How Much Do You Know About Arthur's Conjectures?

10 Questions | By Lindsay_k | Last updated: Mar 18, 2022 | Total Attempts: 104

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Arthur's conjectures is a noted mathematical theorem paper written in 1989 by James Arthur. Arthur is the former president of the American Mathematical Society and a Professor at the University of Toronto. His conjectures deal with Unipotent Automorphic Representations on local and global scales. Test your knowledge with this quiz!

Questions and Answers

1.

What is the overall purpose of Arthur's Conjectures?
- A.
  
  Make sense of semisimple and unipotent representations of automorphic forms.
- B.
  
  Describe how conjectures relate to the spectral side of the trace formula.
- C.
  
  Show the functorial lifting of a reductive group.
- D.
  
  Prove Z(H) is the center of group H.
2.

What does Arthur mean by unipotent?
- A.
  
  Dealing only with local conjectures.
- B.
  
  Dealing only with global conjectures.
- C.
  
  A subgroup where all elements remain unchanged in value when operated on by themselves.
- D.
  
  A subgroup having only one element that is unchanged in value when operated on by itself.
3.

A simplistic form of Arthur's conjectures split automorphic representations into ________, and a refined form used ________ representations.
- A.
  
  Individuals, Group
- B.
  
  Packets, Individual
- C.
  
  Groups, Parabolic
- D.
  
  Parabolas, Packet
4.

Which formula are Arthur's Conjectures based on?
- A.
  
  Derivative Formula
- B.
  
  Trace Formula
- C.
  
  Inverse Function Formula
- D.
  
  Law of Sines Formula
5.

Arthur's mentor, Robert Langlands, created a notion of Functoriality that Arthur worked off of. Simply put, what was the difference in the mens' work?
- A.
  
  Langlands worked with functorial lifting of unitary characters. Arthur worked with lifting of a reductive group
- B.
  
  Langlands proved functoriality for non-Archimedean fields. Arthur worked on Archimedean fields
- C.
  
  Langlands worked with functorial lifting of a reductive group. Arthur worked with lifting of unitary characters
- D.
  
  Langlands and Arthur's work have no differences because they always worked together
6.

What is a reductive group?
- A.
  
  Parabolic algebraic group of points centering on a line
- B.
  
  Linear algebraic group over a field
- C.
  
  Clustered algebraic group over a field
- D.
  
  Geometric points forming a line over a field
7.

A large amount of Arthur's work deals with Endoscopy. Who is responsible for the theory of endoscopy?
- A.
  
  Langlands and Shelstad
- B.
  
  Langlands and Bell
- C.
  
  Arthur and Shelstad
- D.
  
  Friberg and Grioux
8.

Why did Arthur need to consider endoscopy?
- A.
  
  Endoscopy hasn't been proven with tempered representations and he had a hypothesis on how to do so
- B.
  
  In order to understand linear groups better he needed to use elements of endoscopy
- C.
  
  He wanted to apply his theory to non-tempered representations, where he needed to consider endoscopic groups.
- D.
  
  He worked with endoscopic groups to create reductive groups
9.

By changing the parameters of Langlands research, Arthur's Conjectures encountered which problems (according to a 2004 paper by Laurent Clozel)?
- A.
  
  Algebraic and linear
- B.
  
  Combinatorial, representation-theoretic and arguably geometric
- C.
  
  Automorphic-representative and geometric
- D.
  
  Functoriality
10.

A 2009 study by O. Paniagua- Taboada also found what issue with an elliptic Arthur parameter?
- A.
  
  Cannot calculate the twisted characters
- B.
  
  The ellipse cannot be complete
- C.
  
  The parameters cannot be cuspidal
- D.
  
  The parameters cannot be linear