Automorphisms of the Jacobian

Here, $$A$$ is any abelian variety. This post consists of the backstory of this paper, and something interesting I learned about the relationship between the size of $$Aut(A)$$, and the number of principal polarizations $$A$$ has.

Backstory

In the summer of 2017, I wanted to compute a period matrix of a particular genus 3 curve, and I found a wonderful team of low dimensional geometers: Dami Lee and Matthias Weber. After they helped me, I read Dami’s work out of curiosity, and found her calculations of automorphism groups of curves fascinating due to their geometric simplicity. Further, it seemed that her methods of using a tesselation to compute or visualize and automorphism group would generalize naturally to higher dimensions. So, we set to work trying to figure that out. This led us to generalize her prior work on automorphism groups of curves (with equal weight weierstrass points) to a broader class of curves (with nonequal weight weierstrass points).

I was interested in her lovely descriptions of the automorphism groups and their actions as I was trying to use her work to model the action of a subgroup of the Morava Stabilizer group acting on Lubin Tate space. Turns out I will likely have to do this separately, because so much information changes about the automorphism group when you base change your curve from $$\mathbb{C}$$ to $$\mathbb{F}_p$$, and her methods only work of $$\mathbb{C}$$ (relying on properties of Weierstrass points and so on).

In the interim, I separately looked into other ways to compute automorphisms of curves and their Jacobians, and ran into the work of Magma and Sage contributors Edgar Costa, John Voight, Nicolas Mascot, and Jeroen Sijsling. They had a program and paper to compute endomorphism groups of Jacobians, and I wanted to calculate automorphism groups (because, in the end, this is all for modeling subgroups of the Morava Stabilizer group, the automorphism group of a height n formal group law in char p). Together, we wrote a program that calculates the automorphism group of a Jacobian, given its period matrix. Several glitches later, and we realized that we were finding several different automorphism groups for the same Jacobian (as a period matrix), because there were several different automorphism groups — for the different principal polarizations. So, I found the “correct one” by putting in the information of the original plane curve as well. But this detour led us to find several interesting different principal polarizations….

Automorphism Groups and Narasimhan-Nori

…It also led me to learn about the following magic relationship between the size of an automorphism group of an abelian variety $$A$$ and the number of its principle polarizations (this is from a paper of Lange: Abelian varieties with several principal polarizations).

I still do not understand the computability of the order of the set of interest $$\Pi(A)$$ (the number of principal polarizations up to iso of $$A$$) according to Theorem 1.5:
Some notation:
Fix a principal polarization $$L_0: A \to \hat{A}$$ ,
Given a map $$r \in Aut(A)$$, let $$\hat{r}$$ be the dual map in $$Aut(\hat{A})$$
Let $$’$$ indicate the Rosati involution wrt $$L_0$$, that is $$r’ := L_0^{-1} \circ r^ \circ L_0$$

Let us first look at the set of $$r \in Aut(A)$$ such that the following two conditions are met:

(1) $$r’ = r$$ (i.e., $$r$$ is preserved under the Rosati involution wrt $$L_0$$)
(2) the zeros of the minimal polynomial of $$r$$ (wrt the rational embedding) are all positive.

Side comment:
(1) $$\Leftrightarrow$$ $$(r^g) = g!$$ [lemma 1.2] (2) $$\Leftrightarrow$$ $$(L_0^{g-i} \cdot r^i) > 0$$ [lemma 1.3] These conditions give us that $$r$$ is a principal polarization by lemma 1.1

Once we have this set, call it $$U(A)$$, let $$Aut(A)$$ act on it:

$$Aut(A) \times U(A) \to U(A)$$
$$(g, a) \mapsto g’ a g$$

where $$’$$ indicates the rosati involution wrt $$L_0$$.

We mod out the set $$U(A)$$ by the above action of $$Aut(A)$$, call this set $$\Pi(A)$$.

And there we are!