  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.
