The weighted projective Serre bound conjecture

Let $q$ be a prime power, let $a_0,ots,a_n$ be positive weights, and let $e_q(d;a_0,a_1,ots,a_n)$ be the maximum number of ${\mathbb F}_q$-rational points on a hypersurface of weighted degree $d$ in ${\mathbb P}(a_0,ots,a_n)$. Let $p_n=q^n+ots+q+1$ for $n\geq 0$ and $p_n=0$ for $n<0$. The weighted projective Serre bound conjecture. If $a_0=1$, $\operatorname{lcm}(a_1,a_2,ots,a_n)\mid d$, and $a_1\leq a_2\leqots\leq a_n$, then

$$e_q(d;1,a_1,a_2,ots,a_n)=\min\left\{p_n,\frac{d}{a_1}q^{n-1}+p_{n-2}\right\}.$$

The conjecture generalizes the Serre bound from ordinary projective spaces to weighted projective spaces. The paper proves it in every dimension when the second weight is also one, so the general case remains open.