How Much Do You Know About Arthur's Conjectures?
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!
- 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.
Correct Answer
A. Make sense of semisimple and unipotent representations of automorphic forms.Explanation
While the theorem is very long with many separate proofs and hypotheses, Arthur's main point dealt with semisimple and unipotent representations of automorphic forms.Rate this question:
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- 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.
Correct Answer
D. A subgroup having only one element that is unchanged in value when operated on by itself.Explanation
An unipotent element remains unchanged in a subgroup of elements being changed by an outside force such as multiplication.Rate this question:
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- 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
Correct Answer
B. Packets, IndividualExplanation
The field is split into packets on a great scale, and Arthur uses individual representations on a more refined level to try to prove his conjectures.Rate this question:
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- 4.
Which formula are Arthur's Conjectures based on?
- A.
Derivative Formula
- B.
Trace Formula
- C.
Inverse Function Formula
- D.
Law of Sines Formula
Correct Answer
B. Trace FormulaExplanation
Arthur states in the Foreword that it is based on the trace formula.Rate this question:
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- 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
Correct Answer
C. Langlands worked with functorial lifting of a reductive group. Arthur worked with lifting of unitary charactersExplanation
Arthur took the next step from Langlands reductive groups and extended functoriality to see if it worked for a class of naturally unitary representations.Rate this question:
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- 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
Correct Answer
B. Linear algebraic group over a fieldExplanation
A reductive group is linear.Rate this question:
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- 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
Correct Answer
A. Langlands and ShelstadExplanation
Langlands and Shelstad are responsible for the theory of endoscopy.Rate this question:
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- 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
Correct Answer
C. He wanted to apply his theory to non-tempered representations, where he needed to consider endoscopic groups.Explanation
Arthur needed to consider endoscopy because he wanted to apply his theory to non-tempered representations. Endoscopic groups were necessary in order to understand linear groups better and to create reductive groups. Although endoscopy hadn't been proven with tempered representations, Arthur had a hypothesis on how to do so.Rate this question:
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- 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
Correct Answer
B. Combinatorial, representation-theoretic and arguably geometricExplanation
There were many issues that would would need to find theorised solutions. With the larger parameters the geometry of the reductive groups came into question as well.Rate this question:
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- 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
Correct Answer
C. The parameters cannot be cuspidalExplanation
When bigger eigenvalues are used than the discrete spectrum in Arthur's original work, the parameters cannot be cuspidal.Rate this question:
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