We find explicit models for the PSL2(C)-and SL2(C)-character varieties of the fundamental groups of complements in S 3 of an infinite family of two-bridge knots that contains the twist knots. We compute the genus of the components of these character varieties, and deduce upper bounds on the degree of the associated trace fields. We also show that these knot complements are fibered if and only if they are commensurable to a fibered knot complement in a Z/2Z-homology sphere, resolving a conjecture of Hoste and Shanahan.

We investigate a connection between two important classes of Euclidean lattices: well-rounded and ideal lattices. A lattice of full rank in a Euclidean space is called well-rounded if its set of minimal vectors spans the whole space. We consider lattices coming from full rings of integers in number fields, proving that only cyclotomic fields give rise to well-rounded lattices. We further study the well-rounded lattices coming from ideals in quadratic rings of integers, showing that there exist infinitely many real and imaginary quadratic number fields containing ideals which give rise to well-rounded lattices in the plane.

The classical theorem of Bombieri and Vinogradov is generalized to a non-abelian, non-Galois setting. This leads to a prime number theorem of "mixed-type" for arithmetic progressions "twisted" by splitting conditions in number fields. One can view this as an extension of earlier work of M. R. Murty and V. K. Murty on a variant of the Bombieri-Vinogradov theorem. We develop this theory with a view to applications in the study of the Euclidean algorithm in number fields and arithmetic orbifolds.

Infrared and Raman spectra are reported for cis and trans 1,2-13 C 2 H 2 F 2 . Quantum chemical calculations at B3LYP and MP2 levels have been made for cis and trans 1,2-difluoroethylenes, using 6-311G** and 6-311++G** bases. The resulting harmonic force fields for each compound were scaled with nine independent factors, using frequency data corrected, where necessary, for Fermi resonances and for liquid/gas shifts. The previously accepted assignments for ν 7 and ν 12 in the trans isomer are interchanged. Several scale factors for bending motions differ markedly between the cis and trans compounds. Centrifugal distortion constants observed with significant accuracy are predicted within 6%. Harmonic contributions to the vibrational dependence of the rotational constants and perpendicular amplitudes are calculated. The CdC and C-H bonds in the cis and trans isomers are essentially identical in respect to length, force constant, and isolated CH stretching frequency. However, the C-F bond is slightly stronger in the cis compound. A revised allocation of observed infrared intensity between the overlapping trans ν 7 and ν 12 bands is needed. Directions of ∂p/∂Q for the trans B u bands are given, which differ from those reported earlier. Magnitudes and directions of the bond dipole derivatives ∂µ/∂r for the CH and CF bonds are obtained from calculated atomic polar tensors. In both cis and trans isomers, the vector ∂µ/∂r CF lies within 17°of the bond direction, but for the CH bond, ∂µ/∂r is roughly perpendicular to the CH direction.