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Asymptotics of Hecke polynomial coefficients on the Atkin-Lehner eigenspaces

This paper investigates the asymptotic behavior of the coefficients of the $m$ -th Hecke polynomial acting on spaces of cusp forms with specific Atkin-Lehner sign patterns, demonstrating that these coefficients eventually stabilize to a single sign in certain settings while exhibiting non-convergent sign behavior in others.

Original authors: Timothy Nelson, Erick Ross, Maya Wassercug, Hui Xue

Published 2026-08-20

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CC BY 4.0

Original authors: Timothy Nelson, Erick Ross, Maya Wassercug, Hui Xue

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

In the vast landscape of number theory, mathematicians often study patterns that hide within the most fundamental building blocks of arithmetic: the integers. One powerful way to uncover these hidden structures is through objects called modular forms. You can think of a modular form as a highly complex, multi-layered map that encodes deep relationships between numbers. These maps are not static; they possess a rich internal symmetry that allows mathematicians to probe them with specific tools known as Hecke operators. When these tools are applied, they reveal a set of numbers, or coefficients, that act like a fingerprint for the map. These fingerprints tell us about the size of the space the map occupies, how it twists and turns, and whether it contains any "zeroes" that might signal a break in the pattern. For decades, mathematicians have been fascinated by the behavior of these coefficients, particularly whether they follow predictable rules or if they wander into chaos.

A recent paper by Timothy Nelson, Erick Ross, Maya Wassercug, and Hui Xue takes a closer look at these fingerprints, but with a twist. They decided to examine the modular forms not just as a whole, but broken down into smaller, more specific groups based on how they react to a particular kind of symmetry operation. Imagine a large room full of people; the researchers didn't just count the total number of people, but instead grouped them by how they responded to a specific signal, separating those who raised their hands from those who did not. In the world of modular forms, these groups are defined by "sign patterns," which describe whether the form stays the same or flips its sign when subjected to certain transformations. The authors asked a simple but profound question: if we look at the coefficients of these specific, smaller groups, do they behave as predictably as the larger, mixed group?

The researchers focused on the coefficients of a mathematical object called the Hecke polynomial, which is essentially a summary of all the fingerprints for a given group. They investigated how these coefficients change as the complexity of the modular forms increases, either by making the forms more intricate or by looking at higher levels of the underlying number system. Their first major discovery was that for certain types of inputs, the coefficients settle into a very predictable rhythm. As the complexity grows, the sign of these numbers—whether they are positive or negative—becomes fixed. For example, if you look at a specific type of coefficient in a group where the input number is a perfect square, it will eventually always be positive, or always negative, depending on its position in the sequence. This holds true for almost every case, with only a tiny, finite number of exceptions that disappear as the complexity increases.

However, the story becomes much more surprising when the researchers looked at inputs that are not perfect squares. In the broader, unbroken group of modular forms, mathematicians had long suspected that the coefficients would eventually settle into a single, consistent sign as the complexity grew. The new study confirms this for the smaller, sign-pattern groups, but only for the even-indexed coefficients (such as the second, fourth, and sixth coefficients) when the complexity increases in one specific direction. The odd-indexed coefficients behave differently, as their signs depend on the trace of the operator, which does not necessarily settle into a fixed pattern. But when they looked at how the coefficients behave as the level of the number system changes, the pattern breaks down completely. The authors constructed two infinite families of these sign-pattern groups where the same even-indexed coefficient flips back and forth between positive and negative. In one family, the coefficient is always positive for large levels; in the other, it is always negative. This is a stark contrast to the behavior seen in the larger, unbroken groups, where such flipping does not happen. It reveals that the finer the lens through which we view these mathematical objects, the more erratic and unpredictable their behavior can become.

To reach these conclusions, the team developed a precise method for calculating the "trace" of these operators, which is a way of summing up the effects of the Hecke tools across the entire group. This calculation allowed them to estimate the size of the coefficients with great accuracy. They found that while the main part of the calculation follows a smooth, predictable curve, there is a smaller, fluctuating part that can sometimes overwhelm the main trend. In the case of the non-square inputs, this fluctuating part is strong enough to flip the sign of the coefficient, creating the two opposing families they discovered. The authors also provided explicit mathematical bounds, showing exactly how large the numbers need to be before the predictable signs take over, ensuring that their findings are not just theoretical guesses but concrete, verifiable facts.

The implications of this work extend beyond just counting numbers. By understanding how these coefficients behave in these specific subgroups, the researchers have opened the door to better understanding the distribution of values in modular forms. This is crucial for other areas of mathematics, such as proving that certain numbers are never zero, a question that has puzzled mathematicians for generations. The paper also offers a new, explicit formula that can be used to study these objects in ways that were previously impossible. While the authors have solved the question of how the signs behave for most cases, they leave one important question open: they suspect that for sufficiently complex forms, the second coefficient in the sequence never vanishes, but they have not yet been able to prove it for all cases. Their work stands as a testament to the idea that even in the most abstract corners of mathematics, breaking a problem down into its smallest, most specific parts can reveal a world of unexpected complexity and beauty.

Technical Summary: Asymptotics of Hecke Polynomial Coefficients on the Atkin-Lehner Eigenspaces

Problem Statement
This paper investigates the asymptotic behavior of the coefficients of the characteristic polynomial of the normalized $m$ -th Hecke operator, denoted $T'_m$ , acting on spaces of cusp forms with specific Atkin-Lehner sign patterns. Let $S_k(N)$ be the space of cusp forms of weight $k$ and level $N$ . The space decomposes into eigenspaces $S_k^\sigma(N)$ based on the eigenvalues $\sigma(Q) \in \{\pm 1\}$ of the Atkin-Lehner involutions $W_Q$ for exact divisors $Q \parallel N$ .

The characteristic polynomial is defined as:
$T'_m(N, k, \sigma)(x) = \sum_{0 \le r \le d} c_r(m, N, k, \sigma) x^{d-r}$
where $d = \dim S_k^\sigma(N)$ . While the asymptotic behavior of these coefficients $c_r$ over the full space $S_k(N)$ and the newspace $S_k^{new}(N)$ has been studied (notably in [20]), the behavior over the finer decomposition $S_k^\sigma(N)$ and $S_k^{new, \sigma}(N)$ remained open. The authors aim to determine whether the coefficients exhibit stable asymptotic signs in these subspaces and to establish explicit bounds for these behaviors.

Methodology
The authors employ a combination of algebraic identities and analytic trace formulas:

Newton-Girard Identities: The coefficients $c_r$ are related to the power sums (moments) $p_j$ of the eigenvalues of $T'_m$ via the Newton-Girard identities. Specifically, $c_r = -\frac{1}{r} \sum_{j=1}^r c_{r-j} p_j$ . This reduces the problem of estimating coefficients to estimating traces of powers of Hecke operators.
Explicit Trace Formulas: The core technical tool is an explicit trace formula for $T'_m$ over $S_k^\sigma(N)$ , derived in Appendix A (Proposition A.5). This formula provides a main term and an explicit error bound $E_{k, \sigma}(m, N)$ involving the Dedekind psi function $\psi(N)$ , the number of prime divisors $\omega(N)$ , and class numbers.
Asymptotic Analysis:
- For square $m$ , the authors analyze the limit as $N + k \to \infty$ .
- For non-square $m$ , they analyze the limit as $k \to \infty$ (holding $N$ fixed).
- They utilize bounds on Hurwitz class numbers and properties of the divisor function to estimate error terms.
Construction of Counterexamples: To address the behavior in the $N$ -limit for non-square $m$ , the authors construct specific infinite families of pairs $(N, \sigma)$ using prime levels and properties of class numbers to demonstrate sign oscillation.

Key Contributions and Results

Asymptotics for Square $m$ :
Theorem 1.1 establishes that for a square $m$ , as $N+k \to \infty$ , the $r$ -th coefficient behaves as:
$c_r(m, N, k, \sigma) = \frac{(-1)^r}{r!} \left( \frac{1}{\sqrt{m}} \frac{k-1}{12} \frac{\psi(N)}{2^{\omega(N)}} \right)^r + O_{m}(k^{r-1}N^{r-1/2+\varepsilon})$
Corollary 1.3 follows that for square $m$ , $c_r$ has sign $(-1)^r$ for all but finitely many triples $(N, k, \sigma)$ .
Asymptotics for Non-Square $m$ :
Theorem 1.2 determines the behavior as $k \to \infty$ for non-square $m$ . The even coefficients $c_{2r}$ satisfy:
$c_{2r} = \frac{(-1)^r}{(2r)!!} \left( \frac{\sigma_1(m)}{m} \frac{k-1}{12} \frac{\psi(N)}{2^{\omega(N)}} \right)^r + O_{m,N}(k^{r-1})$
The odd coefficients $c_{2r+1}$ are dominated by the trace $c_1$ . Corollary 1.4 states that for non-square $m$ , $c_{2r}$ has sign $(-1)^r$ for all but finitely many $k$ .
Sign Oscillation in the $N$ -Limit:
Contrary to the behavior in the full space (where coefficients stabilize in sign as $N \to \infty$ ), Theorem 1.5 proves that for non-square $m$ and fixed even $k$ , there exist infinite families of pairs $(N, \sigma)$ where $c_2(m, N, k, \sigma)$ is positive, and infinite families where it is negative. This is achieved by analyzing the trace formula for prime levels $N=p$ , where the sign depends on the competition between a term proportional to $H(4mp)^2$ and a term proportional to $-(p+1)$ .
Extension to Newspace:
Section 6 extends all main theorems to the newspace $S_k^{new, \sigma}(N)$ , provided the sign pattern $\sigma$ is "admissible" (excluding cases where $4 \parallel N$ and $\sigma(4)=+1$ ). The asymptotic forms are analogous, involving a modified function $\psi^{new}(N)$ .
Explicit Bounds:
Propositions 7.1 and 7.2 provide explicit, effective inequalities for $N$ and $k$ that guarantee the sign predictions of Corollaries 1.3 and 1.4 hold for the second coefficient $c_2$ . These bounds are derived from the explicit error terms in the trace formula.

Significance and Claims
The paper claims to extend the understanding of Hecke polynomial coefficients from global spaces to the finest natural decomposition respecting Hecke operators and Galois conjugation: the Atkin-Lehner eigenspaces.

Refinement of Global Properties: The work demonstrates that while certain properties (like the asymptotic sign of coefficients for square $m$ ) persist in the finer eigenspaces, others (specifically the stability of signs for non-square $m$ as $N \to \infty$ ) do not. This highlights a fundamental difference between the global space and its eigenspace decompositions.
Explicit Trace Formula: A primary contribution is the derivation of an explicit trace formula for $T'_m$ over $S_k^\sigma(N)$ (Proposition A.5). The authors note this formula is of independent interest, citing its necessity for proving effective equidistribution of Hecke eigenvalues in a forthcoming paper [13] and for extending classifications of vanishing newspace dimensions (as in [16]) to general weights.
Conjecture on Non-Vanishing: The authors propose Conjecture 1.6, asserting that the second coefficient $c_2$ is non-vanishing for $k \ge 8$ in these spaces, a claim they have verified computationally for a significant range of parameters.

The authors remain modest regarding applications, stating that the trace formula "will be useful to others" for problems beyond the immediate scope of coefficient asymptotics, such as equidistribution and dimension classification, without speculating on unverified future implications.

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