# Principally Polarized CM Abelian Variety

I am interested in considering examples of abelian varieties that are principally polarized with CM in dimension three. However, I am struggling to construct or find even a single instance.

In particular, I attempted to take a totally real field of degree 3 (with relatively small discriminant) from LMFDB, construct the corresponding CM field of degree 6, then considering the Hermitian form $(a,b) \mapsto \textrm{Tr} (a\overline{b})$. When restricting to get the Riemann form, I discover that the degree of the corresponding polarizations are powers of the discriminant of the field (probably not an accident). As far as being principally polarized though, this is not helpful.

Searching the literature, I've only found the following result: https://arxiv.org/pdf/1208.5599.pdf which only discusses even dimensions.

I am wondering if there is a reference that might discuss either odd dimension or just $d = 3$.

Edit: I forgot an important criterion that I'd like the variety to have a compatible cyclic action of degree $3$ or degree $4$ (I've focused mainly on degree $3$ so far).

• The Jacobian of the Klein quartic is a principally polarized abelian variety of dimension 3 and it decomposes into three CM elliptic curves. But the CM field does not have degree 6 as the abelian variety is not simple so you may want to exclude this case. – François Brunault Sep 9 '18 at 23:10
• The jacobian of $y^2=x^7-1$ clearly has CM by the field of $7$-th roots of unity. – Felipe Voloch Sep 9 '18 at 23:33
• Another example is $y^3 = x^4 - x$ which has CM by the 9-th roots of unity. If $\zeta$ is a primitive 9-th root of unity, then $(x,y) \mapsto (\zeta^3 x, \zeta y)$ is an automorphism of order 9. – Ari Shnidman Sep 11 '18 at 2:28
• Thanks everyone for your helpful comments. I have what I need from this, and I think the proper protocol now is to leave this up for future references. Thank you all again. – KTT30 Sep 18 '18 at 17:18