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Let ``the simple case" be when the polarized abelian variety does not break up into a product of polarized abelian varieties.

I am trying to get an idea of what is known about abelian varieties with several principal polarizations. What I know of is the following, but I wish to make sure I am not missing any key results hidden in other papers.

There is the paper of Lange "Abelian Varieties with Several Principal Polarizations" (paywalled here), but it seems to discuss only the simple case:

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There are these two papers by E. Howe which give examples of genus 2 curves which are nonisomorphic and give the same (simple) Jacobian.

Question 1: What is known about the simple case (besides Lange and Howe's results)?

There are many other papers discussing the nonsimple case at genus 2 (Ibukiyama-Katsura-Oort, Supersingular curves of genus two and class numbers) and 3 (Brock, Superspecial curves of genera two and three).

Question 2: What is known about genus 3 and above nonsimple curves with multiple principal polarizations?

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    $\begingroup$ It's arguable that products of elliptic curves are the simplest kind of abelian varieties but, unfortunately, it's standard to say that abelian varieties are simple if they are not isogenous to products of abelian varieties of smaller dimension. This makes your post a little confusing. If an abelian variety is not simple (in the standard sense) then its endomorphism ring is not a division algebra. So Lange's result is for simple abelian varieties in the standard sense. $\endgroup$
    – Felipe Voloch
    Jul 26, 2018 at 8:31

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