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. Namely, we have $\displaystyle \sum_{x \in X_0(\kappa)} \mathrm{Tr} F_x|_{\mathcal{F}_{\overline{x}}} = \sum (-1)^i \mathrm{Tr} F|_{H^i_c(X, \mathcal{F})}.$

So on the left, we’re taking a finite sum—over ${\kappa}$-rational points—and on the right we’re computing the traces of the Frobenius action on cohomology.

Historically, the trace formula provided a proof—not the first, though–that the zeta function of a variety is rational, and thus describing how the number of points over a finite field grows with the size of the finite field. This is itself a nice story, but probably best saved for a different post. Additionally, there is a nice interpretation of this business in terms of the “function-sheaf” correspondence, which is explained by Ben Webster in these two posts.

For us, the idea is that this trace formula is a type of local-to-global principle—you get the global data of the trace on cohomology expressed as local terms. And this is how the trace formula is usually used. In proving the zeta function, one takes ${\mathcal{F}_0 = \overline{\mathbb{Q}_l}}$, so the individual local terms are each just ${1}$, but the global term—the count of the number of points—is very very nontrivial. There are many interesting variants on this theme of this, such as:

Theorem 1 (Deligne) Suppose ${\mathcal{F}_0}$ is an ${l}$-adic sheaf on ${X_0}$ such that the eigenvalues of the local Frobenii are algebraic integers. Then the eigenvalues of the Frobenius on the cohomology ${H_c^\bullet(X, \mathcal{F})}$ are themselves algebraic integers.

This is actually not an immediate consequence of the trace formula: it requires more work, and some dévissage. But it is the same principle.

2. The Artin-Schreier sheaf

Anyhow, so far all this seems pretty abstract, unrelated to down-to-earth things like exponential sums. The relation begins with the construction of the so-called Artin-Schreier sheaf ${\mathcal{L}(\chi)}$ on the affine line ${\mathbf{A}^1_0 = \mathbb{A}^1_\kappa}$, associated to a character ${\chi: \kappa \rightarrow \overline{\mathbb{Q}_l}^*}$.

This sheaf has a really awesome property: if you take a point ${x \in \kappa}$, and take the local geometric Frobenius of the Artin-Schreier sheaf, you get ${\chi^{-1}(x)}$. To construct this sheaf, one uses the Lang cover ${\mathbf{A}^1_0 \rightarrow \mathbf{A}^1_0}$ given by ${t \mapsto t^q - t}$ for ${q = |\kappa|}$. This is a general procedure that one can do for any commutative algebraic group ${G_0}$ over ${\kappa}$: there is a Lang isogeny ${L: G_0 \rightarrow G_0}$, ${Lt = Ft - t}$ for ${F}$ the global Frobenius, that is an étale cover. The fibers of the Lang isogeny are the ${\kappa}$-rational points. In other words, we have a “fiber sequence” $\displaystyle 1 \rightarrow G_0(\kappa) \rightarrow G_0 \stackrel{L}{\rightarrow} G_0 \rightarrow 1.$

Anyhow, the point is that, since we have an étale cover—indeed, an abelian Galois cover—the local Frobenii act on ${G_0 }$ by automorphisms. It turns out, and this is not too hard to check, that the action of the local Frobenius at ${x \in \kappa}$ is just translation by ${x}$.

So the means of getting the Artin-Schreier sheaf on a general commutative algebraic group ${G_0}$ is the following: you take this étale cover, which is actually a ${G_0(\kappa)}$-torsor, and then “tensor it” over ${\overline{\mathbb{Q}_l}}$ with a character ${G_0(\kappa) \rightarrow \overline{\mathbb{Q}_l}}$. So from the torsor, and the character, you get a sheaf—in fact, a local system of dimension one. It’s not too hard to convince oneself that the local Frobenius actions then are given by ${\chi(x)}$, so the geometric Frobenius actions are given by ${\chi^{-1}(x)}$. This is because the local Frobenius actions on the initial torsor were given by translation by $x$, and under $\chi$ that corresponds by multiplication by $\chi(x)$.

3. Trigonometric sums

So, we can get character values as eigenvalues on sheaves. That’s a start, but we have to do a lot more. Say we have some variety ${X_0 }$ over ${\kappa}$, and a ${\kappa}$-rational polynomial map $\displaystyle P: X_0 \rightarrow \mathbf{A}^1_0.$

Given a character ${\phi}$ of ${\kappa}$, we are interested in sums of the form $\displaystyle S(P; \phi) = \sum_{x \in X_0(\kappa)} \phi(P(x)).$

For instance, if ${X_0}$ is the hypersurface ${\left\{(x_1, \dots, x_n): x_1 \dots x_n = 1\right\}}$, then we get a special case of the Kloosterman sums. To interpret this cohomologically, we start with Artin-Schreier sheaf ${\mathcal{L}(\phi)}$ on ${\mathbf{A}^1_0}$. This is a local system of ${\overline{\mathbb{Q}_l}}$-vector spaces, of dimension one, such that the local action of the geometric Frobenius at ${x \in \kappa}$ is given by ${\phi^{-1}(x)}$. As a result, if we pull back by the map ${P: X_0 \rightarrow \mathbf{A}^1_0}$, we get a sheaf ${P^*\mathcal{L}(\phi)}$ on ${X_0}$, such that the local trace at ${x \in \kappa}$ is ${\phi^{-1}(P(x))}$. In particular, we get $\displaystyle S(P; \phi^{-1}) = \sum_{x \in X_0(\kappa)} \mathrm{Tr} F_x|_{P^*\mathcal{L}(\phi)_{\overline{x}}}.$

Thus the trigonometric sum is a sum of local terms. But by the Grothendieck trace formula, this is $\displaystyle S(P; \phi^{-1}) = \sum_i (-1)^i \mathrm{Tr} F|_{H^i_c(X; P^*\mathcal{L}(\phi)}.$

In particular, if we know how to compute the trace of the action of the Frobenius on ${l}$-adic cohomology, then we can evaluate trigonometric sums.

The problem is that computing ${l}$-adic cohomology is usually hard! Let alone computing the actions of the Frobenius. But there are various tools that one has.

The most powerful is the main result of Deligne’s paper “Weil II,” which provides a bound for individual eigenvalues of the geometric Frobenius on cohomology. Namely, it states that if the local eigenvalues of the (geometric) Frobenius on stalks are of complex absolute value ${\leq w}$ (with respect to some fixed isomorphism ${\overline{\mathbb{Q}_l} \simeq \mathbb{C}}$), then the eigenvalues of the Frobenius on ${H^i_c}$ are of complex absolute value ${\leq q^{i/2} w}$. This is Deligne’s strengthening of the Weil conjectures. So that gives us part of the answer: we can bound the individual eigenvalues on cohomology. Since ${\mathcal{L}(\phi)}$, and thus ${P^*\L(\phi)}$, has local eigenvalues which are roots of unity, we will have that the eigenvalues on the global cohomology are ${\leq q^{i/2} w}$.

But we have a problem: to bound the trace, we need to know how many of the eigenvalues there are. So, if we are interested in bounding trigonometric sums, we need more than just the Weil conjectures: we need to bound the dimensions of the cohomology groups of pull-backs of Artin-Schreier sheaves. There are various techniques and tricks to compute these, which will probably require another post even to begin to discuss.