author proves a surprising rule about these similarity scores, but only under certain "ideal weather conditions" (mathematical assumptions like the Generalized Riemann Hypothesis).

The rule is: If two mixed stories don't share any of the exact same "pure" ingredients in their diagonal (meaning they aren't just the same book squared), then they are almost completely unrelated.

The Analogy: Imagine you mix a red paint with a blue paint to make purple. Then you mix a green paint with a yellow paint to make brown. If you ask, "How much red is in the brown?" the answer is zero.
The Paper's Claim: If you mix two different pairs of pure books, the resulting "messy" stories don't overlap in a significant way. Their "inner product" (similarity score) drops to almost nothing, except for a tiny bit of noise.

The "L4-Norm" Problem: The Volume of the Library

There is a famous, difficult puzzle in this library called the L4-norm problem. It asks: "If you take one pure book, multiply it by itself, and then measure the 'loudness' or 'volume' of the result, how big is it?"

The Old Guess: Mathematicians guessed the volume would settle at a specific, predictable number (like 2).
The Paper's Contribution: The author shows that for mixed pairs of books (where the two books are different), the volume behaves in a very specific, predictable way that matches the "no overlap" rule mentioned above. It's like proving that if you mix two different flavors of ice cream, the resulting swirl doesn't accidentally taste like a third, unrelated flavor.

The "Sparsity" Surprise: Not All Coefficients Are Equal

The paper also looks at the "ingredients list" (coefficients) of these mixed stories.

The Expectation: You might think that if you mix many books, the ingredients would be spread out evenly, like sprinkling sugar uniformly over a cake. Every tiny crumb would have a similar amount of sugar.
The Reality: The author proves this is false. The ingredients are not spread out evenly.
- Most of the "sugar" (the coefficients) is concentrated in very small, tiny amounts.
- However, there are a few "giant" ingredients that are much larger than the average.
- The Metaphor: Imagine a crowd of people where 99% are tiny ants, but one person is a giant. If you try to describe the "average" size of a person in this crowd, you'd get a number that doesn't really describe anyone. The "giant" dominates the view, even though the "ants" are more numerous.

The paper shows that these "giant" ingredients are necessary. You cannot create these mixed stories using only tiny, uniform ingredients. There must be some large, dominant components to make the math work.

The "No Common Diagonals" Rule

The paper has a specific condition for its main result: The two mixed stories must not share any "common diagonal terms."

The Analogy: Imagine you have two recipes.
- Recipe A uses: Flour + Sugar.
- Recipe B uses: Sugar + Eggs.
- They share "Sugar."
The Paper's Rule: The author's main "no overlap" result only applies if the recipes don't share the same single ingredient used twice (like "Flour + Flour" vs. "Sugar + Sugar"). If they don't share these specific "double-ingredient" parts, the recipes are totally different from each other.

Summary

In simple terms, this paper says:

When you mix two different types of pure mathematical "songs," the resulting songs are almost completely different from each other (they don't overlap).
The "ingredients" (coefficients) of these mixed songs are not spread out evenly; instead, they are dominated by a few large, significant parts, with the rest being very small.
This behavior is exactly what we would expect if these mathematical objects were behaving like a random, chaotic system, which is a deep and important insight for understanding how these complex numbers interact.

The author uses advanced tools (like "L-functions," which are like mathematical X-rays that reveal the hidden structure of numbers) and assumes some famous unproven theories (like the Riemann Hypothesis) to make these conclusions. Without those assumptions, the proof is conditional, but the picture it paints is very clear.

Technical Summary of "Quadratic Forms of Modular Forms" by Shenghao Hua

Problem Statement
This paper investigates the statistical properties of quadratic forms within spaces of holomorphic cusp forms, specifically focusing on their expansion coefficients with respect to an orthonormal Hecke basis. The central problem addresses two interconnected areas:

Inner Products of Quadratic Forms: Understanding the asymptotic behavior of the inner product $\langle f_1 f_2, f_3 f_4 \rangle$ as the weights tend to infinity, where $f_i$ are $L^2$ -normalized Hecke eigenforms. This is framed as a "mixed $L^4$ -norm problem."
Coefficient Distribution: Determining the magnitude and distribution of the coefficients of a quadratic form $Q(f_1, \dots, f_N)$ when expanded in the Hecke basis. The author seeks to establish whether these coefficients are uniformly small or if they exhibit specific large-amplitude behaviors consistent with the distribution of orthogonal families of $L$ -functions.

Methodology
The paper employs analytic number theory techniques centered on the study of moments of $L$ -functions. The methodology proceeds as follows:

Decomposition and Reduction: The author reduces the problem of estimating the $\ell^p$ -norm of a general quadratic form to the study of specific terms involving products of Hecke eigenforms, namely $f^2$ and $fg$. This utilizes Minkowski's inequality.
Watson's Formula: To relate the inner products $\langle fg, h \rangle$ to $L$ -functions, the paper utilizes Watson's formula, which expresses the square of these inner products in terms of central values of triple product $L$ -functions ( $L(1/2, f \times g \times h)$ ) and symmetric square $L$ -functions.
Soundararajan's Method: The core technical engine is the application of Soundararajan's method for bounding moments of $L$ $L$ -functions. This involves:
- Establishing upper bounds for the moments of $L(1/2, f \times g \times h)$ and related symmetric square $L$ -functions.
- Using the Generalized Riemann Hypothesis (GRH) and the Generalized Ramanujan Conjecture to control the size of $L$ -values.
- Employing a "short Dirichlet polynomial" approach to approximate the logarithm of $L$ -functions.
- Applying a variance analysis and Gaussian tail bounds (Lemma 2.5) to estimate the number of forms with large $L$ -values.
Petersson's Formula: Lemma 2.1 utilizes Petersson's formula to handle sums of Hecke eigenvalues, which is crucial for the moment calculations.

Key Contributions and Results

Conditional Proof of Conjecture 1.1 (Theorem 1.2):
The paper provides a conditional proof for the asymptotic behavior of the inner product of quadratic forms. Specifically, for $L^2$ -normalized Hecke eigenforms $f_1, f_2, f_3, f_4$ with $k_1+k_2 = k_3+k_4$ , the inner product $\langle f_1 f_2, f_3 f_4 \rangle$ converges to a specific value determined by the equality of the sets $\{f_1, f_2\}$ and $\{f_3, f_4\}$ , provided they are not all identical.
- Conditions: The result assumes the GRH, the Generalized Ramanujan Conjecture (in specific cases), the non-equivalence of Rankin-Selberg convolutions ( $f_1 \times f_2 \not\sim f_3 \times f_4$ ), and the analytic continuation of triple product $L$ -functions involving symmetric squares.
- Significance: This completes the proof for cases where the four forms are not all identical, extending previous work by Huang.
Upper Bounds for $\ell^p$ -Norms (Theorem 1.4 and 1.9):
The author establishes conditional upper bounds for the $\ell^p$ -norm of quadratic forms in the Hecke basis.
- For a quadratic form $Q(f_1, \dots, f_N)$ , the $\ell^p$ -norm is bounded by $O(k^{\frac{1}{p} - \frac{1}{2}} \log^{-\alpha} k)$ .
- A sharper bound is obtained if the quadratic form has no diagonal terms ( $a_{i,i} = 0$ ).
- Crucially, for $p > 2$ , the $\ell^p$ -norm is shown to be $o(1)$ , implying that the coefficients decay as the weight increases.
Non-Uniformity of Coefficients (Theorem 1.6 and 1.11):
By combining the upper bounds on the $\ell^p$ -norm with lower bound arguments involving the first Fourier coefficients, the paper deduces that the coefficients of quadratic forms in the Hecke basis are not uniformly small.
- Theorem 1.6 states that for a sufficiently large weight, a quadratic form cannot be represented by a linear combination of Hecke eigenforms with a small number of non-zero coefficients (specifically, fewer than $L$ non-zero coordinates is impossible for large $k$ ).
- Theorem 1.11 asserts that there exist coordinates in the Hecke basis expansion that are of "larger-than-average order," meaning the distribution is dominated by small-amplitude components but contains specific large outliers.
Corollary on Decorrelation (Corollary 1.8):
The paper shows that if two quadratic forms share no common diagonal terms, their inner product converges to the sum of the products of their common off-diagonal coefficients. This is interpreted as a "decorrelation" phenomenon.

Significance and Claims
The paper claims to interpret the behavior of quadratic forms of modular forms through the lens of mixed $L^4$ -norm problems. The primary significance lies in:

Completing the Classification: Providing the missing cases for the asymptotic inner product conjecture regarding products of Hecke eigenforms.
Coefficient Distribution: Demonstrating that while the "average" coefficient of a quadratic form in the Hecke basis is small (decaying with weight), the distribution is not uniform; it is characterized by a dominance of small components with specific, larger-than-average outliers.
Connection to $L$ -functions: The results are presented as consistent with the expected distribution of orthogonal families of $L$ -functions, reinforcing the deep link between the spectral theory of modular forms and the analytic properties of $L$ -functions.

The author explicitly notes that the results are conditional on standard conjectures in the field (GRH, Ramanujan, analytic continuation). The paper does not propose new experimental applications but rather refines the theoretical understanding of the structure of spaces of modular forms.