$j$-invariants of CM curves $E$ over (say) the complex numbers are known as singular moduli. As the theory of complex multiplication explains, singular moduli are algebraic integers of great arithmetical significance, for they generate the maximal abelian extensions of imaginary quadratic fields.

I would be interested in reading about generalizations of this, where $E$ is replaced by an abelian variety of higher dimension with a big endomorphism ring. What are the singular moduli in this case, and what are their features? Can you recommend some references please? Or better address this question? Thanks!

[ERRATUM: singular moduli only generate the maximal abelian extensions of imaginary quadratic fields K over which ${\rm Gal}(K/\mathbf{Q})$ acts by inversion.]

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    $\begingroup$ For the case of abelian surfaces, Igusa has constructed invariants that generalize the $j$-invariants of elliptic curves. In particular, there is a generalization of Hilbert Class Polynomial using them. I am definitely not an expert, but I've found this paper that can help msp.berkeley.edu/ant/2011/5-4/p03.xhtml $\endgroup$ – Nicolas B. Feb 10 '12 at 9:13
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    $\begingroup$ Here are a few other papers that discuss Igusa invariants: - MR0141643 Igusa, Jun-ichi; On Siegel modular forms of genus two. Amer. J. Math. 84 (1962) - Bröker, Reinier; Lauter, Kristin; Modular polynomials for genus 2. LMS J. Comput. Math. 12 (2009) - Kirsten Eisentraeger, Kristin Lauter; A CRT algorithm for constructing genus 2 curves over finite fields. arXiv:math/0405305. to appear in Proceedings of AGCT-10. $\endgroup$ – Joe Silverman Feb 10 '12 at 12:39

Some standard references on abelian varieties having complex multiplication:

  • MR0125113 Shimura, Goro; Taniyama, Yutaka. Complex multiplication of abelian varieties and its applications to number theory. Publications of the Mathematical Society of Japan, 6 The Mathematical Society of Japan, Tokyo 1961 xi+159 pp.
  • MR0314766 Shimura, Goro. Introduction to the arithmetic theory of automorphic functions. Kanô Memorial Lectures, No. 1. Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. xiv+267 pp. [Sections 5.5 and 7.8]
  • MR0713612 Lang, Serge. Complex multiplication. Grundlehren der Mathematischen Wissenschaften, 255. Springer-Verlag, New York, 1983. viii+184 pp.
  • MR0236190 Serre, Jean-Pierre; Tate, John. Good reduction of abelian varieties. Ann. of Math. (2) 88 1968 492–517.

The article of Serre and Tate contains a proof using the criterion of Neron-Ogg-Shafarevich (described in the article) that an abelian variety with complex multiplication has everyone potential good reduction, which is the analogue of the $j$-invariant of an elliptic curve being an algebraic integer.


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