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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 $N = N(g) < \infty$.

Should we expect, for $g$-dimensional abelian varieties, the count to be always asymptotic to $d^{\alpha}$ for a finite set of exponents $\alpha$ depending on $g$? Are there any clues 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), or at least to the abelian schemes over the spectra of rings of integers of number fields of bounded degree, 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 $< 1/d$?

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 torsion points $x \in E$ 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 $N = N(g) < \infty$.

Should we expect, for $g$-dimensional abelian varieties, the count to be always asymptotic to $d^{\alpha}$ for a finite set of exponents $\alpha$ depending on $g$? Are there any clues 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), or at least to the abelian schemes over the spectra of rings of integers of number fields of bounded degree, 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 $< 1/d$?

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 $N = N(g) < \infty$.

Should we expect, for $g$-dimensional abelian varieties, the count to be always asymptotic to $d^{\alpha}$ for a finite set of exponents $\alpha$ depending on $g$? Are there any clues 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), or at least to the abelian schemes over the spectrum of ring of integers of a number field of bounded degree, 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 $< 1/d$?

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 $N = N(g) < \infty$.

Should we expect, for $g$-dimensional abelian varieties, the count to be always asymptotic to $d^{\alpha}$ for a finite set of exponents $\alpha$ depending on $g$? Are there any clues 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), or at least to the abelian schemes over the spectra of rings of integers of number fields of bounded degree, 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 $< 1/d$?

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 $< 1/d$?

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 $N = N(g) < \infty$.

Should we expect, for $g$-dimensional abelian varieties, the count to be always asymptotic to $d^{\alpha}$ for a finite set of exponents $\alpha$ depending on $g$? Are there any clues 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), or at least to the abelian schemes over the spectrum of ring of integers of a number field of bounded degree, 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 $< 1/d$?