In a question previously asked on MO (Units of Endomorphism Rings of Jacobian Varieties with Real Multiplication), a result from the paper ``On the fields of rationality for curves and for their Jacobian varieties'' by Sekiguchi is cited to give a bound for the number of automorphisms of a Jacobian $J(C)$ of a curve $C$. As mentioned in the answer, the original question seems to mix up whether polarizations are taken into account in different places.

What is the correct way to interpret the main theorem in Sekiguchi's original paper regarding polarizations of $J(C)$ (e.g. when counting automorphisms)? From the restrictions that we get, it does look like they are taken into account in this case, but I just wanted to be careful about how this works. Also, what are some examples of automorphisms that respect polarizations and ones that don't?

On another note: Is $\text{Aut } J(C) = (\text{End } J(C))^\times$ ever infinite? How does the number of automorphisms of $J(C)$ ignoring polarizations compare with the number of automorphisms taking them into account?

In a question previously asked on MO (Units of Endomorphism Rings of Jacobian Varieties with Real Multiplication), a result from the paper ``On the fields of rationality for curves and for their Jacobian varieties'' by Sekiguchi is cited to give a bound for the number of automorphisms of a Jacobian $J(C)$ of a curve $C$. As mentioned in the answer, the original question seems to mix up whether polarizations are taken into account in different places.

What is the correct way to interpret the main theorem in Sekiguchi's original paper regarding polarizations of $J(C)$ (e.g. when counting automorphisms)? From the restrictions that we get, it does look like they are taken into account in this case, but I just wanted to be careful about how this works. Also, what are some examples of automorphisms that respect polarizations and ones that don't?

On another note: Is $\text{Aut } J(C) = (\text{End } J(C))^\times$ ever infinite? How does the number of automorphisms of $J(C)$ ignoring polarizations compare with the number of automorphisms taking them into account (e.g. when $\text{Aut } J(C)$ is finite)?

In a question previously asked on MO (Units of Endomorphism Rings of Jacobian Varieties with Real Multiplication), a result from the paper ``On the fields of rationality for curves and for their Jacobian varieties'' by Sekiguchi is cited to give a bound for the number of automorphisms of a Jacobian $J(C)$ of a curve $C$. As mentioned in the answer, the original question seems to mix up whether polarizations are taken into account in different places.

What is the correct way to interpret the main theorem in Sekiguchi's original paper regarding polarizations of $J(C)$ (e.g. when counting automorphisms)? From the restrictions that we get, it does look like they are taken into account in this case, but I just wanted to be careful about how this works.

On another note: Is $\text{Aut } J(C) = (\text{End } J(C))^\times$ ever infinite? How does the number of automorphisms of $J(C)$ ignoring polarizations compare with the number of automorphisms taking them into account?

In a question previously asked on MO (Units of Endomorphism Rings of Jacobian Varieties with Real Multiplication), a result from the paper ``On the fields of rationality for curves and for their Jacobian varieties'' by Sekiguchi is cited to give a bound for the number of automorphisms of a Jacobian $J(C)$ of a curve $C$. As mentioned in the answer, the original question seems to mix up whether polarizations are taken into account in different places.

What is the correct way to interpret the main theorem in Sekiguchi's original paper regarding polarizations of $J(C)$ (e.g. when counting automorphisms)? From the restrictions that we get, it does look like they are taken into account in this case, but I just wanted to be careful about how this works. Also, what are some examples of automorphisms that respect polarizations and ones that don't?

On another note: Is $\text{Aut } J(C) = (\text{End } J(C))^\times$ ever infinite? How does the number of automorphisms of $J(C)$ ignoring polarizations compare with the number of automorphisms taking them into account?

In a question previously asked on MO (Units of Endomorphism Rings of Jacobian Varieties with Real Multiplication), a result from the paper ``On the fields of rationality for curves and for their Jacobian varieties'' by Sekiguchi is cited to give a bound for the number of automorphisms of a Jacobian $J(C)$ of a curve $C$. As mentioned in the answer, the original question seems to mix up whether polarizations are taken into account in different places.

What is the correct way to interpret the main theorem in Sekiguchi's original paper regarding polarizations of $J(C)$ (e.g. when used automorphisms)? From the restrictions that we get, it does look like they are taken into account in this case, but I just wanted to be careful about how this works.

In a question previously asked on MO (Units of Endomorphism Rings of Jacobian Varieties with Real Multiplication), a result from the paper ``On the fields of rationality for curves and for their Jacobian varieties'' by Sekiguchi is cited to give a bound for the number of automorphisms of a Jacobian $J(C)$ of a curve $C$. As mentioned in the answer, the original question seems to mix up whether polarizations are taken into account in different places.

What is the correct way to interpret the main theorem in Sekiguchi's original paper regarding polarizations of $J(C)$ (e.g. when counting automorphisms)? From the restrictions that we get, it does look like they are taken into account in this case, but I just wanted to be careful about how this works.

On another note: Is $\text{Aut } J(C) = (\text{End } J(C))^\times$ ever infinite? How does the number of automorphisms of $J(C)$ ignoring polarizations compare with the number of automorphisms taking them into account?