All Questions

Let $V \subset \mathbb P^n$ be a nice (smooth, projective?) variety over a finite field $\mathbb F_q$. Let $\chi_0,\chi_1,\dots,\chi_{n}: \mathbb F_q^\times \to \mathbb Q(\mu_{q-1})$ be multiplicative ...

I encounter a tricky sum like the Kloosterman sum $${\sum_{x \mod P}}^\ast e\left(\frac{ax+\overline{x}}{P}+\frac{lx^2}{P^2}\right),$$ where $l$ is a positive integer co-prime with $P$ and here $P$ ...

Let $$ f_{a}(x)=x^3+(u-2-a)x^2+ax+1, $$ where $u\in\mathbb{Z}_p^*$ is fixed. Let $S$ be the set consisting of all $a\in\mathbb{Z}_p$ such that $f_{a}(x)$ factor linearly. Then what is the cardinality ...