Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Computations directly on the cuspidal cohomology of congruence subgroups of $\mathrm{SL}(3, \mathbb{Z})$

Zachary Porat

TL;DR

This work develops a direct, cuspidal-focused method for computing Hecke actions on the third cohomology of prime-level congruence subgroups of SL(3, Z) by translating automorphic cusp data into a tractable vector-space framework. It constructs an explicit annihilator for the non-cuspidal component and works with the quotient W/W^nc to isolate cuspidal cohomology, enabling Hecke computations without contamination from non-cuspidal eigenvalues. The authors implement modular-symbol–based machinery and finite-field linear algebra to determine cuspidal dimensions and eigenvalues for p<3500, obtaining new levels with nonzero cuspidal classes and local L-factor data for five levels, with splitting fields typically imaginary quadratic. The approach provides a practical pathway to extracting local factors of associated GL(3) L-functions and offers insights into the structure and dimension of cuspidal cohomology across low prime levels.

Abstract

Ash, Grayson, and Green [J. Number Theory 19 (1984), pp. 412-436] compute the action of Hecke operators on a certain subspace of the cohomology of low-level congruence subgroups of $\mathsf{SL}(3, \mathbb{Z})$. This subspace contains the cuspidal cohomology, which is of primary interest. We extend their work, introducing a method that allows for computing the action of Hecke operators directly on the cuspidal cohomology. Using this method, we obtain data for prime level less than 3500, finding seven additional levels at which nonzero cuspidal classes appear and calculating local factors for five of these levels.

Computations directly on the cuspidal cohomology of congruence subgroups of $\mathrm{SL}(3, \mathbb{Z})$

TL;DR

This work develops a direct, cuspidal-focused method for computing Hecke actions on the third cohomology of prime-level congruence subgroups of SL(3, Z) by translating automorphic cusp data into a tractable vector-space framework. It constructs an explicit annihilator for the non-cuspidal component and works with the quotient W/W^nc to isolate cuspidal cohomology, enabling Hecke computations without contamination from non-cuspidal eigenvalues. The authors implement modular-symbol–based machinery and finite-field linear algebra to determine cuspidal dimensions and eigenvalues for p<3500, obtaining new levels with nonzero cuspidal classes and local L-factor data for five levels, with splitting fields typically imaginary quadratic. The approach provides a practical pathway to extracting local factors of associated GL(3) L-functions and offers insights into the structure and dimension of cuspidal cohomology across low prime levels.

Abstract

Ash, Grayson, and Green [J. Number Theory 19 (1984), pp. 412-436] compute the action of Hecke operators on a certain subspace of the cohomology of low-level congruence subgroups of . This subspace contains the cuspidal cohomology, which is of primary interest. We extend their work, introducing a method that allows for computing the action of Hecke operators directly on the cuspidal cohomology. Using this method, we obtain data for prime level less than 3500, finding seven additional levels at which nonzero cuspidal classes appear and calculating local factors for five of these levels.
Paper Structure (8 sections, 6 theorems, 23 equations, 3 tables)

This paper contains 8 sections, 6 theorems, 23 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Vector Space Interpretation
  4. Modular Symbols
  5. Main Results
  6. Computational Method
  7. Computation of Dimension of Cuspidal Cohomology
  8. Computation of Hecke Operators

Key Result

Theorem 2.1

There exists an isomorphism of vector spaces $\Phi \colon W \to H_3(\Gamma, \mathbb{C})$, where $W$ denotes the vector space of functions $f \colon \mathbb{P}^2(\mathbb{Z}/p\mathbb{Z}) \to \mathbb{C}$ that satisfy the following properties:

Theorems & Definitions (15)

  • Theorem 2.1: AGG84*Thm. 3.2, Prop. 3.12
  • Definition 2.2: AGG84*Def. 3.10, Def. 3.13
  • Definition 2.3: AGG84*Def. 3.15
  • Definition 2.4: AGG84*Def. 3.16
  • Theorem 2.5: AGG84*Thm. 3.19
  • Theorem 2.6: AGG84*Sum. 3.23
  • Theorem 2.7: AGG84*Prop. 3.24
  • Remark
  • Remark
  • Definition 3.1
  • ...and 5 more