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A discrepancy result for Hilbert modular forms

This paper establishes asymptotic estimates for the Petersson trace formula for Hilbert cusp forms as the minimal weight tends to infinity, which are then applied to derive a weighted discrepancy bound for the distribution of Hecke eigenvalues, thereby generalizing a classical result by Jung and Sardari to the setting of totally real number fields.

Original authors: Baskar Balasubramanyam, Jishu Das, Kaneenika Sinha

Published 2026-06-02

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CC BY 4.0

Original authors: Baskar Balasubramanyam, Jishu Das, Kaneenika Sinha

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 a music conductor standing before a massive orchestra. This orchestra isn't made of violins and flutes, but of mathematical objects called Hilbert modular forms. These are complex, multi-dimensional waves that live in a strange, high-dimensional landscape (a "totally real number field").

Each musician in this orchestra plays a specific note, but they don't just play any note; they play notes determined by a set of rules called Hecke operators. When a musician plays, they produce a specific "eigenvalue"—think of this as the pitch of the note they are singing.

The Big Question: Do the Notes Match the Score?

Mathematicians have long suspected that if you listen to a huge number of these musicians, their pitches will distribute themselves in a very specific, predictable pattern. This pattern is called the Sato-Tate distribution. It's like a score that says, "50% of the notes should be in the middle range, 25% should be high, 25% should be low," following a specific bell-curve shape.

For a long time, we knew that if you listened to the entire orchestra over a long time, the notes would eventually match this score. But the authors of this paper asked a sharper question: How fast do they match? And more importantly, how much do they deviate from the score at any given moment?

This deviation is called discrepancy. If the score says "25% high notes" but the orchestra is actually playing "26% high notes," that 1% difference is the discrepancy.

The Old Way vs. The New Way

Previously, mathematicians had a tool to measure this discrepancy, but it was like trying to measure the distance between two cities with a ruler made of rubber. It worked, but the measurement was fuzzy. They could only prove that the discrepancy was small, roughly shrinking by a factor of $1/\log(k)$ (where $k$ represents the "volume" or complexity of the orchestra). This is a slow shrinkage.

The authors wanted to know: Can we prove the discrepancy shrinks much faster? Can we show it drops like a stone ( $1/k^\theta$ ) rather than a feather?

To answer this, they had to build a new, super-precise measuring tape.

The New Tool: The "Petersson Trace Formula"

The paper introduces a refined version of a mathematical tool called the Petersson trace formula.

The Analogy: Imagine the orchestra is a giant, complex machine. The trace formula is a way to look inside the machine and count exactly how many gears (mathematical terms) are turning.
The Problem: In the past, when they looked inside this machine for these specific high-dimensional forms, the "gears" were too noisy. There was too much static (error terms) to get a clear count.
The Breakthrough: The authors (Balasubramanyam, Das, and Sinha) managed to tune the machine. They developed a way to estimate the "noise" so precisely that they could isolate the true signal. They showed that if you look at the orchestra with a specific, very high "resolution" (letting the weights $k$ get very large), the noise becomes predictable and manageable.

The Main Discovery: A "Discrepancy Result"

Using this new, high-resolution view, the authors proved a surprising result about the lower bound of the discrepancy.

In simple terms, they found a sequence of orchestras where the notes do not match the score as perfectly as we might have hoped. They proved that no matter how you try to smooth things out, there will always be a certain amount of "roughness" or "mismatch" that persists.

The Metaphor: Imagine you are trying to paint a perfect circle on a wall. You might think, "If I use a finer brush, I can get infinitely close to a perfect circle."
The Paper's Claim: The authors found that for these specific mathematical shapes, even with the finest brush, there is a tiny, unavoidable jaggedness. They calculated exactly how big that jaggedness is. It turns out the mismatch is roughly proportional to $1 / (\text{log of the complexity})^2$ .

This is a "discrepancy result" because it sets a limit on how well the distribution of these mathematical notes can ever align with the theoretical ideal. It tells us that while the notes do eventually follow the pattern, they do so with a specific, measurable amount of "wiggle room" that cannot be eliminated.

Why This Matters (According to the Paper)

The paper doesn't talk about curing diseases or building bridges. Its importance is purely in the landscape of pure mathematics:

Generalizing a Classic Result: Previous work by Jung and Sardari had done this for "classical" music (standard modular forms, which are like 1-dimensional waves). This paper takes that result and expands it to "Hilbert modular forms," which are like multi-dimensional, complex waves living in higher dimensions.
The "Omega" Bound: They proved that the discrepancy isn't just small; it is at least this big. This prevents mathematicians from hoping for a "perfect" distribution that is smoother than reality allows.
The Method: They successfully adapted a complex formula (the Petersson trace formula) to work in this high-dimensional setting, showing that the "noise" can be controlled even when the math gets very complicated.

Summary

Think of the paper as a rigorous audit of a massive, multi-dimensional orchestra. The authors built a better microphone (the trace formula estimate) and discovered that while the orchestra generally plays the right tune, there is a specific, unavoidable amount of "static" or "off-key" notes that remains. They calculated exactly how much static there is, proving that the music is never perfectly smooth, but follows a predictable pattern of imperfection. This result extends a known rule from simple music to complex, multi-dimensional music.

Technical Summary: A Discrepancy Result for Hilbert Modular Forms

Problem Statement
The paper addresses the quantitative distribution of Hecke eigenvalues for Hilbert modular forms, specifically focusing on the "vertical" Sato-Tate distribution. While the equidistribution of eigenvalues $\kappa_\pi(v)$ for a fixed prime $v$ as the level $N$ or weight $k$ tends to infinity is a known theorem (established by Li, Blomer, etc.), the rate of convergence—measured by the discrepancy $D(\mu, \nu)$ between the empirical distribution of eigenvalues and the limiting measure—remains a subject of active investigation.

In the classical setting of modular forms over $\mathbb{Q}$ , previous work by Gamburd, Jakobson, Sarnak, and later Jung and Sardari [JS20], established that the discrepancy $D(\mu^*_{k,N}, \mu_p)$ is bounded below by a power of the weight $k$ (specifically $\Omega(k^{-1/3} \log^{-2} k)$ ). This contrasts with upper bounds derived from the Eichler-Selberg trace formula, which typically yield logarithmic savings ( $O((\log k)^{-1})$ ). The central question motivating this work is whether this logarithmic barrier can be broken to achieve a power-saving upper bound, or conversely, whether the lower bound (the $\Omega$ -type estimate) can be generalized to the context of Hilbert modular forms over a totally real number field $F$ of degree $r \geq 2$ .

Methodology
The authors employ the Petersson trace formula for Hilbert modular forms, a tool previously utilized in the classical case by Jung and Sardari. The methodology involves several key technical steps:

Asymptotic Analysis of the Trace Formula: The authors derive an asymptotic version of the Petersson trace formula for the space $A_k(N, \omega)$ of holomorphic Hilbert cusp forms. They analyze the sum over the basis of eigenforms $\sum \lambda_\phi(n) W_\phi^{m_1}(1) W_\phi^{m_2}(1) / \|\phi\|^2$ .
Error Term Estimation: A significant portion of the work is dedicated to estimating the error term in the trace formula. This involves bounding a triple sum over ideal classes, units, and lattice points. The authors utilize precise estimates for the Bessel function of the first kind ( $J_\nu$ ), specifically leveraging the transition behavior of $J_\nu(x)$ near $x \approx \nu$ . They partition the lattice points based on the magnitude of their embeddings and apply different bounds (uniform vs. exponential decay) depending on the region.
Main Term Simplification: To obtain a non-trivial lower bound, the authors impose specific arithmetic conditions on the field $F$ $F$ and the ideal $N$ $N$ :
- $F$ has an odd narrow class number.
- The ideal $N$ is generated by a rational integer $\tilde{s}$ (i.e., $N = \tilde{s}\mathcal{O}$ ).
- Under these conditions, the triple sum in the trace formula collapses to a single dominant term involving a specific Kloosterman sum.
Non-Vanishing of Kloosterman Sums: The authors prove that under the assumption that $N$ is square-free and the character is trivial, the relevant global Kloosterman sum does not vanish. This non-vanishing is crucial for ensuring the main term dominates the error term.
Discrepancy Derivation: Using the asymptotic lower bound on the trace, the authors relate the sum of eigenvalues to the discrepancy between the weighted measure $\tilde{\nu}_{k,N}$ and the Sato-Tate measure $\mu_\infty$ . They utilize the orthogonality of Chebyshev polynomials and integration by parts to translate the lower bound on the trace into a lower bound on the discrepancy.

Key Contributions and Results

Theorem 1 (Asymptotic Petersson Trace Formula): The authors establish an asymptotic formula for the Petersson trace formula acting on $A_k(N, \omega)$ as the minimum weight $k_0 = \min(k_j) \to \infty$ . They provide an explicit error term of order $O(\prod (k_j-1)^{-1/3})$ (with variations depending on the index), valid under specific constraints on the Fourier coefficients $m_1, m_2$ .
Theorem 2 (Lower Bound for the Main Term): Under the assumptions that $F$ has odd narrow class number and $N = \tilde{s}\mathcal{O}$ with $\tilde{s} \in \mathbb{Z}$ , the authors show that the main term of the trace formula is non-zero and bounded below by $\prod (k_j-1)^{-1/3}$ . This relies on the reduction of the sum to a single term and the non-vanishing of the associated Kloosterman sum.
Theorem 3 (Discrepancy Lower Bound): The primary result is the generalization of the $\Omega$ -type discrepancy bound to Hilbert modular forms. For a totally real field $F$ with narrow class number 1, and for a specific sequence of weights $k_l$ constructed from a fixed prime ideal $p$ , the discrepancy satisfies:
$D(\tilde{\nu}_{k_l, N}, \mu_\infty) \gg \frac{1}{(\log (k_l)_0)^2 \prod_{i=1}^r (k_{li}-1)^{1/3}}$
This result confirms that the discrepancy does not decay faster than a power of the weight (specifically $k^{-1/3}$ ), mirroring the behavior found in the classical case by Jung and Sardari.

Significance
The paper claims to initiate the discussion on extending $\Omega$ -type discrepancy bounds from classical modular forms to Hilbert modular forms. By successfully adapting the "weight removal" technique and the analysis of the Petersson trace formula to the higher-dimensional setting of totally real fields, the authors demonstrate that the vertical Sato-Tate distribution for Hilbert modular forms exhibits similar irregularities (discrepancies) as in the classical case.

The significance lies in:

Generalization: Extending the results of [JS20] from $\mathbb{Q}$ to arbitrary totally real number fields $F$ (albeit under the restrictive condition of narrow class number 1).
Methodological Advancement: Providing a detailed asymptotic analysis of the error terms in the Hilbert modular Petersson trace formula, which involves handling the geometry of lattices in $\mathbb{R}^r$ and the behavior of Bessel functions in multiple dimensions.
Quantitative Limits: Establishing that the discrepancy cannot be improved to a power-saving upper bound (i.e., $O(k^{-\theta})$ for $\theta > 1/3$ ) without new techniques, as the lower bound proves the existence of sequences where the discrepancy is at least of the order $k^{-1/3}$ .

The authors note that while they prove the result for the weighted measure $\tilde{\nu}_{k,N}$ , they expect a similar result for the unweighted measure, though this remains a subject for future work. The paper does not claim to resolve the question of whether a power-saving upper bound exists generally, but rather provides evidence (via the lower bound) that such a bound, if it exists, must have an exponent $\theta < 1/3$ .