I have been puzzle by the following question for a while. Suppose that we have an a Shimura variety $Sh(G,h_0)$ given by some datatum $(L, V, \psi, h_0)$ such as in Section 4.9 of "Travaux de Shimura" by Deligne (more details below). Now given an abelian variety $A$ (for example, an abelian variety of CM-type constructed from a given CM-field), how do we know if there is a point on the Shimura variety corresponding to this abelian variety?

More concretely, let $L$ be a finite-dimensional semisimple $\mathbb{Q}$-algebra with a positive involution ($\ell\mapsto \ell^*$). Let $V$ be a faithful $L$-module with a nondegenerate alternating bilinear form $\psi$ such that $$\psi(\ell x, y)=\psi(x, \ell^* y)\qquad \forall \ell \in L,\forall x,y \in V.$$ We write $G$ for the $\mathbb{Q}$-algebraic group of $L$-linear symplectic similitudes of $V$. Let $\mathbb{S}$ be the Deligne torus, and $h_0: \mathbb{S}\to G_\mathbb{R}$ a homomorphism of real algebraic groups such that $V$ is equipped with a Hodge structure of weight $-1$ with polarization $\psi$. In other words, $$V_\mathbb{C}= V^{-1,0}\oplus V^{0, -1},$$ and $\psi(x, h_0(i)y)$ is symmetric and positive definite. Let $t: L\to \mathbb{C}$ denote the trace function $$t(\ell)=Tr(\ell, V^{-1, 0}).$$ Suppose that $K$ is a compact open subgroup of $G(\mathbb{A}^f)$, we write $Sh_K(G, h_0)$ for the (complex) Shimura variety given by the above datum.

Then it is known that the Shimura variety parametrizes isomorphism class of abelian varieties up to isogeny with the following structures:

(a) $\rho: L\to End(A)$ such that $Tr(\rho(\ell), Lie(A))=t(\ell)$.

(b) a homogenous polarization $\bar{p}$ on $A$ that induces the involution on $L$.

(c) a class mod $K$ of $L$-linear symplectic similitudes $H_1(A, \mathbb{Q})\otimes \mathbb{A}^f \to V\otimes \mathbb{A}^f.$

(d)There exists an $L$-linear isomorphism $i: V\to H_1(A, \mathbb{Q})$ such that the push forward of $\psi$ coincides with the Riemann form of $\bar{p}$ up to a factor.

(e)Let $h: \mathbb{S}\to GL(H_1(A, \mathbb{Q}))$ be the homomorphism determined by $A$, then $i^{-1}h i $ is conjugate with $h_0$ by an element of $G(\mathbb{R})$.

Very often, it is easy to construct abelian vareities that satisfies both conditions (a) and (b). For example, if $L$ is either a total real field or CM-field. We may take an CM-algebra $E$ containing $L$ with a chosen CM-type such that (a) is satisfied. Then is there a point on the Shimura variety corresponding to $A$ or not? Could there be an abelian variety $A$ satisfying condition (a) (b) yet the push forward of $\psi$ is never a polarization on $A$ for any $L$-linear isomorphism $i: V\to H_1(A, \mathbb{Q})$?

Edit: My motivation is the following. Well, it seems to me that a lot of times, Shimura varieties arises implicitly, so both $\psi$ and $h_0$ are not really available for concrete calculations. For example, you have a polarized abelian scheme over an complex manifold, and then take the generic Mumford-Tate group and construct a Shimura variety from there. Suppose there is a triple $(A, \rho, \bar{p})$. By knowing that there is a point on the Shimura variety corresponding to it (probably with a different polarization), one might have something nice to say about the original family. How do one check that this is the case? Clearly, (a) and (b) are necessary conditions to check, but then what? In the most simple case, can one determine if there is a CM-point on the Shimura variety with complex multiplication by $E$ or not?

Edit2: I guess here is a concrete example of what I am asking. Say take the family of curves $C_\lambda: y^5=x(x-1)(x-\lambda)$ over the punctured complex plane $S:=\mathbb{C}-\{0, 1\}$. This is one of the curves considered by A. J. de Jong and R. Noot in their paper "Jacobians with complex multiplication". They were able to prove that there are infinitely many $\lambda\in \mathbb{C}$ such that the Jacobian $J_\lambda$ of the corresponding curve has complex multiplication. This follows from the fact that the Shimura variety $Sh$ of PEL-type that naturally arises from this family is of dimension one, and thus the image of the map $S\to Sh$ is dense. So my question is for what quadratic extensions $E$ of $\mathbb{Q}(\zeta_5)$ may appear for these complex multiplications? ($E$ is a CM-field, of course.) Do all of them appear or are there some conditions that $E$ has to satisfy?

  • $\begingroup$ I don't understand this question. Isn't the whole point that every point on the symmetric space consisting of conjugates of $h_0$ gives an abelian variety satisfying the conditions, by the general complex analytic uniformisation theory? I must confess I've never studied the details of this but that's how I'd always imagined it worked. On the other hand if you're just going to take a random ab var satisfying some of the conditions and ask whether it satisfies some of the others then presumably the answer is "not always"? $\endgroup$ – user30035 Mar 19 '13 at 7:29

Your Answer

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

Browse other questions tagged or ask your own question.