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Periodicity of traces of Hecke operators modulo prime powers

Jonas Bergström, Sjoerd de Vries

TL;DR

The paper proves that traces of Hecke operators on spaces of cusp forms—both elliptic and Drinfeld—are periodic in the weight modulo prime powers, extending Serre’s primes-to-primes results to prime powers and to function-field settings. The method hinges on Deligne’s trace formula and a careful decomposition of traces into Eisenstein-like and geometric contributions, together with Lucas-type congruences for binomial sums to control the weight-shift. The authors develop renormalised Hecke operators in the Drinfeld setting to align Frobenius eigenvalues with level data, enabling parallel periodicity results across characteristics. The findings bridge classical and function-field arithmetic, giving explicit periods dependent on the prime and the residue characteristics, and they provide concrete examples and computational methods for checking congruences and analyzing Eichler–Shimura-type relations across a broad range of levels and weights.

Abstract

We study traces of Hecke operators on spaces of elliptic cusp forms and Drinfeld cusp forms and show that, modulo any prime power, these traces are periodic in the weight.

Periodicity of traces of Hecke operators modulo prime powers

TL;DR

The paper proves that traces of Hecke operators on spaces of cusp forms—both elliptic and Drinfeld—are periodic in the weight modulo prime powers, extending Serre’s primes-to-primes results to prime powers and to function-field settings. The method hinges on Deligne’s trace formula and a careful decomposition of traces into Eisenstein-like and geometric contributions, together with Lucas-type congruences for binomial sums to control the weight-shift. The authors develop renormalised Hecke operators in the Drinfeld setting to align Frobenius eigenvalues with level data, enabling parallel periodicity results across characteristics. The findings bridge classical and function-field arithmetic, giving explicit periods dependent on the prime and the residue characteristics, and they provide concrete examples and computational methods for checking congruences and analyzing Eichler–Shimura-type relations across a broad range of levels and weights.

Abstract

We study traces of Hecke operators on spaces of elliptic cusp forms and Drinfeld cusp forms and show that, modulo any prime power, these traces are periodic in the weight.
Paper Structure (19 sections, 34 theorems, 120 equations)

This paper contains 19 sections, 34 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. The main theorems
  3. Trace formulae
  4. Modular curves
  5. Drinfeld modular curves
  6. Renormalising the Hecke operators
  7. Proof of the main theorem for elliptic modular forms
  8. The case of $\ell \geq 3$ and $\ell \nmid q$
  9. The case of elltwo
  10. The case of ellmidq
  11. The case when $H$ is not representable and ell23
  12. Proof of the main theorem for Drinfeld modular forms
  13. The case of $\mathfrak{l} \neq \mathfrak{p}$
  14. The case of $\mathfrak{l} = \mathfrak p$
  15. Traces modulo infty
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $N\geq 1$ and let $H$ be a subgroup of $\mathrm{GL}_2(\mathbb Z/N\mathbb Z)$. Let $\ell$ be a prime and if $H$ is not representable then let $\ell \geq 5$. Let $q=p^a$ be a prime power with $\gcd(q,N)=1$. Let $s \geq 1$, and If $k \geq k_0$ then where and where $\epsilon_k=0$ for any $k>0$ and $\epsilon_0=-(q+1) \mathop{\mathrm{Tr}}\nolimits(F_q \, | \, H_c^0(\mathcal{X}_H\otimes \overline{

Theorems & Definitions (81)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • ...and 71 more