Distribution properties of certain exponential sums
Let \(f(x,y)=0\) be a nonsingular affine curve of genus g. For an appropriate prime p, one can associate the number - which we denote by \(N_ p\)- of solutions in \({\mathbb{F}}_ p\times {\mathbb{F}}_ p\) as well as the ``error term'' \(a_ p=N_ p-p\). By the Riemann hypothesis for curves, \(| a_ p| \geq 2g\sqrt{p}\). We call \(d\mu_{g,f}(t)\) the asymptotic density function if \[ \lim_{x\to \infty}\frac{n\{p\geq x| \quad p\in U,\quad a_ p/\sqrt{p}\in (\alpha,\beta)\}}{n\{p\geq x| p\in U\}}=\int^{\beta}_{\alpha}d\mu_{g,f}(t) \] holds. The determination of the asymptotic density function is in general a difficult problem. However, we can get the following result in the case of a certain family of hyperelliptic curves. Let \(f_{q^ k}(t)\) be the asymptotic density function associated to the curve \(y^ 2=x^{q^ k}+1\), with genus \(g=(q^ k-1)/2\). Then we have \[ \lim_{g\to \infty}\sqrt{2q}\int^{y}_{-\infty}f_{q^ k}(\sqrt{2g}t)dt=(1/\sqrt{2\pi})\int^{y}_{-\infty}\exp (-t^ 2/2)dt. \]
scientific article; zbMATH DE number 4202378