  Let $A$ be an abelian variety defined over a number field $K$. For a finite
extension $L/K$, the cardinality of the group $A(L)_{\operatorname{tors}}$ of
torsion points in $A(L)$ can be bounded in terms of the degree $[L:K]$. We
study the smallest real number $\beta_A$ such that for any finite extension
$L/K$ and $\varepsilon>0$, we have $|A(L)_{\operatorname{tors}}| \leq C \cdot
[L:K]^{\beta_A+\varepsilon}$, where the constant $C$ depends only on $A$ and
$\varepsilon$ (and not $L$). Assuming the Mumford--Tate conjecture for $A$, we
will show that $\beta_A$ agrees with the conjectured value of Hindry and
Ratazzi. We also give a similar bound for the maximal order of a torsion point
in $A(L)$.
