André Weil -Number of Solutions of Equations over Finite Fields

We want to count solutions to the equation \displaystyle a_{0} x_{0}^{n_{0}}+a_{1}x_{1}^{n_{1}}+\cdots+a_{r} x_{r}^{n_{r}}=b over a finite field.

Detecting non-zero powers of x in terms of characters, we get the following expression for point count in terms of the characters and Jacobi sums.

\displaystyle N=q^{r}+(q-1) \sum_{\alpha} \chi_{\alpha_{0}}\left(a_{0}^{-1}\right) \cdots \chi_{\alpha_{r}}\left(a_{r}^{-1}\right) \cdot j(\alpha)
\displaystyle j(\alpha)=\sum_{1+v_{1}+\cdots+v_{r}-0} \chi_{\alpha_{1}}\left(v_{1}\right) \cdots \chi_{\alpha_{r}}\left(v_{r}\right)

We can write Jacobi sums in terms of Gauss sums by expanding in terms of additive characters.

\displaystyle j(\alpha)=\frac{1}{q} g\left(\chi_{\alpha_{0}}\right) \cdots g\left(\chi_{\alpha_{r}}\right)

Now square-root cancellation in Gauss sums gives the required square-root cancellation for the point counts- (Riemann Hypothesis for the hypersurface)- Thus the products of characters and the Jacobi sums in the point count expression are the root numbers appearing in the trace-formula.

\displaystyle \left|N_{1}-q^{r}\right| \leqq \left(d_{0}-1\right) \cdots\left(d_{r}-1\right) q^{r / 2}

If the above expression in terms of Jacobi sums is the fixed point counting trace formula expansion- the expressions for different finite fields have to be related through the powers of these root numbers. That is we need relations between the Jacobi sums and Gauss sums of a character over different extensions of fixed base field.- What is that relation? Indeed Gauss sums have such relations between them: Hasse Davenport relation- This is finite field analogue of the following Gauss Multiplication formula for the Gamma function.
\displaystyle \Gamma(z) \Gamma\left(z+\frac{1}{k}\right) \Gamma\left(z+\frac{2}{k}\right) \cdots \Gamma\left(z+\frac{k-1}{k}\right)=(2 \pi)^{\frac{k-1}{2}} k^{1 / 2-k z} \Gamma(k z)

\displaystyle (-1)^{r} \cdot \tau(\chi, \psi)^{r}=-\tau\left(\chi^{\prime}, \psi^{\prime}\right)

https://www.ams.org/journals/bull/1949-55-05/S0002-9904-1949-09219-4/S0002-9904-1949-09219-4.pdf

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