63.25 Exponential sums

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

$S_{a, b}(p) = \sum _{x\in \mathbf{F}_ p - \left\{ 0, 1\right\} } e^{\frac{2\pi ix^ a(x - 1)^ b}{p}}.$

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

$f_*(\bar{\mathbf{Q}}_\ell ^*) = \bigoplus _{\chi : \mathbf{F}_ p^*\to \bar{\mathbf{Q}}_\ell ^*} \mathcal{F}_\chi$

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

$H_ c^1(X_{\bar k}, \mathbf{Q}_\ell ) = \bigoplus _\chi H^1(\mathbf{P}_{\bar k}^1-\{ 0, 1, \infty \} , \mathcal{F}_\chi )$

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

$S_{a, b}(p) = -\text{Tr}(\pi _ X^* |_{H^1(\mathbf{P}_{\bar k}^1-\{ 0, 1, \infty \} , \mathcal{F}_\chi )}).$

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

$2g_ X-2 \approx -2 (p-1) + 3(p-2) \approx p$

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

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