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Definability of restricted theta functions and families of abelian varieties. (English) Zbl 1284.03215
In [Ann. Math. (2) 173, No. 3, 1779–1840 (2011; Zbl 1243.14022)], J. Pila gave an unconditional proof of the André-Oort conjecture for the Shimura variety $$\mathbb C^n$$. One of the central ingredients to this proof is that the Weierstrass $$\wp$$ function is, in a suitable sense, definable in the o-minimal structure $$\mathbb R_{\mathrm{an}, \mathrm{exp}}$$. The present paper proves the analogue of this ingredient for Siegel modular varieties $$\text{Sp}(2g, \mathbb Z) \backslash \mathbb H_g$$, $$g \geq 1$$. Note however that to prove the André-Oort conjecture in this generality along the same lines, several other ingredients are still missing.
More precisely, fix $$g \geq 1$$ and a diagonal matrix $$D = \text{Diag}(d_1, \dots, d_g)$$ with $$d_1 \mid d_2 \mid \dots \mid d_g$$ non-negative integers. Then all the complex abelian varieties of polarization type $$D$$ are of the form $$\mathcal{E}_\tau^D := \mathbb C^g/(\tau \mathbb Z^g + D \mathbb Z^g)$$, where $$\tau$$ runs over the set $$\mathbb H_g$$ of symmetric $$(g \times g)$$-matrices with a positive definite imaginary part. Write $$h_\tau^D : \mathbb C^g \to \mathbb P^k (\mathbb C)$$ for a suitable holomorphic map inducing an embedding $$\mathcal{E}_\tau^D \to \mathbb P^k (\mathbb C)$$ (where $$k$$ only depends on $$D$$). Morally, the main result of the paper (Theorem 1.2) is that the map $$\mathbb H_g \times \mathbb C^g \to \mathbb P^k (\mathbb C), (\tau, z) \mapsto h_\tau^D(z)$$ is definable in $$\mathbb R_{\mathrm{an}, \mathrm{exp}}$$.
Formulated like this, this is not possible for two reasons: $$h_\tau^D$$ is periodic for each fixed $$\tau$$, and, moreover, $$\tau \mapsto \mathcal{E}_\tau^D$$ is periodic in the sense that $$\mathcal{E}_\tau^D$$ and $$\mathcal{E}_{\tau'}^D$$ are isomorphic as polarized abelian varieties iff $$\tau$$ and $$\tau'$$ lie in the same orbit of the action of $$G_D$$ on $$\mathbb H_g$$, where $$G_D$$ is a discrete subgroup of $$\text{Sp}(2g, \mathbb Q)$$. A more precise formulation of Theorem 1.2 is that the restriction of $$(\tau, z) \mapsto h_\tau^D(z)$$ to a subset $$U$$ of $$\mathbb H_g \times \mathbb C^g$$ is definable in $$\mathbb R_{\mathrm{an}, \mathrm{exp}}$$, where the projection of $$U$$ to $$\mathbb H_g$$ contains a fundamental domain of the action of $$G_D$$ on $$\mathbb H_g$$, and for each $$\tau$$ in this projection, the corresponding fiber of $$U$$ contains a fundamental domain of the map $$\mathbb C^g \to \mathcal{E}_\tau^D$$.
The paper also contains several related results that might be of independent interest. In particular, one of the main ingredients to the proof of Theorem 1.2 is the definability of certain Riemann theta functions $$\vartheta[{a \atop b}](z, \tau)$$.

##### MSC:
 03C64 Model theory of ordered structures; o-minimality 03C98 Applications of model theory 11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms 11G18 Arithmetic aspects of modular and Shimura varieties 14G35 Modular and Shimura varieties 14K25 Theta functions and abelian varieties
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