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Two results on Eisenstein part of homology and cohomology groups of Bianchi modular groups

Debargha Banerjee, Pranjal Vishwakarma

Abstract

We explicitly write down the {\it Eisenstein cycles} in the first homology groups of quotients of the hyperbolic three spaces as linear combinations of Cremona symbols (a generalization of Manin symbols) for imaginary quadratic fields. They generate the Eisenstein part of the homology groups. We also study the Eisenstein part of the cohomology groups. As an application, we find an asymptotic dimension formula (level aspect) for the cuspidal cohomology groups of congruence subgroups of the form $\Ga_1(N)$ inside the full {\it Bianchi groups}.

Two results on Eisenstein part of homology and cohomology groups of Bianchi modular groups

Abstract

We explicitly write down the {\it Eisenstein cycles} in the first homology groups of quotients of the hyperbolic three spaces as linear combinations of Cremona symbols (a generalization of Manin symbols) for imaginary quadratic fields. They generate the Eisenstein part of the homology groups. We also study the Eisenstein part of the cohomology groups. As an application, we find an asymptotic dimension formula (level aspect) for the cuspidal cohomology groups of congruence subgroups of the form inside the full {\it Bianchi groups}.
Paper Structure (24 sections, 18 theorems, 141 equations)

This paper contains 24 sections, 18 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Introduction
  3. Bianchi modular symbols and Cremona symbols
  4. Fundamental domains
  5. Differential forms
  6. Bianchi modular forms
  7. Eisenstein differential forms
  8. Ito’s differential forms MR870309 for full subgroup ${\mathrm {SL}}_2(\mathcal{O}_K)$
  9. Generlization of Ito’s differential forms MR870309 for subgroups of ${\mathrm {SL}}_2(\mathcal{O}_K)$
  10. Eisenstein differential forms of weight two
  11. Hida's differential forms
  12. Inner product formula for Bianchi modular forms
  13. Quasi-periods
  14. Eisenstein elements for $\Gamma_0(p)$
  15. Different pairings of homology and cohomology groups
  16. ...and 9 more sections

Key Result

Theorem 2

For any imaginary field $K$ with class number one that is also an Euclidean domain, the modular symbol is the Eisenstein element corresponding to the Eisenstein series $E \in E_2(\Gamma)$.

Theorems & Definitions (39)

  • Definition 1
  • Theorem 2
  • Theorem 3
  • Definition 4
  • Definition 5
  • Theorem 1
  • Proposition 6
  • proof
  • Corollary 7
  • proof
  • ...and 29 more