# Automorphisms of Generic Abelian Varieties

Automorphism groups of elliptic curves are very well understood. Of course, every elliptic curve has the automorphism $[-1]$ of order $2$. If we are over a (algebraically closed) field, this is the only non-trivial automorphism iff the $j$-invariant of our elliptic curve is neither $0$ nor $1728$. Over $\mathbb{C}$ the only other possibilities for automorphism groups are $\mathbb{Z}/4$ and $\mathbb{Z}/6$ occuring for $E = \mathbb{C}/\Lambda$ where $\Lambda$ is the square lattice or the "honeycomb" lattice respectively. In particular, the generic elliptic curve has automorphism group $\mathbb{Z}/2$, i.e. the locus of elliptic curves with bigger automorphism group in the moduli space of elliptic curves has codimension bigger than $0$.

I am interested in the analogous question for abelian varieties. To force all the automorphism groups to be finite, it seems sensible to consider (principally) polarized abelian varieties.

Let $\mathcal{A}_g$ be the moduli space of (principally) polarized complex abelian varieties of dimension $g$. What is the codimension of the locus of such varieties with automorphism group bigger than $\mathbb{Z}/2$ ?

Likewise, one can ask the same question not about arbitrary principally polarized abelian varieties, but such equipped with an endomorphism and level structure. So, in addition to the polarization on our (complex) abelian variety $A$ we also want a ring homomorphism $\mathcal{O}_F \to End(A)$ (for a fixed, say, quadratic imaginary field $F$), compatible with the Rosatti involution, and a level structure (plus some determinant/trace-condition). For a generic such $A$, its automorphism group may contain now $\mathcal{O}_F^\times$, depending on the level structure.

Let $Sh_g$ be the moduli space of such complex abelian varieties with PEL-structure (with the choices of $F$ and level implicit) of dimension $g$. Let $G$ be the smallest automorphism group of a point in $Sh_g$. What is the codimension of the locus of points of $Sh_g$ having automorphism group bigger than $G$?

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The integral symplectic group $Sp_{2g}\mathbb{Z}$ acts biholomorphically on the Siegel upper half space $\mathfrak{h}_g$. So for $H$ a finite subgroup of $Sp_{2g}\mathbb{Z}$ the fixed point sets $\mathfrak{h}_g^H$ are totally geodesic contractible holomorphic subvarieties, hence all have even dimension. Explicitly we need to determine the dimension of the centralizer $Z(H)$ of $H$ in $Sp_{2g}\mathbb{R}$. For genus $g=2$, we can exhibit explicit finite subgroups whose fixed point sets are 4-dimensional subspaces of the 6-dimensional $\mathfrak{h}_2$. –  J. Martel Apr 23 '13 at 21:45
For instance, these finite subgroups (all of which are cyclic) are computed in Connelly and Kozniewski's "Finiteness properties of classifying spaces for $\Gamma$-actions", wherein they refer to a paper by K.Ueno "On fibre spaces of normally polarized abelian varieties of dimension 2", I, J. Fac. Sci. Univ. Tokyo 18 (1971) 37-95. I was unable to locate a copy of this Ueno paper. –  J. Martel Apr 23 '13 at 21:50
Moreover if $H$ is a finite subgroup of $Sp_{2g}\mathbb{R}$ acting irreducibly on $(\mathbb{R}^{2g}, \omega)$ then the corresponding fixed point set in $\mathbb{h}_g$ will be a point. We'll find that finite subgroups which are very $\mathbb{R}$-reducible yield larger dimensional fixed point sets. –  J. Martel Apr 23 '13 at 22:07
I would guess that for your first question if $g>1$ then the ppavs that are products of an elliptic curve and a ppav of dimension $g-1$ would give a maximal codimensional component of the locus of ppavs with extra automorphisms (and this is probably the unique component of this dimension if $g>2$). –  ulrich Apr 24 '13 at 5:54
@J. Martel: There is no problem with the polarisation at all; one just takes the product polarisation! –  ulrich Apr 25 '13 at 4:46