Definability of restricted theta functions and families of abelian varieties

Ya’acov Peterzil; Sergei Starchenko

doi:10.1215/00127094-2080018

Duke Mathematical Journal

Duke Math. J.
Volume 162, Number 4 (2013), 731-765.

Definability of restricted theta functions and families of abelian varieties

Ya’acov Peterzil and Sergei Starchenko

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Abstract

We consider some classical maps from the theory of abelian varieties and their moduli spaces, and we prove their definability on restricted domains in the o-minimal structure $\mathbb {R}_{\mathrm {an,\,exp}}$ . In particular, we prove that the projective embedding of the moduli space of the principally polarized abelian variety $\operatorname{Sp}(2g,\mathbb {Z})\backslash \mathbb {H}_{g}$ is definable in $\mathbb {R}_{\mathrm {an,\,exp}}$ when restricted to Siegel’s fundamental set $\mathfrak {F}_{g}$ . We also prove the definability on appropriate domains of embeddings of families of abelian varieties into projective spaces.

Article information

Source
Duke Math. J., Volume 162, Number 4 (2013), 731-765.

Dates
First available in Project Euclid: 15 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1363355692

Digital Object Identifier
doi:10.1215/00127094-2080018

Mathematical Reviews number (MathSciNet)
MR3039679

Zentralblatt MATH identifier
1284.03215

Subjects
Primary: 03C64: Model theory of ordered structures; o-minimality 03C98: Applications of model theory [See also 03C60] 14K25: Theta functions [See also 14H42] 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
Secondary: 14G35: Modular and Shimura varieties [See also 11F41, 11F46, 11G18] 11G18: Arithmetic aspects of modular and Shimura varieties [See also 14G35]

Citation

Peterzil, Ya’acov; Starchenko, Sergei. Definability of restricted theta functions and families of abelian varieties. Duke Math. J. 162 (2013), no. 4, 731--765. doi:10.1215/00127094-2080018. https://projecteuclid.org/euclid.dmj/1363355692

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