The Jiang–Rallis integral point-count conjecture

Let $p$ be a prime with $p\equiv 5\pmod{6}$. Fix $b,c\in(\mathbf{Z}/p\mathbf{Z})^\times$ such that

$$g(u):=-u^3+bu+c$$

is irreducible. Consider the triples $(r,y,u)\in(\mathbf{Z}/p\mathbf{Z})^*\times(\mathbf{Z}/p\mathbf{Z})\times(\mathbf{Z}/p\mathbf{Z})$ satisfying

$$g(u)\equiv 3uyr+y^3r^2-r\pmod p.$$

The Jiang–Rallis integral point-count conjecture. The number of such triples is $p^2-1$. This conjecture gives the remaining finite-field count needed to evaluate one of the sub-integrals arising in the Jiang–Rallis computation for $F=\mathbf{Q}$ and $p\equiv5\pmod6$. It has since been proved by Victor Scharaschkin, so the conjecture is solved.