scholarly journals Rationality of twists of the Siegel modular variety of genus 2 and level 3

2021 ◽  
Author(s):  
Frank Calegari ◽  
Shiva Chidambaram
Keyword(s):  
Modular Variety ◽  
Siegel Modular Variety ◽  
Genus 2 ◽  
Level 3

scholarly journals On the Oort conjecture for Shimura varieties of unitary and orthogonal types

2016 ◽  
Vol 152 (5) ◽  
pp. 889-917 ◽  
Author(s):  
Ke Chen ◽  
Xin Lu ◽  
Kang Zuo
Keyword(s):  
Real Number ◽  
Number Field ◽  
Higgs Bundles ◽  
Modular Variety ◽  
Spin Groups ◽  
Siegel Modular Variety ◽  
Hermitian Space ◽  
Totally Real ◽  
Formula Type ◽  
Real Number Field

In this paper we study the Oort conjecture concerning the non-existence of Shimura subvarieties contained generically in the Torelli locus in the Siegel modular variety${\mathcal{A}}_{g}$. Using the poly-stability of Higgs bundles on curves and the slope inequality of Xiao on fibered surfaces, we show that a Shimura curve$C$is not contained generically in the Torelli locus if its canonical Higgs bundle contains a unitary Higgs subbundle of rank at least$(4g+2)/5$. From this we prove that a Shimura subvariety of$\mathbf{SU}(n,1)$type is not contained generically in the Torelli locus when a numerical inequality holds, which involves the genus$g$, the dimension$n+1$, the degree$2d$of CM field of the Hermitian space, and the type of the symplectic representation defining the Shimura subdatum. A similar result holds for Shimura subvarieties of$\mathbf{SO}(n,2)$type, defined by spin groups associated to quadratic spaces over a totally real number field of degree at least$6$subject to some natural constraints of signatures.

scholarly journals The Siegel modular variety of degree two and level three

2000 ◽  
Vol 353 (08) ◽  
pp. 3267-3306 ◽  
Author(s):  
J. William Hoffman ◽  
Steven H. Weintraub
Keyword(s):  
Modular Variety ◽  
Siegel Modular Variety
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