Question

Given a CM field we can use its maximal order (and a choice of CM type) to construct an abelian variety $\mathbb{C}^g/\Lambda$ with complex multiplication by the maximal order.

How do I (or where can I find information on) explicitly write down equations for a projective embedding of this variety, and the action of the CM order on points? Is this implemented anywhere?

For genus one we can use the Eisenstein series to find the coefficients of a Weierstrass model for the elliptic curve. So I'm looking for a generalization of this.

Answer 1

This is not the answer. I am adding some relevant papers (some possessing good examples) which won't fit in the comments section.

I have been wanting to know the answer to this question as well. It seems one has to find an ample line bundle, and then calculate the Riemann theta relations which define the projective embedding. The equation(s) of the embedding is decided by relation between the theta functions. The wikipedia page on equations for abelian varieties was written Charles Matthews. Perhaps he might have more to say on this question...

MR0946234 (89i:14038) Barth, Wolf . Abelian surfaces with $(1,2)$-polarization. Algebraic geometry, Sendai, 1985, 41--84, Adv. Stud. Pure Math., 10, North-Holland, Amsterdam, 1987.

MR1257320 (95e:14033) Barth, W. ; Nieto, I. Abelian surfaces of type $(1,3)$ and quartic surfaces with $16$ skew lines. J. Algebraic Geom. 3 (1994), no. 2, 173--222.

MR1336597 (96h:14064) Barth, W. Quadratic equations for level-$3$ abelian surfaces. Abelian varieties (Egloffstein, 1993), 1--18, de Gruyter, Berlin, 1995.

MR1602020 (99d:14046) Gross, Mark ; Popescu, Sorin . Equations of $(1,d)$-polarized abelian surfaces. Math. Ann. 310 (1998), no. 2, 333--377.

MR2194379 (2006m:14054) Gunji, Keiichi . Defining equations of the universal abelian surfaces with level three structure. Manuscripta Math. 119 (2006), no. 1, 61--96.

MR0611469 (82h:14028) Sasaki, Ryuji . Some remarks on the equations defining abelian varieties. Math. Z. 177 (1981), no. 1, 49--60.

MR1048533 (91e:14043) Birkenhake, Ch. ; Lange, H. Cubic theta relations. J. Reine Angew. Math. 407 (1990), 167--177.