The Stacks project

63.25 Exponential sums

A standard problem in number theory is to evaluate sums of the form

In our context, this can be interpreted as a cohomological sum as follows. Consider the base scheme $S = \mathop{\mathrm{Spec}}(\mathbf{F}_ p[x, \frac{1}{x(x - 1)}])$ and the affine curve $f : X \to \mathbf{P}^1-\{ 0, 1, \infty \} $ over $S$ given by the equation $y^{p - 1} = x^ a(x - 1)^ b$. This is a finite étale Galois cover with group $\mathbf{F}_ p^*$ and there is a splitting

where $\chi $ varies over the characters of $\mathbf{F}_ p^*$ and $\mathcal{F}_\chi $ is a rank 1 lisse $\mathbf{Q}_\ell $-sheaf on which $\mathbf{F}_ p^*$ acts via $\chi $ on stalks. We get a corresponding decomposition

and the cohomological interpretation of the exponential sum is given by the trace formula applied to $\mathcal{F}_\chi $ over $\mathbf{P}^1 - \{ 0, 1, \infty \} $ for some suitable $\chi $. It reads

The general yoga of Weil suggests that there should be some cancellation in the sum. Applying (roughly) the Riemann-Hurwitz formula, we see that

so $g_ X\approx p/2$, which also suggests that the $\chi $-pieces are small.