What are the possible automorphism groups of a principally polarized abelian variety $(A,\lambda)$ of dimension $g,$ say an abelian surface ($g=2$) over the complex numbers or algebraic closure of a finite field? The fact that the moduli stack $A\_g$ is of finite diagonal (over the integers) implies that the automorphism groups are all finite, but do we know more? Like the size.


When $g=1$ this is given in Silverman I, p.103.


**Edit:** Let me make the question more specific. Let $(A,\lambda)$ be an $\mathbb F\_q$-point of $A\_g$ (i.e. an abelian variety $A$ over $\mathbb F\_q$ of dimension $g$ and a principal polarization $\lambda$). We want to consider its automorphism group (over $\mathbb F\_q$).


Let $\pi:A\_{g,N}\to A\_g$ be the natural projection, where $A\_{g,N}$ is the moduli stack of p.p.a.v. of dimension $g$ with a level $N$ structure (a symplectic isomorphism $H^1(A,Z/N)\to(Z/N)^{2g}$). We always assume $q$ is prime to $N.$ Note that $\pi$ is a $G$-torsor, for $G=GSp(2g,Z/N),$ so it gives a surjective homomorphism $\pi\_1(A\_g)\to G.$ The sheaf $\pi\_\*\mathbb Q\_l$ on $A\_g$ is lisse (even locally constant), corresponding to the representation of $\pi\_1(A\_g)$ obtained from the regular representation $\mathbb Q\_l[G]$ of $G$ and the projection $\pi\_1(A\_g)\to G.$ For any $\mathbb F\_q$-point $x$ of $A\_g,$ the local trace $\text{Tr}(Frob\_x,(\pi\_\*\mathbb Q\_l)\_{\overline{x}})$ is either $|G|$ or 0, depending on $Frob\_x\in\pi\_1(A\_g)$ is mapped to 1 in $G$ or not. 


We have isomorphisms $H^i\_c(A\_{g,N},\mathbb Q\_l)=H^i\_c(A\_g,\pi\_\*\mathbb Q\_l).$ By Lefschetz trace formula, applied to both $\mathbb Q\_l$ on $A\_{g,N}$ and $\pi\_\*\mathbb Q\_l$ on $A\_g,$ we have 


$$|A\_{g,N}(\mathbb F\_q)|=|G|\sum\_{x\in S} 1/\#Aut(A\_x,\lambda\_x),$$


where $S$ is the subset of $[A\_g(\mathbb F\_q)]$ consisting of points $x$ such that all $N$-torsion points of the abelian variety $A\_x$ are rational over $\mathbb F\_q$ (i.e. $|A\_x[N](\mathbb F\_q)|=N^{2g}$), and $(A\_x,\lambda\_x)$ is the pair corresponding to $x.$ This equation gives some constraints (one for each $N$) that $|Aut(A,\lambda)|$ must satisfy. In particular, when $g=N=2$ and $q=3,$ we have $|A\_{2,2}(\mathbb F\_3)|=10$ and $|G|=720$ (in this case $G$ is the symmetric group $S\_6$), and this becomes a puzzle of solving
$$
1/72 = \sum 1/n\_i,
$$
and the $n\_i$'s satisfy some additional conditions. Any idea on how to solve it? I'm considering the contributions of the two parts in $A\_2,$ one for Jacobians of smooth genus 2 curves and one for Jacobians of stable singular ones $E\_1\times E\_2$. Any suggestion is appreciated.


**Edit**: Maybe it's easier to solve it over $\mathbb F\_5,$ since the (orders of the) automorphism groups of smooth genus 2 curves over finite fields of characteristic 5 is known. 



What are the automorphism groups of (principally polarized) abelian varieties?
The standard proof of finiteness goes as follows: the polarization defines a positive involution \* on the endomorphism algebra, and so the automorphisms are the elements of End(A) tensor the reals that are in End(A) and satisfy a\*a=1. Thus the set of automorphisms is the intersection of a discrete set and a compact set.
Let phi(n) be the degree of the field you get by adjoining an nth root of 1 to Q. Then certainly there exist abelian varieties of dimension g on which the nth roots of 1 act if phi(n) divides 2g (for any CM-field E there is an abelian variety with complex multiplication by E). However, unlike the elliptic curve case, there are other possibilities. 


