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.