Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Traces of Hecke operators on Drinfeld modular forms for $\mathbb{F}_q[T]$

Sjoerd de Vries

TL;DR

This work develops a comprehensive trace-theoretic framework for Hecke operators on Drinfeld cusp forms of level 1 with $A=\\mathbb{F}_q[T]$, providing closed-form traces for primes of degree at most 2 and algorithmic strategies for higher degrees via isogeny-class data and Hurwitz class numbers. It reveals symmetry phenomena in weights, sharpens Ramanujan-type bounds, and connects traces to spectral properties, including eigenvalues, slopes, and injectivity results, with special treatment of characteristic 2 and odd characteristics. The paper also discusses the oldforms/newforms decomposition at level $\\Gamma_0(\\mathfrak p)$ under a multiplicity hypothesis, and it presents extensive computations (and Magma implementations) illustrating power-eigensystems, $A$-expansions, and explicit eigenvalues for primes of degree 1 and 2. Altogether, the results advance both the theoretical understanding of Drinfeld modular forms and the computational toolkit for their Hecke action, offering new avenues for decompositions and conjectural bounds in positive characteristic. The findings have implications for the structure of Hecke algebras in the function-field setting and for translating spectral data into arithmetic information via isogeny counts and Hurwitz-type class numbers.

Abstract

In this paper, we study traces of Hecke operators on Drinfeld modular forms of level 1 in the case $A = \mathbb{F}_q[T]$. We deduce closed-form expressions for traces of Hecke operators corresponding to primes of degree at most 2 and provide algorithms for primes of higher degree. We improve the Ramanujan bound and deduce the decomposition of cusp forms of level $Γ_0(\mathfrak{p})$ into oldforms and newforms, as conjectured by Bandini-Valentino, under the hypothesis that each Hecke eigenvalue has multiplicity less than $p$.

Traces of Hecke operators on Drinfeld modular forms for $\mathbb{F}_q[T]$

TL;DR

This work develops a comprehensive trace-theoretic framework for Hecke operators on Drinfeld cusp forms of level 1 with , providing closed-form traces for primes of degree at most 2 and algorithmic strategies for higher degrees via isogeny-class data and Hurwitz class numbers. It reveals symmetry phenomena in weights, sharpens Ramanujan-type bounds, and connects traces to spectral properties, including eigenvalues, slopes, and injectivity results, with special treatment of characteristic 2 and odd characteristics. The paper also discusses the oldforms/newforms decomposition at level under a multiplicity hypothesis, and it presents extensive computations (and Magma implementations) illustrating power-eigensystems, -expansions, and explicit eigenvalues for primes of degree 1 and 2. Altogether, the results advance both the theoretical understanding of Drinfeld modular forms and the computational toolkit for their Hecke action, offering new avenues for decompositions and conjectural bounds in positive characteristic. The findings have implications for the structure of Hecke algebras in the function-field setting and for translating spectral data into arithmetic information via isogeny counts and Hurwitz-type class numbers.

Abstract

In this paper, we study traces of Hecke operators on Drinfeld modular forms of level 1 in the case . We deduce closed-form expressions for traces of Hecke operators corresponding to primes of degree at most 2 and provide algorithms for primes of higher degree. We improve the Ramanujan bound and deduce the decomposition of cusp forms of level into oldforms and newforms, as conjectured by Bandini-Valentino, under the hypothesis that each Hecke eigenvalue has multiplicity less than .
Paper Structure (30 sections, 68 theorems, 197 equations, 3 figures, 3 tables, 1 algorithm)

This paper contains 30 sections, 68 theorems, 197 equations, 3 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Drinfeld modular forms for Fq[T]
  4. Hecke operators
  5. A-expansions
  6. Drinfeld modules over finite fields
  7. Traces
  8. Rewriting the trace formula
  9. Symmetry
  10. Primes of degree 1
  11. Traces in characteristic 2
  12. Traces in odd characteristic: primes of degree 2
  13. Odd characteristic: primes of higher degree
  14. Traces modulo P
  15. The strong Ramanujan bound
  16. ...and 15 more sections

Key Result

Theorem 1.1

Fix $k \geq 0$ and $l \in \mathbb Z$ such that $k+2 \equiv 2l \pmod{q-1}$. Then we have

Figures (3)

  • Figure 1: $\log_5( 1 + (k-6)/2 - \deg \mathop{\mathrm{Tr}}\nolimits(\mathbf{T}_{T}\space | \space \operatorname{S}_{k,3}))$ for $q=5$ and $18 \leq k \leq 1258$.
  • Figure 2: $\log_5 (1 + (k-6) - \deg \mathop{\mathrm{Tr}}\nolimits(\mathbf{T}_{T^2+T+2} \space | \space \operatorname{S}_{k,3}))$ for $q=5$ and $18 \leq k \leq 3950$.
  • Figure 3: $\deg \mathop{\mathrm{Tr}}\nolimits(\mathbf{T}_{T^5+2T+1} \space | \space \operatorname{S}_{k,0})$ for $q=3$ and $2 \leq k \leq 200$ with the strong Ramanujan bound.

Theorems & Definitions (175)

  • Theorem 1.1: Thm. \ref{['thm:deg1']}
  • Theorem 1.2: Thm. \ref{['thm:char2traces']}
  • Theorem 1.3: Thm. \ref{['thm:char2eigs']} and Thm. \ref{['thm:char2_repetition']}
  • Theorem 1.4: Thm. \ref{['thm:injectivity']}
  • Theorem 1.5: Cor. \ref{['cor:oldnew']}
  • Theorem 1.6: Thm. \ref{['thm:trivial_eigs']}
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Lemma 2.4
  • ...and 165 more