It is an easy consequence of the Serre open image theorem that for the torsion point count on elliptic curves, the following possibilities arise.

1. If $E/\bar{\mathbb{Q}}$ is an elliptic curve without CM, then the number of $x \in E(\bar{\mathbb{Q}})$ with $[\mathbb{Q}(x):\mathbb{Q}] \leq d$ is $\asymp d^{3/2}$ as $d \to \infty$.

2. If $E/\bar{\mathbb{Q}}$ is an elliptic curve with CM, then this number is $\asymp d^2$ as $d \to \infty$.

This appears on page 44 of Serre's book, *Lectures on Mordell-Weil*. As stated there, it is easy to show that, for a $g$-dimensional abelian variety, the corresponding count is bounded by $\leq O(d^{N})$ with $N = N(g) < \infty$.

Has there been any progress since on the precise torsion point count for higher dimensional abelian varieties? Should we expect the count to be always asymptotic to $d^{\alpha}$ for a finite set of exponents $\alpha$ depending on $g$? Is there any conjecture as to the spectrum of those exponents?

Second, I wanted to ask about any results on the uniform torsion count problem giving bounds that are uniform in $E$ but still polynomial in $d$. (Thus I *do not* ask about Merel's uniform boundedness theorem and its relatives). For example, restricting to $g$-dimensional abelian varieties over $\bar{\mathbb{Z}}$ (that is, with integral moduli, or with everywhere potentially good reduction), it is clear a priori that there is a uniform exponential bound in $d$. Should we expect this uniform bound to be strengthened to a polynomial bound, or even (for elliptic curves) to $Cd^{2+\epsilon}$? What about the generalization where we count the small algebraic points, say those of (canonical) height $< 1/d$?