scholarly journals A lower bound for the rank of a universal quadratic form with integer coefficients in a totally real number field

2019 ◽  
Vol 94 (2) ◽  
pp. 221-239 ◽  
Author(s):  
Pavlo Yatsyna
Keyword(s):  
Lower Bound ◽  
Quadratic Form ◽  
Real Number ◽  
Number Field ◽  
Integer Coefficients ◽  
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.

scholarly journals ON NUMBER FIELDS WITH EQUIVALENT INTEGRAL TRACE FORMS

2012 ◽  
Vol 08 (07) ◽  
pp. 1569-1580 ◽  
Author(s):  
GUILLERMO MANTILLA-SOLER
Keyword(s):  
Quadratic Form ◽  
Real Number ◽  
Number Field ◽  
Number Fields ◽  
Trace Form ◽  
Integral Quadratic Form ◽  
Trace Forms ◽  
Fundamental Discriminant ◽  
Totally Real

Let K be a number field. The integral trace form is the integral quadratic form given by tr k/ℚ(x2)|OK. In this article we study the existence of non-conjugated number fields with equivalent integral trace forms. As a corollary of one of the main results of this paper, we show that any two non-totally real number fields with the same signature and same prime discriminant have equivalent integral trace forms. Additionally, based on previous results obtained by the author and the evidence presented here, we conjecture that any two totally real quartic fields of fundamental discriminant have equivalent trace zero forms if and only if they are conjugated.

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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