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][1]), but it seems to discuss only the simple case:

[![enter image description here][2]][2]

There are these [two][4] [papers][5] 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*][3]) 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? 


  [1]: https://projecteuclid.org/euclid.dmj/1077306167
  [2]: https://i.sstatic.net/Qymg2.png
  [3]: http://www.numdam.org/article/CM_1986__57_2_127_0.pdf
  [4]: https://www.sciencedirect.com/science/article/pii/S0022314X96900268
  [5]: https://arxiv.org/pdf/math/0304471.pdf