Bianchi Period Polynomials

• Bianchi period polynomials are algebraic objects attached to Bianchi modular forms, encoding arithmetic data over imaginary quadratic fields.
• They are constructed via explicit period integrals, with an Eichler–Shimura–Harder isomorphism linking cusp forms to cohomology of hyperbolic 3-manifolds.
• Their structured Hecke module action and congruence properties facilitate computational advances and reveal deep connections to special L-values.

Bianchi period polynomials are algebraic objects associated to Bianchi modular forms—automorphic forms for $\mathrm{SL}_2(\mathcal{O}_K)$, where $\mathcal{O}_K$ is the ring of integers of an imaginary quadratic field $K$. They arise as analogues of classical period polynomials for modular forms, now capturing the intricate arithmetic and cohomological structure governing automorphic forms over hyperbolic $3$-manifolds. Bianchi period polynomials encode special values of $L$-functions, reflect dualities in cohomology, and provide a natural setting for exploring congruences, Hecke module structure, and rationality phenomena analogous to those in the classical theory.

1. Cohomological Foundations and Eichler–Shimura–Harder Isomorphism

The theoretical underpinning of Bianchi period polynomials is provided by an explicit Eichler–Shimura–Harder isomorphism which identifies cusp forms of parallel weight $(k,k)$ on Bianchi groups $\mathrm{SL}_2(\mathcal{O}_K)$ with systems in the cohomology of the associated hyperbolic $3$-manifold $Y_\Gamma$. The correspondence is realized as

$$S_{k,k}(\Gamma) \cong H^1_{\mathrm{cusp}}(Y_\Gamma,\mathcal{V}_{k,k}),$$

where $\mathcal{V}_{k,k}$ is the local system of bi-homogeneous polynomials $V_{k,k}$ in $(X, Y, \overline{X}, \overline{Y})$.

Given a cusp form $F$, its period polynomial is explicitly constructed via a period integral of the attached differential form $\omega_F$:

$$r(F)(X,Y,\overline{X},\overline{Y}) = \int_0^\infty \omega_F = \sum_{p,q=0}^k \binom{k}{p}\binom{k}{q} r_{p,q}(F) X^{k-p}Y^p\overline{X}^{k-q}\overline{Y}^q,$$

where each coefficient $r_{p,q}(F)$ encodes analytic data via integrals of Fourier–Bessel components of $F$.

The isomorphism $F \mapsto r(F)$ gives a canonical identification with a quotient of the period polynomial space $\widetilde{W}_{k,k}$, after modding out by the coboundary subspace. Evaluation at the order-two element $S$ in the Hecke–Tamagawa group yields a concrete realization of the period map (Anderson et al., 21 Sep 2025).

2. Algebraic Structure and Defining Relations

Bianchi period polynomials are characterized by a system of linear relations imposed on $V_{k,k}$, reflecting the parabolic cocycle conditions for $\mathrm{SL}_2(\mathcal{O}_K)$. For Euclidean fields $$K = \mathbb{Q}(i), \mathbb{Q}(\sqrt{-2}), \mathbb{Q}(\sqrt{-3}), \mathbb{Q}(\sqrt{-7}), \mathbb{Q}(\sqrt{-11})$$, these relations generalize classical modular ones and are of the form

$$P|_{I + S} = P|_{I - L} = P|_{I + U + U^2} = P|_{I + T_\omega SL + (T_\omega SL)^2} = 0,$$

where the notation reflects group relations adapted to the specific arithmetic of $\mathcal{O}_K$ (Combes, 2023, Anderson et al., 21 Sep 2025).

This system defines a subspace $W_{k,k} \subset V_{k,k}$ of admissible Bianchi period polynomials. Upon quotient by the coboundary space, one obtains $\widetilde{W}_{k,k}$ as the true period polynomial domain parametrizing cohomological data of Bianchi forms.

3. Hecke Operators: Heilbronn Matrices and Module Structure

Bianchi period polynomials admit a natural Hecke module structure, analogous to the classical case but adapted to the geometry of $\mathcal{O}_K$. The Hecke action is described explicitly via sums over Heilbronn matrices:

$$T_\mathfrak{p}(P) = \sum_{g \in H_\mathfrak{p}} P|g,$$

where $H_\mathfrak{p}$ consists of matrices $$g = \begin{pmatrix} a & b \ c & d \end{pmatrix}$$ with norm and determinant conditions corresponding to $\mathfrak{p}$ (a prime in $\mathcal{O}_K$).

This action descends to $\widetilde{W}_{k,k}$ and is compatible with the adjunction property:

$$(P|T_\mathfrak{p}, Q) = (P, Q|T_\mathfrak{p}^\sharp),$$

where $T_\mathfrak{p}^\sharp$ is defined using conjugate-transpose matrices (Combes, 2023).

Hecke equivariance allows direct transport of Hecke eigenvalue information, and enables numerical calculation and congruence detection in the period polynomial space.

4. Special Values of L-functions and Rationality

Each Bianchi period encodes special values of the associated $L$-function. The integral formula

$$r_{p,q}(F) = 2 \cdot {2k + 2 \choose k + p - q + 1}^{-1} (-1)^{k+q+1} \int_0^\infty t^{p+q} F_{k + p - q + 1}(0, t) dt$$

relates the analytic data of the cusp form $F$ to critical $L$-values, in analogy with the formula $r_n(f) = n! (2\pi)^{-n-1} L(f, n+1)$ for classical forms.

A principal theorem asserts that, after normalization by a period $\Omega$, the coefficients $(1/\Omega) r_{p,q}(F)$ lie in $K(F)$, the extension generated by $K$ and the Fourier coefficients of $F$. This is the Bianchi analogue of Manin's rationality theorem, and is proven via explicit module-theoretic arguments leveraging Hecke action matrices (Anderson et al., 21 Sep 2025).

5. Duality with Modular Symbols and Cohomological Implications

There exists a duality between Bianchi period polynomials and weight $k$ modular symbols. Notably,

$W_{k,k} = (\ker \Phi_k)^\perp$

with orthogonality taken with respect to a natural bilinear pairing $(P, Q) = (ad - bc)^k$.

This duality is a direct manifestation of Borel–Serre duality between $H^1$ and $H^2$ cohomology for Bianchi groups, and mirrors the modular symbol–period polynomial correspondences in the classical setting (Combes, 2023). The computational transfer of Hecke actions via modular symbols to periods is established.

6. Dimension Bounds, Multiple Zeta Relations, and Motives

The interplay between Bianchi period polynomials and multiple zeta values (MZVs) mirrors that in the classical case. Structural relations in the period polynomial space provide upper bounds for the dimension of spaces spanned by triple zeta values of even weight:

$$\dim \langle \zeta(s_1, s_2, s_3) : s_1 + s_2 + s_3 = 2k \rangle \leq \text{(bound from period polynomials)}$$

as derived by establishing isomorphisms between kernel spaces of explicit combinatorial matrices and period polynomial relations (Ma et al., 2016, Ma et al., 2017).

This correspondence, though technical, demonstrates that the algebraic constraints in the Bianchi setting impose arithmetic dependencies among values of generalized MZVs, reflecting automorphic symmetries and motives.

7. Congruences, Computational Applications, and Extended Frameworks

Numerical investigations—via platforms such as Magma—have been undertaken to paper Hecke eigenstructure and congruences in Bianchi period polynomial spaces. The explicit computation of period polynomials and their reduction modulo prime ideals has enabled the detection of congruences between genuine Bianchi cusp forms, their base-changes, and Eisenstein series (Combes, 2023).

Moreover, practical routines for calculating Hecke action matrices and analyzing period vector spaces have yielded evidence for new types of arithmetic congruences, unifying cohomological and analytic frameworks in the modular forms landscape.

---

Bianchi period polynomials function as central algebraic objects in the paper of automorphic forms over imaginary quadratic fields. They provide an explicit bridge between cohomology, special $L$-values, modular symbol theory, and arithmetic algebraicity. Their structure, symmetries, and computational tractability reveal deep parallels with—and substantial extensions of—the classical period polynomial theory, and place them as keystones in modern explorations of arithmetic geometry and the theory of motives.