Manin’s Rationality in Bianchi Periods

• Manin’s Rationality Theorem for Bianchi Periods is a framework that generalizes classical modular rationality to modular forms and cohomological data from Bianchi groups.
• It leverages explicit Hecke operator actions and period polynomial constructions to connect period integrals, L-values, and the topology of hyperbolic 3-manifolds.
• After normalization by a complex period, the theorem implies that the critical L-values and period coefficients are algebraic, deepening the link between arithmetic and geometry.

Manin’s Rationality Theorem for Bianchi Periods is a conjectural framework and a collection of results—now proven in several cases—that generalizes Manin’s classical rationality theorem for modular forms over $\mathbb{Q}$ to the setting of modular forms and cohomological structures arising from Bianchi groups, i.e., $\mathrm{PSL}_2(\mathcal{O}_K)$ for imaginary quadratic fields $K$. This theorem—along with its computational, geometric, and arithmetic foundations—connects period integrals, cohomology of hyperbolic 3–manifolds, Hecke operators, automorphic forms, and special values of $L$-functions in the non-cocompact, higher-dimensional setting.

1. Foundations: Bianchi Groups, Cohomology, and Periods

Bianchi groups $\Gamma = \mathrm{PSL}_2(\mathcal{O}_K)$, where $\mathcal{O}_K$ is the ring of integers in an imaginary quadratic field $K$, act properly discontinuously on hyperbolic 3–space $\mathbb{H}^3$. The quotient $Y_\Gamma = \Gamma \backslash \mathbb{H}^3$ forms a noncompact 3–orbifold, with its (co)homology exhibiting rich arithmetic and geometric properties. For any finite-dimensional (complex) representation $V$ of $\Gamma$, one has: $H^i(\Gamma, V) \cong H^i(Y_\Gamma, \mathcal{V}),$ where $\mathcal{V}$ is the locally constant sheaf on $Y_\Gamma$ attached to $V$.

The cuspidal cohomology, consisting of classes restricting trivially to the boundary, encodes the most significant number-theoretic information. The virtual cohomological dimension is two, and so the interesting classes occur in degrees 1 and 2. Periods in this setting refer to the integrals of Bianchi modular forms along 1–cycles in $Y_\Gamma$, or, equivalently, to the cohomological data computed with explicit fundamental domains, leading to realizations in homological invariants $H_1(\Gamma, \mathbb{Z})$ (Sengun, 2012).

2. Hecke Operators and the Structure of Period Spaces

Analogous to the classical modular group, double coset correspondences define Hecke operators on the cohomology: $$T_g = s_g^* \circ r_g^*: H^i(M, \mathcal{V}) \to H^i(M, \mathcal{V}),$$ for $g \in \mathrm{GL}_2(K)$. The Hecke algebra generated by such operators is commutative and acts semisimply on cuspidal cohomology, organizing the eigenvalue systems and connecting (co)homology classes to automorphic forms.

A central element in the algebraic structure is the space of Bianchi period polynomials:

• For Euclidean fields $K$, Karabulut constructs a concrete subspace $W_{k,k} \subset V_{k,k}$, bihomogeneous polynomials in $(X,Y,\overline{X}, \overline{Y})$, with $V_{k,k}$ the space of bidegree $(k,k)$.
• The quotient $$\widetilde{W}_{k,k} = W_{k,k} / \langle X^k\overline{X}^k - Y^k \overline{Y}^k \rangle$$ provides the algebraic receptacle for period polynomials (Anderson et al., 21 Sep 2025), mirroring classical period polynomial theory for modular forms.

Hecke actions on periods are explicitly described and remain compatible with the module structure. For $\mathfrak{p} = (\pi)$ a prime ideal, the Hecke operator acts via matrices $A(n)$ with entries in $\mathcal{O}_K$ mapping periods as $r(T_{\mathfrak{n}}F) = A(n) \cdot r(F)$ (Anderson et al., 21 Sep 2025).

3. Eichler–Shimura–Harder Isomorphism and Explicit Period Formulas

The Eichler–Shimura–Harder Isomorphism in the Bianchi setting provides an explicit correspondence: $$S_{k,k}(\Gamma) \cong H^1_{\mathrm{cusp}}(Y_\Gamma, \mathcal{V}_{k,k}) \cong H^1_{\mathrm{par}}(\Gamma, V_{k,k}) \cong \widetilde{W}_{k,k}.$$ Here, $S_{k,k}(\Gamma)$ is the space of Bianchi cusp forms of weight $(k,k)$.

Period coefficients $r_{p,q}(F)$ associated to a normalized cusp form $F$ are given by

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

indexed by $0 \leq p, q \leq k$, with $F_m(0,t)$ denoting suitable Fourier–Bessel coefficients. These enter the expansion: $$r(F)(X, Y, \overline{X}, \overline{Y}) = \sum_{p,q=0}^k \binom{k}{p}\binom{k}{q} r_{p,q}(F) X^{k-p}Y^p \overline{X}^{k-q} \overline{Y}^q.$$ This explicit period polynomial captures the critical $L$-values $L(F, m)$ via the periods $r_{p,q}(F)$ (Anderson et al., 21 Sep 2025).

4. Rationality Theorem: Statement and Key Ingredients

Manin’s Rationality Theorem for Bianchi Periods (Euclidean case and certain base change situations):

Let $F$ be a normalized Bianchi Hecke eigenform of weight $(k,k)$ with Fourier coefficients in $K(F)$. Then

$$\frac{1}{\Omega} r_{p,q}(F) \in K(F) \quad \text{for all } 0 \leq p,q \leq k$$

for some nonzero complex period $\Omega$ (depending algebraically on $F$). Equivalently, after normalizing by $\Omega$, all periods are algebraic numbers, and, consequently, the corresponding critical $L$-values are algebraic multiples of $\Omega$.

This result is established using:

• The explicit Eichler–Shimura–Harder isomorphism to connect cohomology and periods.
• The construction of period polynomials in $\widetilde{W}_{k,k}$ satisfying invariance under generators (e.g., $P|_{I+S}=0$ and $P|_{I+U+U^2}=0$).
• Detailed calculation of Hecke operators on these period spaces, showing their $\mathcal{O}_K$-linearity and compatibility with period data (Anderson et al., 21 Sep 2025).

Moreover, in the case of base-change forms from classical modular forms $f$ over $\mathbb{Q}$, the rationality properties of the Bianchi periods are reduced to those of $f$, as in the proof via Stark-Heegner cycles and comparison of étale Abel–Jacobi images (Venkat, 2021).

5. Connections to Cohomology, Torsion, and Galois Representations

The arithmetic of Bianchi periods is further illuminated by their role in cohomology and arithmetic geometry:

• Homological Realization: Bianchi periods are contained in the first homology (or abelianization) of the group; the kernel of restriction to the boundary—cuspidal cohomology—contains the periods with arithmetic significance (Sengun, 2012).
• Hecke Modules and Galois Representations: The Hecke eigensystems arising in cohomology are attached to Galois representations of $G_K = \mathrm{Gal}(\overline{K}/K)$, generalizing the Eichler–Shimura formalism. Thus, the rationality of periods reflects algebraicity properties of eigenvalues and special values of associated $L$-functions.
• Torsion: Computational evidence indicates the existence of large $p$-torsion in $H_1(\Gamma, \mathbb{Z})$ for $p > 3$ even though the corresponding groups have only 2- or 3-torsion elements (Sengun, 2012). This phenomenon, though not entirely explained in all cases, may correspond to mod $p$ forms invisible to characteristic zero cohomology, and raises subtle questions about the realization of periods in torsion classes.

Such structures are essential for any period rationality theorem encompassing non-characteristic zero phenomena and congruences (Combes, 2023).

6. Implications for $L$-values, Modular Symbols, and Congruences

Manin’s rationality philosophy, as extended to Bianchi periods, has direct impact on the algebraicity of special $L$-values of Bianchi modular forms:

• The period polynomial construction encodes critical values of $L$-functions attached to $F$. The rationality theorem ensures that, up to a period, these $L$-values are algebraic over the field of coefficients.
• Modular symbols, dual to period polynomials (Combes, 2023), provide an explicit computational tool for extracting periods and congruence relations between eigenforms. The Hecke-equivariant isomorphism guarantees that period polynomial congruences reflect congruences between modular forms, including between genuine and base-change forms as shown through explicit numerical experiments.
• Formalism for higher-depth (bi-period) relations and irrationality of period ratios, as in the work on Manin’s relations of order 2 (Provost, 2017), provides deeper structural understanding relevant to the rationality theorem.

These interrelations extend to Stark–Heegner cycles and $p$-adic $L$-functions, where rationality statements about period images in appropriate Galois cohomology can be reduced to explicit period data (Venkat, 2021, 1908.10095).

7. Broader Framework, Open Questions, and Generalizations

The rationality results for Bianchi periods—building on computational topology, algebraic number theory, and automorphic forms—anchor a suite of conjectures and further questions:

• Quantitative conjectures relating the asymptotics of torsion growth in cohomology to the volume of Bianchi quotients and to the expected rationality structure (Sengun, 2012).
• The search for a full Rademacher symbol theory for Bianchi groups, akin to the Manin–Drinfeld approach for modular curves, as a path toward explicit period rationality (Burrin, 2020).
• The extension of mixed Tate motive and period structures toward Bianchi IX cosmological models in mathematical physics, highlighting that the arithmetic of periods is not limited to group cohomology but applies to periods of geometric and physical significance (Fan et al., 2017).
• The full generalization to non-Euclidean imaginary quadratic fields, where the computational and theoretical landscape remains more complex and several ingredients are less explicit or conjectural.

The framework laid out by Manin’s Rationality Theorem for Bianchi periods, and the computational-arithmetic machinery brought to bear in recent results, has become central for understanding $L$-values, rationality phenomena, and hidden structures in three-dimensional arithmetic geometry.