We exhibit a closed hyperbolic 3-manifold which satisfies a very strong form
of Thurston's Virtual Fibration Conjecture. In particular, this manifold has
finite covers which fiber over the circle in arbitrarily many ways. More
precisely, it has a tower of finite covers where the number of fibered faces of
the Thurston norm ball goes to infinity, in fact faster than any power of the
logarithm of the degree of the cover, and we give a more precise quantitative
lower bound. The example manifold M is arithmetic, and the proof uses detailed
number-theoretic information, at the level of the Hecke eigenvalues, to drive a
geometric argument based on Fried's dynamical characterization of the fibered
faces. The origin of the basic fibration of M over the circle is the modular
elliptic curve E=X_0(49), which admits multiplication by the ring of integers
of Q[sqrt(-7)]. We first base change the holomorphic differential on E to a
cusp form on GL(2) over K=Q[sqrt(-3)], and then transfer over to a quaternion
algebra D/K ramified only at the primes above 7; the fundamental group of M is
a quotient of the principal congruence subgroup of level 7 of the
multiplicative group of a maximal order of D. To analyze the topological
properties of M, we use a new practical method for computing the Thurston norm,
which is of independent interest. We also give a non-compact finite-volume
hyperbolic 3-manifold with the same properties by using a direct topological
argument.Comment: 42 pages, 7 figures; V2: minor improvements, to appear in Amer. J.
Mat