Arithmetic Compactifications of PEL-Type Shimura Varieties

2013 ◽  
Author(s):  
Kai-Wen Lan
Keyword(s):  
Graduate Students ◽  
Automorphic Forms ◽  
Abelian Varieties ◽  
Shimura Varieties ◽  
Modular Curves ◽  
Automorphic Representations ◽  
Langlands Program ◽  
Arithmetic Properties ◽  
Sufficient Detail ◽  
Recent Developments

By studying the degeneration of abelian varieties with PEL structures, this book explains the compactifications of smooth integral models of all PEL-type Shimura varieties, providing the logical foundation for several exciting recent developments. PEL-type Shimura varieties, which are natural generalizations of modular curves, are useful for studying the arithmetic properties of automorphic forms and automorphic representations, and they have played important roles in the development of the Langlands program. As with modular curves, it is desirable to have integral models of compactifications of PEL-type Shimura varieties that can be described in sufficient detail near the boundary, which this book explains in detail. Through the discussion, the book generalizes the theory of degenerations of polarized abelian varieties and the application of that theory to the construction of toroidal and minimal compactifications of Siegel moduli schemes over the integers (as developed by Mumford, Faltings, and Chai). The book is designed to be accessible to graduate students who have an understanding of schemes and abelian varieties.

On some arithmetic properties of automorphic forms ofGLmover a division algebra

2014 ◽  
Vol 10 (04) ◽  
pp. 963-1013 ◽  
Author(s):  
Harald Grobner ◽  
A. Raghuram
Keyword(s):  
Division Algebra ◽  
Automorphic Forms ◽  
Local Version ◽  
Automorphic Representations ◽  
Langlands Correspondence ◽  
Central Division ◽  
Arithmetic Properties ◽  
Central Division Algebra ◽  
Archimedean Place ◽  
Split Form

In this paper we investigate arithmetic properties of automorphic forms on the group G' = GLm/D, for a central division-algebra D over an arbitrary number field F. The results of this article are generalizations of results in the split case, i.e. D = F, by Shimura, Harder, Waldspurger and Clozel for square-integrable automorphic forms and also by Franke and Franke–Schwermer for general automorphic representations. We also compare our theorems on automorphic forms of the group G′ to statements on automorphic forms of its split form using the global Jacquet–Langlands correspondence developed by Badulescu and Badulescu–Renard. Beside that we prove that the local version of the Jacquet–Langlands transfer at an archimedean place preserves the property of being cohomological.

scholarly journals The Chowla–Selberg formula for abelian CM fields and Faltings heights

2015 ◽  
Vol 152 (3) ◽  
pp. 445-476 ◽  
Author(s):  
Adrian Barquero-Sanchez ◽  
Riad Masri
Keyword(s):  
Gamma Function ◽  
Abelian Varieties ◽  
Modular Function ◽  
Rational Numbers ◽  
Explicit Formulas ◽  
Arithmetic Properties ◽  
Hilbert Modular ◽  
Cm Points ◽  
Analogous Function

In this paper we establish a Chowla–Selberg formula for abelian CM fields. This is an identity which relates values of a Hilbert modular function at CM points to values of Euler’s gamma function ${\rm\Gamma}$ and an analogous function ${\rm\Gamma}_{2}$ at rational numbers. We combine this identity with work of Colmez to relate the CM values of the Hilbert modular function to Faltings heights of CM abelian varieties. We also give explicit formulas for products of exponentials of Faltings heights, allowing us to study some of their arithmetic properties using the Lang–Rohrlich conjecture.

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.

scholarly journals Arithmetic gauge theory: A brief introduction

2018 ◽  
Vol 33 (29) ◽  
pp. 1830012 ◽  
Author(s):  
Minhyong Kim
Keyword(s):  
Moduli Spaces ◽  
Automorphic Forms ◽  
Galois Representations ◽  
Arithmetic Geometry ◽  
Quantum Field ◽  
Principal Bundles ◽  
Langlands Program ◽  
Effective Version ◽  
Diophantine Geometry ◽  
Higher Genus

Much of arithmetic geometry is concerned with the study of principal bundles. They occur prominently in the arithmetic of elliptic curves and, more recently, in the study of the Diophantine geometry of curves of higher genus. In particular, the geometry of moduli spaces of principal bundles holds the key to an effective version of Faltings’ theorem on finiteness of rational points on curves of genus at least 2. The study of arithmetic principal bundles includes the study of Galois representations, the structures linking motives to automorphic forms according to the Langlands program. In this paper, we give a brief introduction to the arithmetic geometry of principal bundles with emphasis on some elementary analogies between arithmetic moduli spaces and the constructions of quantum field theory.

AUTOMORPHIC LEFSCHETZ PROPERTIES FOR NONCOMPACT ARITHMETIC MANIFOLDS

2021 ◽  
pp. 1-48
Author(s):  
Arvind N. Nair ◽  
Ankit Rai
Keyword(s):  
Symmetric Spaces ◽  
Automorphic Forms ◽  
Hodge Theory ◽  
Hyperbolic Manifolds ◽  
Shimura Varieties ◽  
Restriction Maps ◽  
Locally Symmetric Spaces ◽  
Compact Congruence ◽  
Lefschetz Properties ◽  
Invent Math

Abstract We prove the injectivity of Oda-type restriction maps for the cohomology of noncompact congruence quotients of symmetric spaces. This includes results for restriction between (1) congruence real hyperbolic manifolds, (2) congruence complex hyperbolic manifolds, and (3) orthogonal Shimura varieties. These results generalize results for compact congruence quotients by Bergeron and Clozel [Quelques conséquences des travaux d’Arthur pour le spectre et la topologie des variétés hyperboliques, Invent. Math.192 (2013), 505–532] and Venkataramana [Cohomology of compact locally symmetric spaces, Compos. Math.125 (2001), 221–253]. The proofs combine techniques of mixed Hodge theory and methods involving automorphic forms.

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 Galois representations attached to automorphic forms on over fields

2014 ◽  
Vol 150 (4) ◽  
pp. 523-567 ◽  
Author(s):  
Chung Pang Mok
Keyword(s):  
Automorphic Forms ◽  
Galois Representations ◽  
Suitable Condition ◽  
Two Dimensional ◽  
Automorphic Representations ◽  
Cuspidal Automorphic Representations ◽  
Central Characters

AbstractIn this paper we generalize the work of Harris–Soudry–Taylor and construct the compatible systems of two-dimensional Galois representations attached to cuspidal automorphic representations of cohomological type on ${\rm GL}_2$ over a CM field with a suitable condition on their central characters. We also prove a local-global compatibility statement, up to semi-simplification.

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