q-Analogues of Wilson's theorem

Electronic version of an article published in International Journal of Number Theory Volume 04, Issue 04, August 2008, pp. 539-547. DOI: 10.1142/S1793042108001511. Copyright © 2008 World Scientific Publishing Company: http://www.worldscientific.com/worldscinet/ijntWe give q-analogues of Wilson's theorem for the primes congruent to 1 and 3 modulo 4, respectively. Also q-analogues of two congruences due to Mordell and Chowla are established

Determinants on lens spaces and cyclotomic units

The Laplacian functional determinants for conformal scalars and coexact one-forms are evaluated in closed form on inhomogeneous lens spaces of certain orders, including all odd primes when the essential part of the expression is given, formally as a cyclotomic unitComment: 18 pages, 1 figur

Symmetries and reversing symmetries of toral automorphisms

Toral automorphisms, represented by unimodular integer matrices, are investigated with respect to their symmetries and reversing symmetries. We characterize the symmetry groups of GL(n,Z) matrices with simple spectrum through their connection with unit groups in orders of algebraic number fields. For the question of reversibility, we derive necessary conditions in terms of the characteristic polynomial and the polynomial invariants. We also briefly discuss extensions to (reversing) symmetries within affine transformations, to PGL(n,Z) matrices, and to the more general setting of integer matrices beyond the unimodular ones.Comment: 34 page