  Recent observations have revealed evidence for the global oscillations of
Jupiter and Saturn, which can potentially provide a new window into the
interior structure of giant planets. Motivated by these observations, we study
the non-radial oscillation modes of giant planets containing a solid core. Our
calculations include the elastic response of the core and consider a wide range
of possible values of the core shear modulus. While the elasticity of the core
only slightly changes the frequencies of acoustic modes, which reside mostly in
the fluid envelope, it adds two new classes of shear modes that are largely
confined to the core. We also calculate the effects of the Coriolis force on
the planetary oscillation modes. In addition to changing the mode frequencies,
the Coriolis force can cause the shear modes to mix with the f-modes. Such
mixing occurs when the frequencies of the shear mode and the f-mode are close
to each other, and results in "mixed modes" with similar properties that are
slightly split in frequency. We discuss our results in light of the recent work
by Hedman & Nicholson (2013), which revealed the presence of density waves in
Saturn's C-ring that appear to be excited by the gravitational perturbations
associated with the f-mode oscillations within Saturn. We find that the fine
splitting in wave frequencies observed in the rings can in principle be
explained by the rotation-induced mixing between core shear modes and f-modes,
possibly indicating the presence of a solid core within Saturn. However, in our
current calculations, which assume rigid-body rotation and include only
first-order rotational effects, significant fine-tuning in the planetary model
parameters is needed in order to achieve these mode mixings and to explain the
observed fine frequency splitting. We briefly discuss other effects that may
modify the f-modes and facilitate mode mixing.
