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Catherine Ray
Catherine Ray
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Let ``the simple case" be when we examine the Jacobian of the curve which breakspolarized abelian variety does not break up into a product of elliptic curvespolarized 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:

enter image description here

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 simplenonsimple 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 1: What is known about the simple case at genus greater than 3 (besides Lange's result)?

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

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

Let ``the simple case" be when we examine the Jacobian of the curve which breaks up into a product of elliptic curves.

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:

enter image description here

There are many other papers discussing the simple 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 1: What is known about the simple case at genus greater than 3 (besides Lange's result)?

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

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

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:

enter image description here

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?

Source Link
Catherine Ray
Catherine Ray
  • 3.5k
  • 12
  • 37

What is known about abelian varieties with several principal polarizations?

Let ``the simple case" be when we examine the Jacobian of the curve which breaks up into a product of elliptic curves.

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:

enter image description here

There are many other papers discussing the simple 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 1: What is known about the simple case at genus greater than 3 (besides Lange's result)?

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

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