We compute the E-polynomials of a family of twisted character varieties M [superscript g] (Sl [subscript n]) by proving they have polynomial count, and applying a result of N. Katz on the counting functions. To compute the number of F [subscript q]-points of these varieties as a function of q, we used a formula of Frobenius. Our calculations made use of the character tables of Gl [subscript n](q) and Sl subscript n](q), previously computed by J. A. Green and G. Lehrer, and a result of Hanlon on the Möbius function of a subposet of set-partitions. The Euler Characteristics of the M [superscript g] (Sl [subscript n]) are calculated then with these polynomial. / text

The central theme of this thesis is the application of mirror symmetry to the study of the arithmetic geometry of Calabi-Yau threefolds. It formulates a conjecture about the properties of the limit mixed Hodge structure at the large complex structure limit of an arbitrary mirror threefold, which is supported by a two-parameter example of a self-mirror Calabi-Yau threefold. It further studies the connections between this conjecture with Voevodsky's mixed motives. This thesis also studies the connections between the conifold transition and Beilinson's conjecture on the values of the L-functions at integral points. It carefully studies the arithmetic geometry of the conifold in the mirror family of the quintic Calabi-Yau threefold and its L-function, which is shown to provide a very interesting example to Beilinson's conjecture.