scholarly journals On the Nielsen root theory of n-valued maps

2021 ◽  
Vol 23 (4) ◽  
Author(s):  
Robert F. Brown
Keyword(s):  
Fixed Point ◽  
Lower Bound ◽  
Lie Group ◽  
Hausdorff Distance ◽  
Root Number ◽  
Root Numbers ◽  
Fixed Point Result

AbstractLet $$\phi :X \multimap Y$$ ϕ : X ⊸ Y be an n-valued map of connected finite polyhedra and let $$a \in Y$$ a ∈ Y . Then, $$x \in X$$ x ∈ X is a root of $$\phi $$ ϕ at a if $$a \in \phi (x)$$ a ∈ ϕ ( x ) . The Nielsen root number $$N(\phi : a)$$ N ( ϕ : a ) is a lower bound for the number of roots at a of any n-valued map homotopic to $$\phi $$ ϕ . We prove that if X and Y are compact, connected triangulated manifolds without boundary, of the same dimension, then given $$\epsilon > 0$$ ϵ > 0 , there is an n-valued map $$\psi $$ ψ homotopic to $$\phi $$ ϕ within Hausdorff distance $$\epsilon $$ ϵ of $$\phi $$ ϕ such that $$\psi $$ ψ has finitely many roots at a. We conjecture that if X and Y are q-manifolds without boundary, $$q \ne 2$$ q ≠ 2 , then there is an n-valued map homotopic to $$\phi $$ ϕ that has $$N(\phi : a)$$ N ( ϕ : a ) roots at a. We verify the conjecture when $$X = Y$$ X = Y is a Lie group by employing a fixed point result of Schirmer. As an application, we calculate the Nielsen root numbers of linear n-valued maps of tori.

scholarly journals Gamma Factors, Root Numbers, and Distinction

2018 ◽  
Vol 70 (3) ◽  
pp. 683-701 ◽  
Author(s):  
Nadir Matringe ◽  
Omer Offen
Keyword(s):  
Linear Group ◽  
General Linear Group ◽  
Quadratic Extension ◽  
Root Number ◽  
Converse Problem ◽  
Root Numbers ◽  
Special Values ◽  
Distinguished Representations ◽  
Local Invariants ◽  
Gamma Factor

AbstractWe study a relation between distinction and special values of local invariants for representations of the general linear group over a quadratic extension of p-adic fields. We show that the local Rankin–Selberg root number of any pair of distinguished representation is trivial, and as a corollary we obtain an analogue for the global root number of any pair of distinguished cuspidal representations. We further study the extent to which the gamma factor at 1/2 is trivial for distinguished representations as well as the converse problem.

scholarly journals Genetic Screening for Mutants with Altered Seminal Root Numbers in Hexaploid Wheat Using a High-Throughput Root Phenotyping Platform

2019 ◽  
Vol 9 (9) ◽  
pp. 2799-2809 ◽  
Author(s):  
Oluwaseyi Shorinola ◽  
Ryan Kaye ◽  
Guy Golan ◽  
Zvi Peleg ◽  
Stefan Kepinski ◽  
...  
Keyword(s):  
High Throughput ◽  
Genetic Screening ◽  
Hexaploid Wheat ◽  
Seminal Root ◽  
Root Numbers ◽  
Root Phenotyping

On the local Langlands conjecture forGL(2)

1978 ◽  
Vol 46 (2) ◽  
pp. 179-200 ◽  
Author(s):  
Jerrold B. Tunnell
Keyword(s):  
Local Langlands Conjecture
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