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…)