10 problems

Helleseth's vanishing conjecture. If $q>2$ and $s\equiv 1 \pmod{p-1}$, then $s$ is singular.

Helleseth's three-valued conjecture. If $[F:\mathbb{F}_p]$ is a power of $2$, then, for every invertible exponent $s$, the Weil spectrum of $W_{F,s}(a)$ is not three-valued.

Nonexistence conjecture. The polynomial $f(x)$ is not a permutation polynomial over $\mathbb{F}_{q^2}$.

Five-valued conjecture. If either $d_1+d_2\geq 5$, or $d_1+d_2=3$ and $p^n\equiv 11\pmod{12}$, then the Weil spectrum over $L$ is at least five-valued. In the first case its five v…

Let $F$) be a finite field of order $4^m$, where $m$ is even, and set … Let $W_{F,d}(a)$ denote the Weil sum associated with this decimation. Niho's five-value conjecture. The set…

Symmetry conjecture. $W_{F,d}$ is symmetric.

Katz–Langevin–Lee–Sapozhnikov conjecture.

Sarwate–Pursley conjecture. There is some $a\in F^*$ such that

Let $F$ be a finite field of characteristic $p$, write $q=|F|$, and let $V_{F,d}$ denote the normalized $p$-adic valuation associated with the Weil sum of the binomial exponent $d$…

Let $F$ be a finite field of order $q=3^n$, where $n$ is odd and $n>1$, and let $d=3^r+2$ satisfy … For $a\in F$, define the Weil sum … where $\psi\colon F\to\mathbb C$ is the cano…