🔢 mathematics

Traces of Hecke operators on Drinfeld modular forms for $\mathrm{GL}_2(\mathbb{F}_q[T])$

This paper investigates traces of Hecke operators on Drinfeld modular forms of level 1 over $\mathbb{F}_q[T]$ , deriving closed-form expressions for low-degree primes, providing algorithms for higher degrees, improving the Ramanujan bound, and establishing the newform-oldform decomposition under a specific multiplicity hypothesis.

Original authors: Sjoerd de Vries

Published 2026-03-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Sjoerd de Vries

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 detective trying to solve a mystery in a world made of numbers, but instead of the usual integers (1, 2, 3...), this world is built from polynomials (like $T^2 + T + 1$ ). This is the world of Drinfeld Modular Forms.

In the "classical" world of math (like the one Ramanujan and Euler lived in), mathematicians study shapes called elliptic curves. These shapes have hidden patterns, and to find them, they use special tools called Hecke operators. Think of a Hecke operator as a magical scanner that scans a shape and gives you a number (an eigenvalue) that tells you something deep about its structure.

This paper, written by Sjoerd de Vries, explores what happens when we swap the classical world for this polynomial world. Here is the story of what he found, explained simply.

1. The Setting: A Different Kind of Number Line

In our normal world, numbers go on forever in a line. In this paper's world, the "numbers" are polynomials over a finite field (a tiny, closed universe of numbers).

The Analogy: Imagine a library where books are made of Lego blocks. In the classical library, the books are heavy, solid stone. In this Drinfeld library, the books are made of colorful, snapping Lego bricks. They behave similarly, but the rules of how they snap together are slightly different and stranger.

2. The Mystery: The "Scanner" (Hecke Operators)

The main characters in this story are the Hecke operators.

The Analogy: Imagine you have a machine that takes a complex Lego sculpture and runs it through a filter. The machine spits out a single number. If you run the same sculpture through the machine again, you get the same number. That number is the sculpture's "fingerprint."
The Problem: In the classical world, these machines are very well-behaved. They always give unique fingerprints, and we know exactly how to predict them. In this Lego world, the machines are a bit glitchy. Sometimes they give the same fingerprint to two different sculptures (repeated eigenvalues), and sometimes they don't behave nicely at all.

3. The Detective's Tool: The Trace Formula

To solve the mystery, the author uses a "Trace Formula."

The Analogy: Imagine you can't see the individual Lego sculptures in a dark room. But you can count how many shadows they cast on the wall. The Trace is the total sum of all those shadows.
The Breakthrough: The author figured out how to calculate these shadows (traces) very precisely. He found that for certain types of "scanners" (primes of degree 1), the shadows follow a beautiful, predictable pattern, almost like a song. He wrote down a "closed-form expression"—a simple recipe—to calculate these shadows without having to count them one by one.

4. The Special Case: When the World is "Even" (Characteristic 2)

The paper discovers something amazing when the underlying number system has an "even" nature (like binary code, 0 and 1).

The Analogy: In the "odd" world, the shadows are chaotic and hard to predict. But in the "even" world, the shadows line up perfectly in a straight, rhythmic line.
The Result: In this even world, the author can predict exactly which fingerprints appear an odd number of times. It's like knowing exactly which notes in a song will be played a single time, and which will be repeated.

5. The Big Question: Old vs. New

In classical math, there's a concept of "Old" forms (patterns that come from simpler, smaller shapes) and "New" forms (patterns that are truly unique to the current size).

The Analogy: Imagine a family tree. "Old" forms are like cousins you already know from a smaller branch of the family. "New" forms are like a baby born today who has a unique face.
The Conjecture: Mathematicians Bandini and Valentino guessed that you can always separate the "Old" from the "New" in this Lego world, provided the "scanner" doesn't get confused and give the same fingerprint to too many things at once.
The Paper's Contribution: The author proves that if the "family" (the space of forms) is small enough (smaller than the size of the number system), then yes, you can perfectly separate the Old from the New. He also improved the "Ramanujan Bound," which is a speed limit on how fast these patterns can grow. He showed the speed limit is actually lower than we thought!

6. The Twist: A Counter-Example

One of the most exciting parts is a discovery about "A-expansions."

The Analogy: Some Lego sculptures have a special "blueprint" (an A-expansion) that makes them easy to understand. Mathematicians thought: "If a sculpture has a fingerprint that looks exactly like a blueprint, it must have a blueprint."
The Twist: The author found a specific sculpture (in the world where $q=3$ ) that has a perfect fingerprint, but no blueprint. It's a "fake" blueprint. This proves that in this Lego world, things aren't always as straightforward as they seem in the classical world.

Summary

Sjoerd de Vries took a complex, abstract mathematical problem about polynomial shapes and:

Mapped the shadows: He created a recipe to calculate the "total shadows" (traces) of these shapes.
Found the rhythm: He showed that in "even" worlds, these shadows follow a perfect, predictable pattern.
Fixed the speed limit: He proved that these patterns grow slower than previously thought.
Found a trickster: He discovered a shape that looks like it has a secret code but actually doesn't.

This paper is a bridge between the familiar world of classical numbers and the strange, fascinating world of polynomial numbers, showing us that while the rules are different, there is still a beautiful, hidden order waiting to be found.

1. Problem Statement and Context

The paper investigates the arithmetic properties of Drinfeld modular forms over the ring $A = \mathbb{F}_q[T]$ , specifically focusing on the action of Hecke operators $T_p$ on spaces of cusp forms $S_{k,l}$ of level 1.

The Analogy: Drinfeld modular forms are function-field analogues of classical elliptic modular forms. However, significant differences exist:
- There is no natural Petersson inner product in positive characteristic, making the decomposition of forms into oldforms and newforms non-trivial.
- Hecke operators are not always diagonalizable.
- Eigenvalues do not generally arise as coefficients of a $q$ -expansion (except for forms with specific $A$ -expansions).
The Core Challenge: While trace formulas exist (relating the trace of a Hecke operator to a sum over isomorphism classes of Drinfeld modules), explicit evaluation is difficult because it requires counting isomorphism classes within specific isogeny classes over finite fields. Furthermore, recovering individual eigenvalues from traces is obstructed in characteristic $p$ if eigenvalues have algebraic multiplicities divisible by $p$ .

2. Methodology

The author employs a combination of geometric trace formulas, combinatorial number theory, and computational algebra.

The Trace Formula: The paper utilizes a geometric trace formula (from [Vri25a]) which expresses the trace of $T_p^n$ as a sum over isomorphism classes of rank-2 Drinfeld modules over $\mathbb{F}_{p^n}$ :
$\text{Tr}(T_p^n | S_{k+2,l}) = \sum_{[\phi]} h_k(\pi_\phi, \bar{\pi}_\phi) \left( \frac{\pi_\phi \bar{\pi}_\phi}{\wp^n} \right)^{l-k-1}$
where $h_k$ is the $k$ -th homogeneous symmetric polynomial and $\pi_\phi$ is the Frobenius endomorphism.
Counting Isomorphism Classes: A major technical contribution is the explicit determination of the number of isomorphism classes of Drinfeld modules in a fixed isogeny class, denoted $\# \text{Iso}_{p^n}(a,b)$ . The author extends results by Yu and Gekeler to express these counts in terms of Hurwitz class numbers and points on Jacobians of hyperelliptic curves (Appendix A).
Symmetry and Binomial Identities: The paper exploits symmetries in the weights $k$ of the form $k = p^m + 1$ . Using Lucas's theorem and binomial identities, the author derives recurrence relations for traces that relate weights $k$ and $k'$ symmetrically around $p^m + 1$ .
Characteristic 2 Analysis: A distinct approach is taken for characteristic 2 ( $2 \mid q$ ), where the behavior of traces simplifies significantly, allowing for closed-form expressions for all primes.
Computational Verification: The author implements algorithms in Magma to compute traces for higher-degree primes and low weights, verifying conjectures and discovering patterns in eigenvalues.

3. Key Contributions and Results

A. Closed-Form Trace Expressions

Degree 1 Primes: Theorem 1.1 provides a closed-form expression for the trace of $T_p$ where $p$ is a prime of degree 1. The trace is a polynomial in the generator $\wp$ of $p$ , involving binomial coefficients:
$\text{Tr}(T_p | S_{k+2,l}) = \sum_{0 \le j < k/2, j \equiv l-1} (-1)^j \binom{k-j}{j} \wp^j$
Characteristic 2: Theorem 1.2 and Theorem 3.26 show that if $q$ is even, the trace formula simplifies for any prime $p$ and any power $n$ , effectively modeling the trace of any Hecke operator after the degree 1 case.
Higher Degree Primes: For odd $q$ and primes of degree $\ge 2$ , the author provides an explicit (though more complex) formula for degree 2 primes (Theorem 3.33) and an algorithm (Algorithm 3.38) to compute traces for arbitrary degrees using Hurwitz class numbers.

B. Eigenvalue Analysis in Characteristic 2

Odd Multiplicity Eigenvalues: In characteristic 2, the author completely characterizes the eigenvalues of $T_p$ that occur with odd algebraic multiplicity. Theorem 5.9 states that these eigenvalues are exactly $\wp^j$ for specific indices $j$ determined by binomial coefficients modulo 2.
Repeated Eigenvalues: Theorem 5.13 proves that for $q$ even, there are only finitely many weights where the Hecke action has no repeated eigenvalues. This implies that for large weights, trace methods alone cannot recover all eigenvalues due to the "vanishing" of even-multiplicity contributions in characteristic 2.

C. The Strong Ramanujan Bound

The paper refines the classical Ramanujan bound on the degree of Hecke traces.
Conjecture 4.3 (Strong Ramanujan Bound): The author conjectures that $\deg \text{Tr}(T_p^n | S_{k,l}) \le \frac{nd(k - (q+1))}{2}$ .
Proofs: This bound is proven for all primes where $n \deg(p) \le 3$ and for all primes in characteristic 2 (Theorem 4.9).
Significance: This bound is crucial for understanding the distribution of slopes of Hecke operators.

D. Oldforms and Newforms Decomposition

The paper addresses the decomposition of cusp forms of level $\Gamma_0(p)$ into oldforms and newforms, a conjecture by Bandini and Valentino.
Theorem 1.5: It is proven that if $\dim S_{k,l} < p$ , then the decomposition $S_{k,l}(\Gamma_0(p)) = S^{\text{new}} \oplus S^{\text{old}}$ holds. This relies on the fact that if the decomposition fails, a specific eigenvalue must have multiplicity divisible by $p$ , which contradicts the Strong Ramanujan bound under the dimension constraint.

E. $A$ -Expansions and Eigenforms

The paper investigates the converse of Petrov's theorem: Does an eigenform with a "power eigensystem" (eigenvalues $\wp^{n-1}$ ) necessarily have an $A$ -expansion?
Theorem 6.1: The author provides a counterexample. For $q=3$ , the form $g_2 h_2 \in S_{12,0}$ has the eigenvalue $\wp^3$ for all $T_p$ , yet it does not possess an $A$ -expansion. This resolves an open question regarding the rigidity of $A$ -expansions.

4. Significance and Impact

Explicit Computability: The paper moves the theory of Drinfeld modular forms from abstract existence to explicit computation. By linking isogeny class counts to Hurwitz class numbers and hyperelliptic curves, it provides a concrete algorithmic framework for calculating traces.
Resolution of Conjectures: It proves the Bandini-Valentino conjecture on old/newform decomposition under specific dimension constraints and provides strong evidence (and proofs in special cases) for the Strong Ramanujan bound.
Characteristic 2 Insights: The detailed analysis of characteristic 2 reveals a stark difference from the odd characteristic case, particularly regarding the prevalence of repeated eigenvalues and the simplicity of trace formulas.
Counterexamples: The discovery of eigenforms with power eigensystems lacking $A$ -expansions challenges the intuition that spectral properties strictly dictate the analytic structure of the form.

5. Conclusion

Sjoerd de Vries's work provides a comprehensive toolkit for analyzing Hecke operators on Drinfeld modular forms. By combining geometric trace formulas with combinatorial number theory and computational verification, the paper establishes closed-form trace formulas, improves bounds on eigenvalue degrees, and clarifies the structure of the space of cusp forms, particularly in the context of the old/newform decomposition and the behavior of eigenvalues in positive characteristic.