Units of Endomorphism Rings of Jacobian Varieties with Real Multiplication

Question

Let $(A,a)$ be a principally polarised (with indecomposable polarisation) Abelian variety over $\mathbb C$. Assume that End(A) contains an order $R$ of a totally real number field of degree $>1$ over $\mathbb Q$ and that $R$ respects the polarisation of $A$.

The units of $R^*$ are therefore automorphisms of $(A,a)$.

If $g = \dim(A)$ and $g = \text{rank}_{\mathbb Z}(R)$ then my understanding is that such $(A,a)$ are parameterised by a Hilbert Modular variety of dimension $g$. Therefore if $g$ is small, the Hilbert modular variety will intersect the locus of Jacobians in the moduli space of principally polarised Abelian varieties. (The locus of Jacobians being 3g-3, that of the moduli space being g(g+1)/2. The locus of hyperelliptic curves has locus 2g-1).

On other hand: i) the Dirichlet unit theorem applies to all orders of number rings and implies that $R^*$ contains elements of infinite order. ii) a Theorem of Sekiguchi implies that Aut(C) = Aut(\Jac(C)) when C is hyperelliptic, and that Aut(C) x { \pm 1} = \Aut(\Jac(C)) otherwise.
iii) Hurwitz's theorem implies that # \Aut(C) <= 84(g-1).

This suggests a contradiction: namely that if $C$ is hyperelliptic and End(\Jac(C)) contains $R$ (e.g. take $g= 2$ and $R$ an order in $\mathbb Q(\sqrt{D})$), then it will have an automorphim group containing $R^*$ which is infinite.

I presume I am making a very basic error somewhere but I don't see where. Thanks.

Answer 1

I presume I am making a very basic error somewhere but I don't see where.

Yes: you are confusing the automorphism group $\operatorname{Aut} A = (\operatorname{End} A)^{\times}$ with the automorphism group of the polarized abelian variety $(A,a)$. The former can be infinite as soon as $g = \operatorname{dim} A \geq 2$ but the latter is always finite.

Let me also say that the part about Jacobians looks like a red herring to me: to define a moduli space of Hilbert modular varieties, you do need to include data on the "polarization type" (which is probably part of your confusion: the polarization is definitely a key part of the Hilbert modular picture; it just gets neglected in the definition of the endomorphism ring): in particular, the most common definition of Hilbert modular varieties constructs them as subvarieties of the Siegel moduli space $\mathcal{A}_{g,1}$ of principally polarized abelian $g$-folds. These standard constructions do give endomorphisms by the full ring of integers of your degree $g$ totally real number field. (For that matter, every complex abelian variety is isogenous to a principally polarized abelian variety and whether the automorphism group is infinite depends only on the isogeny class.)