As far as I know, we expect that the image of the Galois group in $GL_{2g}(\mathbb A_\mathbb Q)$ is open in $G(\mathbb A_\mathbb Q)$ for $G$ the monodromy group of the Galois representation (which is expected to be independent of $l$. Then the asymptotic for this function clearly depends only on the connected component of the identity of this monodromy group, which is expected to be the Mumford-Tate group.

(1) For each possible Mumford-Tate group $G$, is the number of points in $(\mathbb Q/\mathbb Z)^{2g}$ fixed by an index $d$ subgroup in $G(\mathbb A_\mathbb Q)$ asymptotic to $d^\alpha$ for some $\alpha$?

(2) Are there finitely many possible Mumford-Tate groups?

The first one is not too hard to see. The index of the subgroup fixing an $n$-torsion point is the size of the orbit of that point, which is approximately the number of $\mathbb Z/n$ points of some algebraic variety, which is approximately polynomial in $n$. Then we can sum over all points $n$-torsion for $n\leq N$, another polynomial. If there are different sorts of orbits of different dimensions, then we can just sum over the different sorts of orbits and still get something of the form $d^\alpha$ up to some constant error.

The second one is clear since the Mumford-Tate group should be semisimple, and thus there is a nice classification.

As far as I know, we expect that the image of the Galois group in $GL_{2g}(\mathbb A_\mathbb Q)$ is open in $G(\mathbb A_\mathbb Q)$ for $G$ the monodromy group of the Galois representation (which is expected to be independent of $l$. Then the asymptotic for this function clearly depends only on the connected component of the identity of this monodromy group, which is expected to be the Mumford-Tate group.

(1) For each possible Mumford-Tate group $G$, is the number of points in $(\mathbb Q/\mathbb Z)^{2g}$ fixed by an index $d$ subgroup in $G(\mathbb A_\mathbb Q)$ asymptotic to $d^\alpha$ for some $\alpha$?

(2) Are there finitely many possible Mumford-Tate groups?

The first one is not too hard to see. The index of the subgroup fixing an $n$-torsion point is the size of the orbit of that point, which is approximately the number of $\mathbb Z/n$ points of some algebraic variety, which is approximately polynomial in $n$. Then we can sum over all points $n$-torsion for $n\leq N$, another polynomial. If there are different sorts of orbits of different dimensions, then we can just sum over the different sorts of orbits and still get something of the form $d^\alpha$ up to some constant error.

The second one is clear since the Mumford-Tate group should be reductive, and thus there is a nice classification.

As far as I know, we expect that the image of the Galois group in $GL_{2g}(\mathbb A_\mathbb Q)$ is open in $G(\mathbb A_\mathbb Q)$ for $G$ the monodromy group of the Galois representation (which is expected to be independent of $l$. Then the asymptotic for this function clearly depends only on the connected component of the identity of this monodromy group, which is expected to be the Mumford-Tate group.

(1) For each possible Mumford-Tate group $G$, is the number of points in $(\mathbb Q/\mathbb )Z^{2g}$ fixed by an index $d$ subgroup in $G(\mathbb A_\mathbb Q)$ asymptotic to $d^\alpha$ for some $\alpha$?

(2) Are there finitely many possible Mumford-Tate groups?

The first one is not too hard to see. The index of the subgroup fixing an $n$-torsion point is the size of the orbit of that point, which is approximately the number of $\mathbb Z/n$ points of some algebraic variety, which is approximately polynomial in $n$. Then we can sum over all points $n$-torsion for $n\leq N$, another polynomial. If there are different sorts of orbits of different dimensions, then we can just sum over the different sorts of orbits and still get something of the form $d^\alpha$ up to some constant error.

The second one is clear since the Mumford-Tate group should be semisimple, and thus there is a nice classification.

As far as I know, we expect that the image of the Galois group in $GL_{2g}(\mathbb A_\mathbb Q)$ is open in $G(\mathbb A_\mathbb Q)$ for $G$ the monodromy group of the Galois representation (which is expected to be independent of $l$. Then the asymptotic for this function clearly depends only on the connected component of the identity of this monodromy group, which is expected to be the Mumford-Tate group.

(1) For each possible Mumford-Tate group $G$, is the number of points in $(\mathbb Q/\mathbb Z)^{2g}$ fixed by an index $d$ subgroup in $G(\mathbb A_\mathbb Q)$ asymptotic to $d^\alpha$ for some $\alpha$?

(2) Are there finitely many possible Mumford-Tate groups?

The first one is not too hard to see. The index of the subgroup fixing an $n$-torsion point is the size of the orbit of that point, which is approximately the number of $\mathbb Z/n$ points of some algebraic variety, which is approximately polynomial in $n$. Then we can sum over all points $n$-torsion for $n\leq N$, another polynomial. If there are different sorts of orbits of different dimensions, then we can just sum over the different sorts of orbits and still get something of the form $d^\alpha$ up to some constant error.

The second one is clear since the Mumford-Tate group should be semisimple, and thus there is a nice classification.