Processing math: 100%

The Stacks project

64.25 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.


Comments (0)


Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.