L-Functions for Symmetric Products of Kloosterman Sums

The classical Kloosterman sums give rise to a Galois representation of the function field unramfied outside 0 and $\infty$. We study the local monodromy of this representation at $\infty$ 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.