Bianchi Cusp Forms

• Bianchi cusp forms are automorphic forms for GL₂ over imaginary quadratic fields, generalizing classical modular forms with significant implications in arithmetic geometry.
• They are studied using a blend of analytic, algebraic, and topological methods to compute cohomology, establish dimension bounds, and derive explicit Hecke-module results.
• Their analysis underpins major aspects of the Langlands program, informs the theory of Galois representations, and distinguishes genuine forms from base-change or twist phenomena.

Bianchi cusp forms are automorphic forms for the group $\mathrm{GL}_2$ over imaginary quadratic fields, generalizing classical modular forms to the non-totally real case. These forms constitute a rich source of examples for the Langlands program, arithmetic geometry, the theory of Galois representations, and the study of special values of $L$-functions. Bianchi cusp forms realize the cuspidal cohomology of arithmetic hyperbolic 3-manifolds and admit arithmetic significance via the Eichler–Shimura–Harder isomorphism. Their study involves intricate intertwining of analytic, algebraic, and topological tools. The subject has seen major advances in dimension bounds, explicit cohomological and Hecke-module computations, $p$-adic $L$-functions, rationality theorems, and the detection of non-base-change ("genuine") phenomena.

1. Definition and Fundamental Properties

Let $F$ be an imaginary quadratic field with ring of integers $\mathcal{O}_F$. For a nonzero ideal $\mathfrak{n} \subset \mathcal{O}_F$, define the congruence subgroup: $$\Gamma_0(\mathfrak{n}) = \left\{ \begin{pmatrix} a & b \ c & d \end{pmatrix} \in \mathrm{GL}_2(\mathcal{O}_F) : c \equiv 0 \pmod{\mathfrak{n}} \right\}.$$ The complex three-dimensional hyperbolic space $$\mathbb{H}^3 \cong \mathrm{SL}_2(\mathbb{C})/\mathrm{SU}_2(\mathbb{C})$$ underpins the geometry, with $\Gamma_0(\mathfrak{n})$ acting discontinuously.

A (parallel weight $k$) Bianchi modular form of level $\Gamma_0(\mathfrak{n})$ is, analytically, a function $F : \mathbb{H}^3 \to V$ (for a suitable finite-dimensional $V$) transforming by the appropriate automorphy factor and satisfying a system of Casimir differential equations, cuspidality (vanishing of all constant Fourier–Bessel coefficients at cusps), and moderate growth. Weight-$2$ forms with values in $\mathbb{C}^3$ are of particular arithmetic interest. The space of weight-$k$ Bianchi cusp forms is denoted $S_{k,k}^F(\Gamma_0(\mathfrak{n}))$.

Cohomologically, the space is realized as: $$S_{k,k}^F(\Gamma_0(\mathfrak{n})) \cong H^1_{\mathrm{cusp}}(Y_{\Gamma_0}, \mathscr{V}_{k,k}),$$ where $Y_{\Gamma_0}$ is the relevant arithmetic 3-manifold and $\mathscr{V}_{k,k}$ is the canonical local system (Banerjee et al., 2023, Rahm et al., 2017).

2. Cohomology, Eisenstein and Cuspidal Decomposition

The Borel–Serre compactification yields: $$H^1(\Gamma, E) = H^1_{\mathrm{Eis}}(\Gamma, E) \oplus H^1_{\mathrm{cusp}}(\Gamma, E)$$ for any finite-dimensional coefficient $E$ (Banerjee et al., 2023, Sengun et al., 2012). The Eisenstein part is governed by boundary cohomology and explicit cycle computations (such as and Eisenstein cycles constructed using generalized Manin/Cremona symbols). The cuspidal part, via Harder’s isomorphism, corresponds to Bianchi cusp forms and is the primary arena for arithmetic phenomena.

For principal congruence subgroups and for $$\Gamma_1(N) := \{ \gamma \in SL_2(\mathcal{O}_K) : a \equiv d \equiv 1, c \equiv 0 \pmod{N} \}$$, explicit asymptotic lower bounds in the level aspect are: $$\dim H^1_{\mathrm{cusp}}(\Gamma_1(p^n), \mathbb{C}) \gg p^{3n}, \qquad \dim H^1_{\mathrm{cusp}}(\Gamma_1(N), \mathbb{C}) \gg N^3,$$ for $N$ coprime to the discriminant (Banerjee et al., 2023, Sengun et al., 2012).

3. Dimension Bounds and Multiplicity Growth

Sharp quantitative estimates for $\dim S_{k,k}(\Gamma_0(\mathfrak{n}))$ play a crucial structural role. In the weight (horizontal) aspect, the best general upper bound for any fixed level $K_f$ and imaginary quadratic $F$ is linear growth in $k$: $\dim_\mathbb{C} S_k(K_f) \ll k,$ confirming conjectures of Finis–Grunewald–Tirao and Calegari–Mazur (Fu, 2022). This is achieved by translating the problem to bounding coinvariants of completed Iwasawa modules under local analytic group action, microlocalization to completed enveloping algebras $U(\mathfrak{g})$, and precise PBW-filtration analysis. The lower bound $\dim S_{k,k}(\Gamma_0(\mathfrak{n})) \gg k$ is also proven for the full Bianchi group (Sengun et al., 2012). In the level (vertical) aspect, the lower bound is cubic in norm, as above.

Results for non-totally real fields show a uniform codimension-$1$ "multiplicity-freeness" phenomenon absent in the totally real case (Fu, 2022). Open directions include strengthening lower bounds and refining the exponent in the sharp upper bound.

4. Genuine and Non-Genuine Bianchi Cusp Forms

Automorphic forms in $S_{k,k}^F(\Gamma_0(\mathfrak{n}))$ split into non-genuine and genuine subspaces. Non-genuine forms comprise (a) base-change forms (Langlands transfer from elliptic modular forms on $\mathbb{Q}$), (b) twists by Hecke characters, and (c) CM-forms arising by automorphic induction from quadratic extensions of $F$. The dimensions of these subspaces admit closed formulas in many cases (Rahm et al., 2017).

The genuine subspace, defined as forms not accounted for by base-change, twist, or CM, is of particular Diophantine significance. Explicit computations reveal that genuine Bianchi cusp forms are exceedingly rare, especially at level one: out of $\sim 5,000$ newform spaces up to weight $30$, only $22$ showed genuine classes (Rahm et al., 2011, Rahm et al., 2017). For higher levels, dimension formulas for non-genuine forms allow identification of new genuine forms; examples include sporadic $2$-dimensional spaces for $F = \mathbb{Q}(\sqrt{-7})$ at $k=12$ and $F=\mathbb{Q}(\sqrt{-11})$ at $k=10$, as well as higher-level and lower-weight cases (Rahm et al., 2017). These genuine forms conjecturally correspond to simple abelian surfaces over $F$ that do not descend to $\mathbb{Q}$.

5. $L$-functions, Rationality, and $p$-adic Aspects

Bianchi cusp forms admit Fourier–Whittaker expansions, with Hecke eigenforms parameterizing global $L$-functions: $$L(s, f) = \prod_{\mathfrak{p}} (1 - a_{\mathfrak{p}} N\mathfrak{p}^{-s} + \cdots)^{-1},$$ where $a_{\mathfrak{p}}$ are Hecke eigenvalues (1908.10095, Thalagoda et al., 31 Jan 2025). Associated Asai $L$-functions, adjoint $L$-functions, and their algebraic and $p$-adic interpolations play central roles in arithmetic applications.

For the Asai $L$-function, Balasubramanyam–Ghate–Vangala proved a rationality result for critical L-values (after normalization by explicit period invariants), and constructed $p$-adic Asai $L$-functions via cohomological distributions, satisfying strong interpolation formulae: $$\int_{\mathbb{Z}_p^\times} \chi(x) x^m\,d\mu_{n-m+2}(x) = c(f, \chi, m) L(2n-m+2, \mathrm{As} f \otimes \chi)$$ (1908.10095). These results facilitate Iwasawa-theoretic attacks on main conjectures for Asai Galois representations.

For full-level forms in Euclidean fields, there exists an explicit Eichler–Shimura–Harder isomorphism relating cusp forms to cohomology, leading to a Manin-style rationality theorem for periods and special $L$-values: after dividing by a unique period $\Omega$, all normalized periods and special $L$-values are algebraic over the Hecke field (Anderson et al., 21 Sep 2025).

Recent work constructs $p$-adic adjoint $L$-functions on the Bianchi eigenvariety via overconvergent cohomology and Hecke-equivariant pairings, showing strong results on their nonvanishing and explicit interpolation to classical adjoint $L$-values (Lee et al., 2023).

6. Sato–Tate and Ramanujan Conjectures, Galois Representations

The Ramanujan–Petersson conjecture for Bianchi modular forms predicts that unramified Hecke eigenvalues are bounded by the optimal weight exponent, and the Sato–Tate conjecture asserts their normalized conjugacy classes in $SU(2)$ become equidistributed with Haar measure. Both conjectures have now been established for all parallel weight $\geq2$ Bianchi cusp forms (i.e., regular algebraic cuspidal automorphic representations of $\mathrm{GL}_2(\mathbb{A}_F)$) (Boxer et al., 2023). The proof leverages potential automorphy theorems for symmetric powers of compatible Galois representations, weight-zero automorphy-lifting, and the Harris tensor-product method. For such forms, the associated system of $\ell$-adic Galois representations is pure, crystalline, and attaches the expected Hodge–Tate weights and Frobenius eigenvalues.

7. Computation, Class Number, and Explicit Databases

Advances in computation have enabled explicit determination of Hecke modules of Bianchi modular forms over fields with higher class numbers, notably $F = \mathbb{Q}(\sqrt{-17})$, class group $C_4$ (Thalagoda et al., 31 Jan 2025). The computational pipeline synthesizes perfect-form tessellations (Ash–Koecher), Voronoi reduction, and cell-complex methods (Cremona–Rahm–Şengün, implemented for Magma) to build and diagonalize sparse Hecke operators, isolate newforms, and determine Hecke fields and twist structure.

Extensive data sets reveal specific patterns: full twist orbits, rare inner-twist systems, base-change matching, and Hecke fields of varying degree (including cubic and quartic). The lack of a simple closed formula for the cuspidal dimension in class number $4$ fields necessitates continued reliance on explicit homology computations. The same approach facilitates modularity proofs for elliptic curves and analysis of arithmetic invariants.

Level Norm	Homology Dimension	Newform Dimension
2.1	1	4
8.1	7	8
16.1	15	16

8. Open Problems and Directions

The rarity of genuine Bianchi cusp forms, the structure of their Galois orbits (a Maeda-type conjecture is suggested), and the properties of their associated motives (simple abelian surfaces over $F$ with everywhere good reduction) remain areas of ongoing investigation (Rahm et al., 2011, Rahm et al., 2017). The cohomological, analytic, and $p$-adic approaches are expected to yield further interactions, particularly in the context of equivariant rationality, Iwasawa theory, and the precise structure of overconvergent cohomology in families. Multiplicity bounds, optimal growth rates, and explicit dimension formulas in higher class number or non-parallel weight $k$ remain significant open directions.