Definability of restricted theta functions and families of abelian varieties

By Ya'acov Peterzil, Sergei Starchenko. 2011

63 citations recorded by the source corpus.

We consider some classical maps from the theory of abelian varieties and their moduli spaces and prove their definability, on restricted domains, in the o-minimal structure $\Rae$. In particular, we prove that the embedding of moduli space of principally polarized ableian varierty, $Sp(2g,\Z)\backslash \CH_g$, is definable in $\Rae$, when restricted to Siegel's fundamental set $\fF_g$. We also prove the definability, on appropriate domains, of embeddings of families of abelian varieties into projective space.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8212dd6470393094eaaa5a36ba8d1ae0012b7794

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Citations in the corpus

Ax-Lindemann for $\mathcal{A}_g$ (2014, methodology): “An essential input to enable the o-minimal machinery to be applied is the following result due to Peterzil-Starchenko [26]: Theorem 2.6.”
Ax-Schanuel for Shimura varieties (2017, unknown): “For a general Shimura variety, this result is due to Klingler-Ullmo-Yafaev [18], generalizing results of Peterzil-Starchenko [29] for moduli spaces of abelian varieties.”
Tame topology of arithmetic quotients and algebraicity of Hodge loci (2018, methodology): “It is worth noticing that the proof of the more general Theorem 1.1 is easier than the one in [PS13] (which uses explicit theta functions) or the one in [KUY16] (which uses the delicate toroidal compactiﬁcations of [AMRT75]): it relies exclusively on classical properties of Siegel sets (see Section…”
Ax–Schanuel for the $j$-function (2014, methodology): “Studying the possible structure of the Zariski closure of the integer points of the stabilizer of V allows us to apply an o-minimal version of Chow’s theorem due to Peterzil-Starchenko [18, 19] to prove that the image of U in Y (1) is algebraic.”
Towards the Andr\'e-Oort conjecture for mixed Shimura varieties: the Ax-Lindemann theorem and lower bounds for Galois orbits of special points (2013, background): “Therefore for any Ag(N), the fundamental set [0, N) 2g × FG ⊂ V2g(R)×Hg ' X 2g,a is the one considered in [45].”
The hyperbolic Ax-Lindemann-Weierstraß conjecture (2015, unknown): “In [26], in part inspired by [34] and relying on [20], Pila and Tsimerman proved Theorem 1.”