Voronoï’s reduction theory of 𝐺𝐿_{𝑛} over a totally real number field

2013 ◽  
pp. 213-232 ◽  
Author(s):  
Takao Watanabe ◽  
Syouji Yano ◽  
Takuma Hayashi
Keyword(s):  
Real Number ◽  
Number Field ◽  
Reduction Theory ◽  
Totally Real ◽  
Real Number Field

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.

scholarly journals Modularity of PGL2(𝔽p)-representations over totally real fields

2021 ◽  
Vol 118 (33) ◽  
pp. e2108064118
Author(s):  
Patrick B. Allen ◽  
Chandrashekhar B. Khare ◽  
Jack A. Thorne
Keyword(s):  
Real Number ◽  
Number Field ◽  
Projective Representations ◽  
Totally Real ◽  
Totally Real Fields ◽  
Real Number Field

We study an analog of Serre’s modularity conjecture for projective representations ρ¯:Gal(K¯/K)→PGL2(k), where K is a totally real number field. We prove cases of this conjecture when k=F5.

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.

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