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 Lower bounds for Maass forms on semisimple groups

2020 ◽  
Vol 156 (5) ◽  
pp. 959-1003
Author(s):  
Farrell Brumley ◽  
Simon Marshall
Keyword(s):  
Real Number ◽  
Number Field ◽  
Lower Bounds ◽  
Positive Curvature ◽  
Semisimple Group ◽  
Maass Forms ◽  
Semisimple Groups ◽  
Cartan Involution ◽  
Totally Real ◽  
Real Number Field

Let $G$ be an anisotropic semisimple group over a totally real number field $F$. Suppose that $G$ is compact at all but one infinite place $v_{0}$. In addition, suppose that $G_{v_{0}}$ is $\mathbb{R}$-almost simple, not split, and has a Cartan involution defined over $F$. If $Y$ is a congruence arithmetic manifold of non-positive curvature associated with $G$, we prove that there exists a sequence of Laplace eigenfunctions on $Y$ whose sup norms grow like a power of the eigenvalue.

A Hypergraph with Commuting Partial Laplacians

2001 ◽  
Vol 44 (4) ◽  
pp. 385-397 ◽  
Author(s):  
Cristina M. Ballantine
Keyword(s):  
Prime Ideal ◽  
Real Number ◽  
Number Field ◽  
Linear Group ◽  
General Linear Group ◽  
Commuting Operators ◽  
Affine Building ◽  
Finite Quotient ◽  
Totally Real ◽  
Real Number Field

AbstractLetFbe a totally real number field and let GLnbe the general linear group of rank n overF. Let р be a prime ideal ofFand Fрthe completion ofFwith respect to the valuation induced by р. We will consider a finite quotient of the affine building of the group GLnover the field Fр. We will view this object as a hypergraph and find a set of commuting operators whose sum will be the usual adjacency operator of the graph underlying the hypergraph.

Sum formula for 𝑆𝐿₂ over a totally real number field

2009 ◽  
Vol 197 (919) ◽  
pp. 0-0 ◽  
Author(s):  
Roelof W. Bruggeman ◽  
Roberto J. Miatello
Keyword(s):  
Real Number ◽  
Number Field ◽  
Sum Formula ◽  
Totally Real ◽  
Real Number Field

Hecke L-Functions and the Distribution of Totally Positive Integers

2007 ◽  
Vol 59 (4) ◽  
pp. 673-695 ◽  
Author(s):  
Avner Ash ◽  
Solomon Friedberg
Keyword(s):  
Real Number ◽  
Number Field ◽  
Expected Value ◽  
Fractional Ideal ◽  
Compact Torus ◽  
Positive Elements ◽  
Totally Positive ◽  
Totally Real ◽  
Positive Integers ◽  
Real Number Field

AbstractLet K be a totally real number field of degree n. We show that the number of totally positive integers (or more generally the number of totally positive elements of a given fractional ideal) of given trace is evenly distributed around its expected value, which is obtained fromgeometric considerations. This result depends on unfolding an integral over a compact torus.

scholarly journals -INVARIANTS AND LOCAL–GLOBAL COMPATIBILITY FOR THE GROUP

2016 ◽  
Vol 4 ◽  
Author(s):  
YIWEN DING
Keyword(s):  
Real Number ◽  
Number Field ◽  
Cohomology Group ◽  
Shimura Curves ◽  
Langlands Correspondence ◽  
Decomposition Group ◽  
Étale Cohomology ◽  
Totally Real ◽  
Local Langlands Correspondence ◽  
Real Number Field

Let $F$ be a totally real number field, ${\wp}$ a place of $F$ above $p$. Let ${\it\rho}$ be a $2$-dimensional $p$-adic representation of $\text{Gal}(\overline{F}/F)$ which appears in the étale cohomology of quaternion Shimura curves (thus ${\it\rho}$ is associated to Hilbert eigenforms). When the restriction ${\it\rho}_{{\wp}}:={\it\rho}|_{D_{{\wp}}}$ at the decomposition group of ${\wp}$ is semistable noncrystalline, one can associate to ${\it\rho}_{{\wp}}$ the so-called Fontaine–Mazur ${\mathcal{L}}$-invariants, which are however invisible in the classical local Langlands correspondence. In this paper, we prove one can find these ${\mathcal{L}}$-invariants in the completed cohomology group of quaternion Shimura curves, which generalizes some of Breuil’s results [Breuil, Astérisque, 331 (2010), 65–115] in the $\text{GL}_{2}/\mathbb{Q}$-case.

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