64.25 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.
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