Geometry and Topology without figures

In this post, I'd like to give a rapid introduction to the theory of $l$-adic Fourier transform developed by Laumon-Deligne-... My goal is not to how can we apply $l$-adic Fourier transform to prove the Weil conjectures but rather to see why their definitions are natural in comparison with the classical theory. My feeling is that it is easier to present Fourier transforms on finite fields than on measurable spaces (which require a lot of work and details) and the $l$-adic one is formally adapted from the one for finite fields. Fourier analysis on finite abelian groups Given a finite abelian group $G$, written additively, what we want to do here is to define a space $L^2(G)$ similar to the Hilbert space $L^2(X)$ of square integrable functions $X \longrightarrow \mathbb{C}$ (modulo equal almost everywhere relation) for $X$ being a measurable space. Then it is possible to develop a Fourier transform on $L^2(G)$. The finiteness seems to be a technical condition that you can see to be u...

This is motivated from a talk presented by prof. Ngo Bao Chau that I attended recently. The subject of the talk was to give an estimate of the so-called Kloosterman sum by transferring it to the language of $l$-adic cohomology. Follow Katz, let me spend some momemt to recall the motivating problem: given a prime $p$ and an integer $a$ s.t. $(a,p)=1$, the Kloosterman sum is defined as the complex number $$\mathrm{Kl}(a,p) = \sum_{(x,y) \in \mathbb{F}_p: xy = a} \operatorname{exp}\left(\frac{2\pi i}{p}(x+y) \right).$$ By an elementary argument, one can see that this sum is a real number and in the early time when Kloosterman studied the Hardy-Littlewood circle method, he wanted to bound this sum by a function of $p$.  Some motivations   Theorem 1 (Kloosterman 1926). For any $\epsilon > 0$, we have $\left |\mathrm{Kl}(a,p) \right| < Cp^{3/4+\epsilon}$.   Kloosterman's proof was quite elementary, however, the bound can be sharpen much more as follows. Theorem 2 (Weil) ...