For any imaginary quadratic field $F$, the Hilbert class field $H$ is generated by the $j$-invariant of any elliptic curve with complex multiplication (CM) by $\mathcal O$, the ring of algebraic integers of $F$.

What is the simplest generalization of this well known and useful fact? Since an imaginary quadratic field is the simplest example of a CM field, one is led to ask:

Is there an (albeit conjectural) invariant of abelian surfaces which generates the Hilbert class field of a CM field $K$ (of degree $4$ over $\mathbb Q$)?

Note that $K$ is a totally imaginary quadratic extension of a real quadratic field.


1 Answer 1


The simplest generalisation to abelian surfaces is, I believe, the statement (which is a theorem, not a conjecture) that the Igusa invariants of an abelian surface with CM by $K$ generate an abelian unramified extension of the reflex field of $K$. Note that the reflex field is not, in general, equal to $K$, and that there is no claim that the full Hilbert class field is obtained in this way. There is also a generalisation of this to higher dimensional principally polarised abelian varieties.

  • $\begingroup$ thanks! could you please point to a reference? also, are there interesting cases where the reflex field is $K$? $\endgroup$
    – guest
    May 19, 2017 at 1:52

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.