An automorphic criterion for fibrations of arithmetic hyperbolic 3-manifolds (Dunfield–Ramakrishnan, Question 1.7)

Dunfield and Ramakrishnan (Amer. J. Math. 132, 2010) constructed a closed arithmetic hyperbolic 3-manifold with a tower of finite covers in which the number of fibered faces of the Thurston norm ball grows to infinity faster than any power of the logarithm of the covering degree. Their mechanism transfers a CM modular form with systematically vanishing Hecke eigenvalues to a quaternionic arithmetic manifold, and combines that arithmetic input with Fried's dynamical characterization of fibered faces; however, the crucial initial fact that the manifold fibers was established by topological computation, not by automorphic reasoning. The paper's explicitly posed Question 1.7 asks for an automorphic criterion: an intrinsic condition on the automorphic data attached to a cohomology class of an arithmetic hyperbolic 3-manifold — for example on its Hecke eigenvalues, local components, or transfer behavior to inner forms — that is sufficient for the class to be represented by a fibration over the circle. Question 1.7 remains open.

Background

A hyperbolic 3-manifold is a quotient $M=\Gamma\backslash\mathbb{H}^3$ of hyperbolic 3-space by a torsion-free discrete subgroup $\Gamma\subset \mathrm{PSL}(2,\mathbb{C})$. It is arithmetic when $\Gamma$ is commensurable with the unit group of an order in a quaternion algebra (a central division algebra of dimension 4) over a number field — over an imaginary quadratic field $K$ for the manifolds relevant here. Arithmetic manifolds have an exceptionally large commensurator: the elements of $\mathrm{PSL}(2,\mathbb{C})$ that conjugate some finite-index subgroup of $\Gamma$ back into $\Gamma$ form a dense subgroup, and a single commensurator element produces a pair of finite covers $p,q\colon N\to M$.

A 3-manifold fibers over the circle when it is a surface bundle, i.e. a mapping torus $M\cong S\times[0,1]/(x,0)\sim(\varphi(x),1)$ for a homeomorphism $\varphi$ of a surface $S$ (the monodromy); by Thurston's hyperbolization, when $M$ is hyperbolic $\varphi$ is pseudo-Anosov (a mapping class with a transversely measured unstable foliation and exponentially growing orbits). A fibration determines an integral cohomology class $\omega\in H^1(M;\mathbb{Z})$, and the fibered classes are organized by the Thurston norm: for $\phi\in H^1(M;\mathbb{R})$, $x(\phi)$ is the minimum of $\sum_i\max(0,-\chi(S_i))$ over properly embedded surfaces dual to $\phi$ (here $\chi$ is the Euler characteristic). The unit ball of $x$ is a finite polytope, and Thurston proved that the fibered classes are exactly the integral classes lying in the open cones over certain top-dimensional faces, called fibered faces. Fried's dynamical characterization states that two fibrations of a closed hyperbolic 3-manifold lie over the same fibered face if and only if the pseudo-Anosov flows transverse to them are isotopic; the number of fibered faces thus measures the number of essentially different fibrations.

For an arithmetic $M$, each pair of covers $p,q\colon N\to M$ defines a Hecke operator $T_{p,q}=q_*\circ p^*$ on $H^1(M)$, where $p^*$ is pullback and $q_*$ is the transfer map. Dunfield and Ramakrishnan proved (their Lemma 1.3): if $\omega$ is a fibered class of a closed hyperbolic $M$ and $T_{p,q}(\omega)=0$, then $p^*(\omega)$ and $q^*(\omega)$ lie on distinct fibered faces of $N$. They then built a closed arithmetic manifold for which this applies infinitely often. Starting from the elliptic curve $X_0(49)$ (the modular curve parametrizing elliptic curves with a cyclic subgroup of order 49, here itself an elliptic curve), which has complex multiplication (CM) by the ring of integers of $\mathbb{Q}(\sqrt{-7})$ — so that the weight-2 modular form $f$ attached to it (its holomorphic differential; the Fourier coefficients $a_p$ at primes $p$ are its Hecke eigenvalues) has $a_p=0$ for every rational prime inert in $\mathbb{Q}(\sqrt{-7})$, a set of density $1/2$; the primes used in the construction, those inert in $\mathbb{Q}(\sqrt{-7})$ and split in the base-change field $K=\mathbb{Q}(\sqrt{-3})$, form a set of density $1/4$ — they applied base change (the passage from an automorphic form over $\mathbb{Q}$ to one over $K$), and then the Jacquet–Langlands correspondence (which transfers automorphic representations of $\mathrm{GL}(2)$ to inner forms such as the quaternion algebra $D/K$ ramified exactly at the two primes above 7), obtaining a cusp form contributing to $H^1(M)$ of the manifold $M=X(7)$, whose fundamental group is a quotient of the principal congruence subgroup of level 7 of a maximal order of $D$. A delicate computation showed that $M$ fibers, with a class of this automorphic type; each prime with vanishing eigenvalue then doubles the number of fibered faces in the associated cover, giving covers $M_n$ of degree $d_n$ in which the number of fibered faces is at least $\exp(0.3\,\log d_n/\log\log d_n)$.

Throughout, the cohomology $H^1(M;\mathbb{C})$ of an arithmetic manifold carries an action of these Hecke operators and decomposes into joint eigenspaces indexed by automorphic representations (irreducible representations of the adelic group attached to $M$); a cohomology class therefore carries intrinsic automorphic data — its Hecke eigenvalues, the local components of its representation, and its transfer behavior to inner forms — even though it is a priori a purely topological object. Crucially, the initial fibration of $M$ was found by topological computation rather than by automorphic reasoning, and the choice of form is guided by a heuristic obstruction noted in the paper: when the congruence group meets an indefinite quaternion algebra over $\mathbb{Q}$ in a congruence subgroup, the associated Shimura surface embeds in $M$ as a totally geodesic surface, and Thurston showed that the cohomology class defined by such a surface can never fiber; the paper therefore suggests that one should not expect to construct fibered classes from forms that transfer to indefinite quaternion algebras over $\mathbb{Q}$, and notes that its example deliberately avoids such forms. Contemporaneously, Agol proved a group-theoretic criterion for virtual fibering: if $\pi_1(M)$ is RFRS (residually finite rationally solvable — admitting a descending chain of finite-index subgroups intersecting trivially, each containing the kernel of the previous subgroup's map to its rational abelianization), then $M$ virtually fibers; this implies in particular that all arithmetic hyperbolic 3-orbifolds defined by quadratic forms, including the Bianchi orbifolds (those of the form $\mathrm{PSL}(2,\mathcal{O}_K)\backslash\mathbb{H}^3$ for the ring of integers $\mathcal{O}_K$ of an imaginary quadratic field $K$), are virtually fibered. Agol's later virtual Haken theorem shows every closed hyperbolic 3-manifold has some finite cover that fibers. These results settle the existence of virtual fibrations in full generality, but they identify no specific cohomology class of a specific manifold as fibered and use no automorphic input. The only known bridge from automorphic data to fibrations remains the single hand-constructed Dunfield–Ramakrishnan example, in which the automorphic input (Hecke vanishing) spreads a known fibration to new faces of covers but does not by itself certify any class as fibered.

Problem Statement

Dunfield and Ramakrishnan's Question 1.7 asks: is there an automorphic criterion which implies that certain cohomology classes of arithmetic hyperbolic 3-manifolds give fibrations over the circle?

For arithmetic hyperbolic 3-manifolds $M=\Gamma\backslash\mathbb{H}^3$, such a criterion is an explicit condition expressed purely in automorphic terms — for example, a condition on the automorphic representation associated to a cohomology class $\omega\in H^1(M;\mathbb{Z})$, on its Hecke eigenvalues, on its local components, or on its transfer behavior to quaternion inner forms — that is sufficient for a cohomology class $\omega$ satisfying it to be represented by a fibration $M\to S^1$. The condition must be intrinsic to the automorphic data: it may not presuppose the fibration, an a priori fibered face, or any topological computation, and the fibering conclusion must be derived from the automorphic hypothesis itself rather than from a fibration already established by computation for a particular example. Any such condition must also be non-vacuous, in the sense that it is satisfied by at least one cohomology class of an arithmetic hyperbolic 3-manifold. The source paper notes that a cohomology class defined by an embedded totally geodesic surface can never fiber (Thurston), and suggests heuristically that one should not expect to construct fibered classes from forms that transfer to indefinite quaternion algebras over $\mathbb{Q}$; its example deliberately avoids such forms.

References

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