Torsion bounds for a fixed abelian variety and varying number field

Let $A$ be an abelian variety defined over a number field $K$. For a finite extension $L\mathrm{/}K$, the cardinality of the group $A(L{)}_{\mathrm{tors}}$ 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 $L\mathrm{/}K$ and $\epsilon >0$, we have $\mathrm{\mid }A(L{)}_{\mathrm{tors}}\mathrm{\mid }\le C\cdot [L:K{]}^{{\beta }_A+\epsilon }$, where the constant $C$ depends only on $A$ and $\epsilon$ (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 $A(L)$.