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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)?

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    $\begingroup$ Too many questions. As for the last one: $\mathrm{Aut}(JC)$ (polarized) is always finite; $\mathrm{Aut}(JC)$ (unpolarized) can very well be infinite: if $E$ is an elliptic curve, $\mathrm{Aut}(E\times E)$ contains $\mathrm{GL}(2,\mathbb{Z})$. $\endgroup$
    – abx
    Commented Aug 1, 2017 at 4:14
  • $\begingroup$ Thanks! Would $\begin{pmatrix} \pm1 & 0 \\ 0 & \pm1 \end{pmatrix}$ give an automorphism of $E \times E$ respecting polarizations? Does being compatible with the polarization depend on the one we take? $\endgroup$
    – modnar
    Commented Aug 1, 2017 at 5:13
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    $\begingroup$ 1: Yes, if you take the product polarization ($(-1)$ preserves the polarization of $E$). 2: Yes. The automorphism group acts on the cone of ample classes, and the stabilizer of a given class depends very much of the class. $\endgroup$
    – abx
    Commented Aug 1, 2017 at 5:20
  • $\begingroup$ Thanks! If we take $J(C)=E×E$ while taking the canonical polarization for$J(C)$, I don't think the product polarization gives something compatible with the one on $J(C)$. Would this automorphism still work? If not, what would be a nontrivial example that would work? Does the answer the first question depend on the curve $C$? I heard a claim that Jacobians can't be isomorphic to products of principally polarized abelian varieties with the product polarization, but I'm not sure why this would be true. $\endgroup$
    – modnar
    Commented Aug 2, 2017 at 5:50
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    $\begingroup$ Again too many questions... What do you mean by $J(C)=E\times E$? What is $C$??? And: Jacobians are not product because their Theta divisor is irreducible. $\endgroup$
    – abx
    Commented Aug 2, 2017 at 6:41

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