Duke Mathematical Journal

Definability of restricted theta functions and families of abelian varieties

Ya’acov Peterzil and Sergei Starchenko

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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 Ran,exp. In particular, we prove that the projective embedding of the moduli space of the principally polarized abelian variety Sp(2g,Z)\Hg is definable in Ran,exp when restricted to Siegel’s fundamental set Fg. We also prove the definability on appropriate domains of embeddings of families of abelian varieties into projective spaces.

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

First available in Project Euclid: 15 March 2013

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Zentralblatt MATH identifier

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]


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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