On the Convex Hull of the Points on Modular Hyperbolas

Given integers $a$ and $m\ge 2$, let $\Hm$ be the following set of integral points $$\Hm= \{(x,y) \ : \ xy \equiv a \pmod m,\ 1\le x,y \le m-1\}$$ We improve several previously known upper bounds on $v_a(m)$, the number of vertices of the convex closure of $\Hm$, and show that uniformly over all $a$ with $\gcd(a,m)=1$ we have $v_a(m) \le m^{1/2 + o(1)}$ and furthermore, we have $v_a(m) \le m^{5/12 + o(1)}$ for $m$ which are almost squarefree.