Timeline for Is there an excplicit number field of definition for an Abelian Variety $A/\mathbb{C}$ with CM?
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| Aug 7, 2014 at 0:22 | comment | added | user27920 | @VesselinDimitrov: No, fields of moduli are not fields of definition. Very far from it. This is related to one reason that coarse moduli spaces (rather than moduli stacks) generally suck: their rational points have very limited meaning. If you don't allow isogeny change over $\mathbf{C}$ then it is false that a field of definition can be predicted by the CM type which is isogeny-invariant. It is a very subtle problem beyond dimension 1 to nail down a specific field of definition for a specific CM abelian variety (equipped with its CM-structure) without moving around in the isogeny class. | |
| Aug 7, 2014 at 0:16 | comment | added | user27920 | @JohnBinder: The specific "accident" with quadratic fields is that all orders are monogenic. The role of polarizations is also more subtle beyond dimension 1. The $K_{\ell}^{\times}$-valued $G_F$-characters arising from abelian varieties over $F$ with CM by K are the $\ell$-adic "avatars" of $K^{\times}$-valued algebraic Hecke characters of $F$ whose "algebraic part" is the reflex norm, by Theorems 2.5.1 and 2.5.2 in that CM lifting book (due to Shimura-Taniyama & Casselman respectively), and by Cor. A.4.6.5 (really Prop. A.4.6.4 and Example A.4.4.1) such a Hecke character always exists. | |
| Aug 6, 2014 at 20:05 | comment | added | John Binder | @user52824 that is weaker than what I asked about, though possibly enough for my purposes. At the end of the day I'm interested in which Galois characters $G_F \to K_\ell^\times$ arise as Tate modules of abelian varieties with CM by $K$. I'll take a look at 'CM and lifting problems' (which happens to be on my desk). Thank you moreover for putting into words my thoughts about the 'homology lattice'; I was thinking about why the proof for curves would fail in higher dimension couldn't phrase it nearly that eloquently. | |
| Aug 6, 2014 at 18:53 | comment | added | Vesselin Dimitrov | [ @user52824: Rather than the field of moduli $k_0$ itself, I meant its compositum with $K^*$, denoted $k_0^* = k_0K^*$ in the Shimura-Taniyama book. It is an unramified class field over $K^*$, not depending on a choice of polarization. Would it be a field of definition?] | |
| Aug 6, 2014 at 17:31 | comment | added | Vesselin Dimitrov | @user52824: My apology. Indeed I was referring to Main Theorem 1 (I was looking at an old edition of the book), which describes the field of moduli as an unramified class field over $K^*$ (and I neglected to mention the restrictions on the CM type...). But isn't a field of moduli a field of definition? So is it, then, definitely false in higher dimension that a CM abelian variety has to be defined over the Hilbert class field of the reflex field? | |
| Aug 6, 2014 at 17:14 | comment | added | user27920 | Keeping track of the CM order (understood to be saturated in the endomorphism ring) is notoriously more difficult beyond dimension 1 because only in dimension 1 does it happen (thanks to a total accident with quadratic fields which fails in all higher degrees) that the homology lattice is always an invertible module over the CM order. How to make a link to class groups of the CM-order when the homology lattice is not invertible over that order? | |
| Aug 6, 2014 at 17:12 | comment | added | user27920 | Although it is weaker than what you want, within the $K$-linear isogeny class of $A$ there is always a member which (together with its CM-structure) is defined over the field of moduli $F$ of the reflex norm for the CM type $\Phi$ of $A$ (equipped with its CM-structure), and such a descent is unique up to $K$-linear $F$-isogeny. This amounts to constructing on the ideles of that field of moduli an algebraic Hecke character with the reflex norm as its algebraic part; see Cor. A.4.6.5 and Theorem 2.5.2 in the book "Complex multiplication and lifting problems". | |
| Aug 6, 2014 at 16:56 | comment | added | user27920 | @VesselinDimitrov: Chapter IV in that book is quite long, consisting of many subsections. At least in the more re-issued version of the book by Shimura, there is no "Theorem 1" in Chapter IV, but there is a "Main Theorem 1" in section 15.3 which says nothing about fields of definition and imposes severe restrictions (CM type must be primitive, and CM order must be maximal). Moreover, that result involves a specification of a $\mathbf{Q}_{>0}^{\times}$-homothety class of polarizations (rather than a specific polarization), so it is not determined by the abelian variety alone. Please clarify. | |
| Aug 6, 2014 at 16:43 | comment | added | Vesselin Dimitrov | You can take a suitable ray class field of the reflex field. Cf. Thm. IV. 1 in the Shimura-Taniyama book (Complex multiplication of abelian varieties and applications to number theory). | |
| Aug 6, 2014 at 15:54 | history | asked | John Binder | CC BY-SA 3.0 |