Periodicity of traces of Hecke operators modulo prime powers

1. Problem Statement

The paper addresses the arithmetic properties of modular forms, specifically focusing on congruences between traces of Hecke operators acting on spaces of cusp forms of different weights.

• Context: Classical results (e.g., by Serre) established congruences for traces modulo a prime $p$. However, extending these results to prime powers ($p^s$) and to Drinfeld modular forms (the function field analogue of classical modular forms) remains a significant challenge.
• Goal: The authors aim to prove that for a fixed prime power modulus $\ell^s$, the trace of a Hecke operator $T_q$ (or $F_q$) acting on the space of cusp forms $S_k$ becomes periodic in the weight $k$ once $k$ is sufficiently large.
• Scope: The results apply to:
  1. Elliptic modular forms over $\mathbb{Z}$ with general level structures.
  2. Drinfeld modular forms over function fields of positive characteristic.

2. Methodology

The authors employ a unified approach based on trace formulae derived from the geometry of modular curves and the Grothendieck–Lefschetz trace formula.

A. Trace Formulae

The core tool is the Eichler–Shimura–Deligne relation combined with the Grothendieck–Lefschetz trace formula.

• Elliptic Case: The trace of Frobenius $F_q$ on the space $S[H, k+2]$ is expressed as a sum over isomorphism classes of elliptic curves with level structure over $\mathbb{F}_q$.
  $$\text{Tr}(F_q | S[H, k+2]) + \epsilon_k = -e_{is} - \sum_{j=0}^{\lfloor k/2 \rfloor} \binom{k-j}{j} (-q)^j [a_1^{k-2j}]$$
  Here, $[a_1^m]$ represents a weighted sum of powers of the trace of Frobenius ($a_1(E)$) over the moduli space.
• Drinfeld Case: A parallel trace formula exists for Drinfeld modular forms, involving sums over Drinfeld modules. The authors introduce a renormalization of Hecke operators ($T_{p^n} = \wp_n^{k-l+1} T'_{p^n}$) to ensure the trace formula behaves well with respect to the weight and type.

B. Periodicity Analysis via Generating Functions

To prove periodicity modulo $\ell^s$, the authors analyze the sums appearing in the trace formula:

1. Decomposition: The moduli space is split into two sets:
   • $N(F_q)$: Curves where the trace $a_1(E) \equiv 0 \pmod \ell$.
   • $U(F_q)$: Curves where $a_1(E) \not\equiv 0 \pmod \ell$.
2. Combinatorial Periodicity: The terms involving binomial coefficients $\binom{k-j}{j}$ and powers of $a_1(E)$ are analyzed using:
   • Lucas's Theorem for Prime Powers: To handle congruences of binomial coefficients modulo $\ell^s$.
   • Generating Functions: The authors construct rational generating functions for the sums. They show that the denominators of these functions divide $x^M - 1$ modulo $\ell^s$ for a specific period $M$.
   • Lemma 3.8: A key lemma stating that if a generating function $g(x) = f(x)/d(x)$ has a denominator $d(x)$ dividing $x^n - 1$, then the coefficients $a_k$ are periodic with period $n$ (for large $k$).

C. Handling Non-Representable Levels

When the level structure $H$ is not representable (i.e., automorphism groups are non-trivial), the authors introduce a correction term $\nu_\ell(H, q)$ (the maximum $\ell$-adic valuation of the automorphism group size). This allows them to adjust the required weight threshold $k_0$ and the period $n$ to maintain the congruence.

3. Key Contributions and Results

A. Main Theorem for Elliptic Modular Forms (Theorem 1.1)

For a subgroup $H \subseteq GL_2(\mathbb{Z}/N\mathbb{Z})$, a prime $\ell$, and a prime power $q$ coprime to $N$:

• There exists a threshold $k_0$ (depending on $s, \ell, q$) such that for all $k \ge k_0$:
  $$\text{Tr}(F_q | S[H, k+2]) + \epsilon_k \equiv_{\ell^s} \text{Tr}(F_q | S[H, k+2+n]) + \epsilon_{k+n}$$
• The Period $n$: The period depends on the Legendre symbol $\left(\frac{q}{\ell}\right)$ and the parity of $\ell$:
  • If $\ell \ge 3$ and $\left(\frac{q}{\ell}\right) = -1$: $n = \ell^{s-1}(\ell^2 - 1)$.
  • If $\ell \ge 3$ and $\left(\frac{q}{\ell}\right) = 1$: $n = \ell^s(\ell^2 - 1)/2$.
  • If $\ell = 2$: The period involves LCMs of powers of 2 and $\ell^2-1$.
• Significance: This generalizes Serre's result (which was modulo primes) to prime powers and arbitrary levels, without assuming $p$-adic analytic methods apply.

B. Main Theorem for Drinfeld Modular Forms (Theorem 1.4)

For Drinfeld modular forms of level $K$ over a function field:

• Similar periodicity holds modulo $\ell^s$ (where $\ell$ is a prime ideal in the function field).
• The period $m$ is explicitly calculated based on the degree of the prime ideal and whether it equals the characteristic $p$.
• Renormalization: The use of renormalized Hecke operators is crucial for the Drinfeld case to align the trace formula with the periodicity of the weight.

C. Characteristic Polynomials and Slopes

• Proposition 5.5 & 5.7: The authors show that the characteristic polynomials of Hecke operators modulo $p$ are periodic in the weight: $f_k(x) \equiv_p f_{k+p-1}(x)$.
• Corollary 5.8: This implies a periodicity in the dimensions of the spaces of cusp forms with slope 0 (related to ordinary forms), recovering Hida's results and supporting the Gouvêa–Mazur conjecture for slope 0.

4. Significance and Impact

1. Unification: The paper provides a unified framework for understanding trace congruences in both classical (number field) and Drinfeld (function field) settings.
2. Extension of Serre's Work: It significantly extends Serre's 1996 results from modulo $p$ to modulo $p^s$, offering a deeper understanding of the arithmetic structure of Hecke traces.
3. Explicit Periods: Unlike previous qualitative results, this paper provides explicit formulas for the periods of congruence, which depend on the Legendre symbol and the prime power modulus.
4. Computational Utility: The results allow for the computation of traces in high weights by reducing them to a finite number of computations in lower weights (as demonstrated in Section 5).
5. Connection to Gouvêa–Mazur: The findings on characteristic polynomials and slope 0 dimensions provide concrete evidence and partial proofs related to the Gouvêa–Mazur conjecture, which predicts the behavior of families of modular forms in $p$-adic families.

5. Conclusion

Bergström and de Vries successfully demonstrate that the traces of Hecke operators exhibit a robust periodicity in the weight variable when considered modulo prime powers. By leveraging the geometry of modular curves and sophisticated combinatorial arguments on generating functions, they establish precise periods for these congruences, bridging the gap between classical modular forms and their Drinfeld counterparts. This work lays a foundation for further exploration of $p$-adic families and the arithmetic of modular forms in positive characteristic.