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Bianchi period polynomials: Hecke action and congruences

Lewis Combes

TL;DR

This work develops a duality between weight $k$ Bianchi period polynomials for Euclidean imaginary quadratic fields and weight $k$ modular symbols, situating period polynomials as a concrete realization of the $H^2$-duality for Bianchi groups. It then endows the period polynomials with a Hecke action via Heilbronn matrices, commuting with the Manin surjection and yielding a Hecke module structure on $W_{k,k}$. Using this framework, the authors perform extensive computations to detect congruences among level 1 Bianchi cusp forms by examining period polynomials, and subsequently prove several congruences with base-change lifts and Eisenstein series. These results include the first demonstrated congruences between genuine higher-weight Bianchi cusp forms and both a base-change Bianchi form and an Eisenstein series, highlighting a novel computational pathway for congruences in the Bianchi setting. The work combines rigorous cohomological dualities with explicit Hecke actions to extract arithmetic information from period polynomials, with important implications for understanding the arithmetic of Bianchi modular forms and their congruence relations.

Abstract

Let $Γ$ be a Bianchi group associated to one of the five Euclidean imaginary quadratic fields. We show that the space of weight $k$ period polynomials for $Γ$ is ``dual'' to the space of weight $k$ modular symbols for $Γ$, reflecting the duality between the first and second cohomology groups of Bianchi groups. Using this result, we describe the action of Hecke operators on the space of period polynomials for $Γ$ via the Heilbronn matrices. In the second part of the paper, we numerically investigate congruences between level 1 Bianchi eigenforms via computer programs which implement the Hecke action on spaces of Bianchi period polynomials. Computations with the Hecke action are used to indicate moduli of congruences between the underlying Bianchi forms; we then prove the congruences using the period polynomials. From this we find congruences between genuine Bianchi modular forms and both a base-change Bianchi form and an Eisenstein series. We believe these congruences are the first of their kind in the literature.

Bianchi period polynomials: Hecke action and congruences

TL;DR

This work develops a duality between weight Bianchi period polynomials for Euclidean imaginary quadratic fields and weight modular symbols, situating period polynomials as a concrete realization of the -duality for Bianchi groups. It then endows the period polynomials with a Hecke action via Heilbronn matrices, commuting with the Manin surjection and yielding a Hecke module structure on . Using this framework, the authors perform extensive computations to detect congruences among level 1 Bianchi cusp forms by examining period polynomials, and subsequently prove several congruences with base-change lifts and Eisenstein series. These results include the first demonstrated congruences between genuine higher-weight Bianchi cusp forms and both a base-change Bianchi form and an Eisenstein series, highlighting a novel computational pathway for congruences in the Bianchi setting. The work combines rigorous cohomological dualities with explicit Hecke actions to extract arithmetic information from period polynomials, with important implications for understanding the arithmetic of Bianchi modular forms and their congruence relations.

Abstract

Let be a Bianchi group associated to one of the five Euclidean imaginary quadratic fields. We show that the space of weight period polynomials for is ``dual'' to the space of weight modular symbols for , reflecting the duality between the first and second cohomology groups of Bianchi groups. Using this result, we describe the action of Hecke operators on the space of period polynomials for via the Heilbronn matrices. In the second part of the paper, we numerically investigate congruences between level 1 Bianchi eigenforms via computer programs which implement the Hecke action on spaces of Bianchi period polynomials. Computations with the Hecke action are used to indicate moduli of congruences between the underlying Bianchi forms; we then prove the congruences using the period polynomials. From this we find congruences between genuine Bianchi modular forms and both a base-change Bianchi form and an Eisenstein series. We believe these congruences are the first of their kind in the literature.
Paper Structure (48 sections, 5 theorems, 109 equations, 3 figures, 1 table)

This paper contains 48 sections, 5 theorems, 109 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Data availability statement
  4. Structure of the paper
  5. Acknowledgements
  6. Cohomology of Bianchi groups
  7. Bianchi groups
  8. The case $d=1$
  9. The case $d=2$
  10. The case $d=3$
  11. The case $d=7$
  12. The case $d=11$
  13. The weight modules
  14. Second cohomology
  15. The case $d=1$
  16. ...and 33 more sections

Key Result

Lemma 2.4

For our Bianchi groups $\Gamma$, we have an isomorphism ${\rm St} \simeq \mathcal{M}_2$ of $\Gamma$-modules.

Figures (3)

  • Figure 1: Congruence graph of Hecke eigenvalue systems captured in $S_{12}(\mathop{\mathrm{\mathrm{PGL}}}\nolimits_2(\mathcal{O}_{11}))$.
  • Figure 2: Hecke eigenvalue systems captured in $S_{12}(\mathop{\mathrm{\mathrm{PGL}}}\nolimits_2(\mathcal{O}_{11}))$.
  • Figure 3: The "normalised" algebraic period polynomial of $\Delta$ over $\mathbb{Q}(\sqrt{-11})$.

Theorems & Definitions (15)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Lemma 3.1
  • Corollary 4.1
  • ...and 5 more