scholarly journals Holomorphic curves in Shimura varieties

2018 ◽  
Vol 111 (4) ◽  
pp. 379-388 ◽  
Author(s):  
Michele Giacomini
Keyword(s):  
Abelian Varieties ◽  
Holomorphic Curves ◽  
Zariski Closure ◽  
Shimura Varieties

Abstract We prove a hyperbolic analogue of the Bloch–Ochiai theorem about the Zariski closure of holomorphic curves in abelian varieties. We consider the case of non compact Shimura varieties completing the proof of the result for all Shimura varieties. The statement which we consider here was first formulated and proven by Ullmo and Yafaev for compact Shimura varieties.

Isogenies Between K3 Surfaces Over

2020 ◽  
Author(s):  
Ziquan Yang
Keyword(s):  
Abelian Varieties ◽  
K3 Surfaces ◽  
Main Step ◽  
Shimura Varieties ◽  
K3 Surface ◽  
Perfect Field ◽  
Finite Height ◽  
Mild Assumption

Abstract We generalize Mukai and Shafarevich’s definitions of isogenies between K3 surfaces over ${\mathbb{C}}$ to an arbitrary perfect field and describe how to construct isogenous K3 surfaces over $\bar{{\mathbb{F}}}_p$ by prescribing linear algebraic data when $p$ is large. The main step is to show that isogenies between Kuga–Satake abelian varieties induce isogenies between K3 surfaces, in the context of integral models of Shimura varieties. As a byproduct, we show that every K3 surface of finite height admits a CM lifting under a mild assumption on $p$.

scholarly journals The value distribution of holomorphic curves into semi-Abelian varieties

2000 ◽  
Vol 331 (3) ◽  
pp. 235-240 ◽  
Author(s):  
Junjiro Noguchi ◽  
Jörg Winkelmann ◽  
Katsutoshi Yamanoi
Keyword(s):  
Abelian Varieties ◽  
Value Distribution ◽  
Holomorphic Curves

scholarly journals Foliations on unitary Shimura varieties in positive characteristic

2018 ◽  
Vol 154 (11) ◽  
pp. 2267-2304 ◽  
Author(s):  
Ehud de Shalit ◽  
Eyal Z. Goren
Keyword(s):  
Tangent Bundle ◽  
Positive Characteristic ◽  
Blow Up ◽  
Level Structure ◽  
Horizontal Component ◽  
Zariski Closure ◽  
Shimura Varieties ◽  
Shimura Variety ◽  
Integral Manifolds ◽  
Imaginary Field

When$p$is inert in the quadratic imaginary field$E$and$m<n$, unitary Shimura varieties of signature$(n,m)$and a hyperspecial level subgroup at$p$, carry a naturalfoliationof height 1 and rank$m^{2}$in the tangent bundle of their special fiber$S$. We study this foliation and show that it acquires singularities at deep Ekedahl–Oort strata, but that these singularities are resolved if we pass to a natural smooth moduli problem$S^{\sharp }$, a successive blow-up of$S$. Over the ($\unicode[STIX]{x1D707}$-)ordinary locus we relate the foliation to Moonen’s generalized Serre–Tate coordinates. We study the quotient of$S^{\sharp }$by the foliation, and identify it as the Zariski closure of the ordinary-étale locus in the special fiber$S_{0}(p)$of a certain Shimura variety with parahoric level structure at$p$. As a result, we get that this ‘horizontal component’ of$S_{0}(p)$, as well as its multiplicative counterpart, are non-singular (formerly they were only known to be normal and Cohen–Macaulay). We study two kinds of integral manifolds of the foliation: unitary Shimura subvarieties of signature$(m,m)$, and a certain Ekedahl–Oort stratum that we denote$S_{\text{fol}}$. We conjecture that these are the only integral submanifolds.

scholarly journals Algebraic flows on abelian varieties

2018 ◽  
Vol 2018 (741) ◽  
pp. 47-66 ◽  
Author(s):  
Emmanuel Ullmo ◽  
Andrei Yafaev
Keyword(s):  
Abelian Variety ◽  
Algebraic Curve ◽  
Abelian Varieties ◽  
Zariski Closure ◽  
Topological Closure

Abstract Let A be an abelian variety. The abelian Ax–Lindemann theorem shows that the Zariski closure of an algebraic flow in A is a translate of an abelian subvariety of A. The paper discusses some conjectures on the usual topological closure of an algebraic flow in A. The main result is a proof of these conjectures when the algebraic flow is given by an algebraic curve.

scholarly journals The field of moduli of quaternionic multiplication on abelian varieties

2004 ◽  
Vol 2004 (52) ◽  
pp. 2795-2808 ◽  
Author(s):  
Victor Rotger
Keyword(s):  
Abelian Varieties ◽  
Number Fields ◽  
Rational Points ◽  
Shimura Varieties ◽  
Field Of Moduli

We consider principally polarized abelian varieties with quaternionic multiplication over number fields and we study the field of moduli of their endomorphisms in relation to the set of rational points on suitable Shimura varieties.

scholarly journals Applications of the hyperbolic Ax–Schanuel conjecture

2018 ◽  
Vol 154 (9) ◽  
pp. 1843-1888 ◽  
Author(s):  
Christopher Daw ◽  
Jinbo Ren
Keyword(s):  
Abelian Varieties ◽  
Shimura Varieties ◽  
Shimura Variety ◽  
Point Counting ◽  
Counting Problem

In 2014, Pila and Tsimerman gave a proof of the Ax–Schanuel conjecture for the$j$-function and, with Mok, have recently announced a proof of its generalization to any (pure) Shimura variety. We refer to this generalization as the hyperbolic Ax–Schanuel conjecture. In this article, we show that the hyperbolic Ax–Schanuel conjecture can be used to reduce the Zilber–Pink conjecture for Shimura varieties to a problem of point counting. We further show that this point counting problem can be tackled in a number of cases using the Pila–Wilkie counting theorem and several arithmetic conjectures. Our methods are inspired by previous applications of the Pila–Zannier method and, in particular, the recent proof by Habegger and Pila of the Zilber–Pink conjecture for curves in abelian varieties.

scholarly journals The second main theorem for holomorphic curves into semi-Abelian varieties

2002 ◽  
Vol 188 (1) ◽  
pp. 129-161 ◽  
Author(s):  
Junjiro Noguchi ◽  
Jörg Winkelmann ◽  
Katsutoshi Yamanoi
Keyword(s):  
Abelian Varieties ◽  
Holomorphic Curves ◽  
Second Main Theorem ◽  
The Second Main Theorem

scholarly journals Holomorphic curves in compact Shimura varieties

2018 ◽  
Vol 68 (2) ◽  
pp. 647-659
Author(s):  
Emmanuel Ullmo ◽  
Andrei Yafaev
Keyword(s):  
Holomorphic Curves ◽  
Shimura Varieties
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