In a question previously asked on MO (https://mathoverflow.net/questions/170330/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.