The classical Kloosterman sums give rise to a Galois representation of the function field unramified outside 0 and ∞. We study the local monodromy of this representation at ∞ using l-adic method based on the work of Deligne and Katz. As an application, we determine the degrees and the bad factors of the L-functions of the symmetric products of the above representation. Our results generalize some results of Robba obtained through p-adic method. © Walter de Gruyter Berlin · New York 2005.

Using ℓ-adic cohomology of tensor inductions of lisse Qℓ-sheaves, we study a class of incomplete character sums.

We provide upper bounds for the cardinality of the value set of a polynomial map in several variables over a finite field. These bounds generalize earlier bounds for univariate polynomials. © 2012 The author 2012. Published by Oxford University Press. All rights reserved.

A new algebraic Cayley graph is constructed using finite fields. It provides a more flexible source of expander graphs. Its connectedness, the number of connected components, and diameter bound are studied via Weil's estimate for character sums. Furthermore, we study the algorithmic problem of computing the number of connected components and establish a link to the integer factorization problem. © 2014 Elsevier Inc.