I’d like to explain what I think is a pretty piece of mathematics.

Deligne’s proof of the Weil conjectures, and his strengthenings of them, in his epic paper “La conjecture de Weil II” have had, needless to say, many applications—far more than the initial paper where he proved the last of the Weil conjectures. One of the applications is to bounding exponential sums. I’d like to begin sketching the idea today. I learned this from the article “Sommes trigonometriques” in SGA 4.5, which is very fun to read.

1. The trace formula

Let’s say you have some variety ${X_0}$ over a finite field ${\kappa}$, say separated. Suppose you have a ${l}$-adic sheaf ${\mathcal{F}_0}$ on ${X_0}$. I’ll denote by “dropping the zero” the base change to ${\overline{\kappa}}$, so ${\mathcal{F}}$ denotes the pull-back to ${X = X_0 \times_{\kappa} \overline{\kappa}}$.

For each point ${x \in X_0}$, we can take the “geometric stalk” ${\mathcal{F}_{\overline{x}}}$ (or ${\mathcal{F}_{0 \overline{x}}}$) which is given by finding a map ${\mathrm{Spec} \overline{\kappa} \rightarrow X_0}$ hitting ${x}$, and pulling ${\mathcal{F}_0}$ back to it. This “stalk” is automatically equivariant with respect to the Galois group ${\mathrm{Gal}(\overline{\kappa}/k(x))}$ for ${k(x)}$ the residue field, and as a result we can compute the trace of the geometric Frobenius ${F_x}$ of ${k(x) \hookrightarrow \overline{\kappa}}$—that’s the inverse of the usual Frobenius—and take its trace on ${\mathcal{F}_{\overline{x}}}$.

So the intuition here is that we’re taking local data of the sheaf ${\mathcal{F}_0}$: just the trace of its Frobenius at each point. One interesting interpretation of this procedure was discussed on this MO question: namely, the process is analogous to “integration over a contour.” Here the contour is ${\mathrm{Spec} k(x)}$, which has cohomological dimension one and is thus the étale version of a curve.

Now, the trace formula says that the local data of these traces all pieces into a simple piece of global data, which is the compactly supported cohomology. (more…)

The topic of this post is a curious functor, discovered by Deligne, on the category of sheaves over the affine line, which is a “sheafification” of the Fourier transform for functions.

Recall that the classical Fourier transform is an almost-involution of the Hilbert space ${L^2(\mathbb{R})}$. We shall now discuss the Fourier-Deligne transform, which is an almost-involution of the bounded derived category of ${l}$-adic sheaves, ${\mathbf{D}^b_c(\mathbb{A}^1_{\kappa}, \overline{\mathbb{Q}_l})}$. The Fourier transform is defined by multiplying a function with a character (which depends on a parameter) and integrating. Analogously, the Fourier-Deligne transform will twist an element of ${\mathbf{D}^b_c(\mathbb{A}^1_{\kappa}, \overline{\mathbb{Q}_l})}$ by a character depending on a parameter, and then take the cohomology.

More precisely, consider the following: let $G$ be a LCA group, $G^*$ its dual. We have a canonical character on $\phi: G \times G^* \to \mathbb{C}^*$ given by evaluation. To construct the Fourier transform $L^2(G) \to L^2(G^*)$, we start with a function $f: G \to \mathbb{C}$. We pull back to $G \times G^*$, multiply by the evaluation character $\phi$ defined above, and integrate along fibers to give a function on $G^*$.

Everything we’ve done here has a sheaf-theoretic analog, however: pulling back a function corresponds to the functorial pull-back of sheaves, multiplication by a character corresponds to tensoring with a suitable line bundle, and integration along fibers corresponds to the lower shriek push-forward. Much of the classical formalism goes over to the sheaf-theoretic case. One can prove an “inversion formula” analogous to the Fourier inversion formula (with a Tate twist).

Why should we care? Well, Laumon interpreted the Fourier transform as a suitable “deformation” of the cohomology of a suitable sheaf on the affine line, and used it to give a simplified proof of the main results of Weil II, without using scary things like vanishing cycles and Picard-Lefschetz theory. The Fourier transform also behaves very well with respect to perverse sheaves: it is an auto-equivalence of the category of perverse sheaves, because of the careful way in which it is calibrated. Its careful use can be used to simplify some of the arguments in BBD that also rely on other scary things.