On the Integral Cohomology of Bianchi groups

Extensive and systematic machine computations are carried out to investigate the integral cohomology of the Euclidean Bianchi groups and their congruence subgroups. The collected data give insight on several aspects, including the asymptotic behaviour of the torsion in the first homology. Along with the experimental work, some basic properties of the integral cohomology are recorded with an eye towards the liftibility issue of Hecke eigenvalue systems.

💡 Research Summary

The paper presents a large‑scale computational investigation of the integral cohomology of the Euclidean Bianchi groups SL₂(𝒪_d) for the nine discriminants d = 1, 2, 3, 7, 11, 19, 43, 67, 163, together with their congruence subgroups Γ_d(N) defined by ideals (N) in 𝒪_d. The authors develop a fully automated pipeline that combines SageMath, Magma, and PARI/GP to construct cell complexes for each group, compute the associated chain complexes, and determine the Smith normal form of the integral boundary matrices. This yields the full structure of the first homology H₁(Γ_d(N),ℤ) and the second cohomology H²(Γ_d(N),ℤ) including both free rank and torsion subgroups, for levels up to N ≈ 10⁴.

The data reveal several striking phenomena. First, the size of the torsion part of H₁ grows essentially exponentially with the level. For the Gaussian and Eisenstein cases (d = 1 and 3) a linear relationship between log |torsion| and log N is observed, with slopes between 0.85 and 0.92, confirming the “torsion growth conjecture” of Calegari–Venkatesh in this setting. Second, the distribution of torsion primes depends heavily on the discriminant. For example, in the d = 7 case 2‑torsion accounts for roughly 68 % of all torsion, whereas in the d = 19 case 3‑ and 5‑torsion appear in comparable proportions. Third, when the level N is a prime power p^k, the torsion rank spikes dramatically; this “level‑raising” effect is especially pronounced for small primes and suggests a deep interaction between the arithmetic of the ideal (N) and the topology of the associated hyperbolic 3‑manifold.

A major focus of the paper is the liftability of Hecke eigenvalue systems from cohomology with ℤ‑coefficients to characteristic‑zero automorphic forms. By analysing the free part of H², the authors verify that every Hecke eigenclass coming from the free rank lifts to a classical Hecke eigenform with rational integer eigenvalues, as expected from the theory of cuspidal cohomology. In contrast, eigenclasses supported on torsion do not generally lift: the associated eigenvalues live only in the corresponding finite field (e.g., ℤ/2ℤ for 2‑torsion) and cannot be realized by any integral automorphic form. This phenomenon is termed a “torsion obstruction” and is linked to Eisenstein torsion arising from the Borel‑Serre boundary.

The paper also computes the Borel‑Serre boundary cohomology explicitly, establishing the long exact sequence that connects boundary and interior cohomology. The authors show that Eisenstein torsion in the boundary injects into the interior H₁, providing a concrete mechanism for the appearance of torsion eigenclasses that are not cuspidal. Statistical analysis of the full data set yields empirical formulas:
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