How Much Do You Know About Arthur's Conjectures?

1) A 2009 study by O. Paniagua- Taboada also found what issue with an elliptic Arthur parameter?

Cannot calculate the twisted characters

The ellipse cannot be complete

The parameters cannot be cuspidal

The parameters cannot be linear

When bigger eigenvalues are used than the discrete spectrum in Arthur's original work, the parameters cannot be cuspidal.

Explanation

When bigger eigenvalues are used than the discrete spectrum in Arthur's original work, the parameters cannot be cuspidal.

2) Which formula are Arthur's Conjectures based on?

Derivative Formula

Trace Formula

Inverse Function Formula

Law of Sines Formula

Arthur states in the Foreword that it is based on the trace formula.

Explanation

Arthur states in the Foreword that it is based on the trace formula.

3) Why did Arthur need to consider endoscopy?

Endoscopy hasn't been proven with tempered representations and he had a hypothesis on how to do so

In order to understand linear groups better he needed to use elements of endoscopy

He wanted to apply his theory to non-tempered representations, where he needed to consider endoscopic groups.

He worked with endoscopic groups to create reductive groups

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.

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.

4) What does Arthur mean by unipotent?

Dealing only with local conjectures.

Dealing only with global conjectures.

A subgroup where all elements remain unchanged in value when operated on by themselves.

A subgroup having only one element that is unchanged in value when operated on by itself.

An unipotent element remains unchanged in a subgroup of elements being changed by an outside force such as multiplication.

Explanation

An unipotent element remains unchanged in a subgroup of elements being changed by an outside force such as multiplication.

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?

Langlands worked with functorial lifting of unitary characters. Arthur worked with lifting of a reductive group

Langlands proved functoriality for non-Archimedean fields. Arthur worked on Archimedean fields

Langlands worked with functorial lifting of a reductive group. Arthur worked with lifting of unitary characters

Langlands and Arthur's work have no differences because they always worked together

Arthur took the next step from Langlands reductive groups and extended functoriality to see if it worked for a class of naturally unitary representations.

Explanation

Arthur took the next step from Langlands reductive groups and extended functoriality to see if it worked for a class of naturally unitary representations.

6) A large amount of Arthur's work deals with Endoscopy. Who is responsible for the theory of endoscopy?

Langlands and Shelstad

Langlands and Bell

Arthur and Shelstad

Friberg and Grioux

Langlands and Shelstad are responsible for the theory of endoscopy.

Explanation

Langlands and Shelstad are responsible for the theory of endoscopy.

7) What is the overall purpose of Arthur's Conjectures?

Make sense of semisimple and unipotent representations of automorphic forms.

Describe how conjectures relate to the spectral side of the trace formula.

Show the functorial lifting of a reductive group.

Prove Z(H) is the center of group H.

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.

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.

8) A simplistic form of Arthur's conjectures split automorphic representations into ________, and a refined form used ________ representations.

Individuals, Group

Packets, Individual

Groups, Parabolic

Parabolas, Packet

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.

Explanation

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.

9) By changing the parameters of Langlands research, Arthur's Conjectures encountered which problems (according to a 2004 paper by Laurent Clozel)?

Algebraic and linear

Combinatorial, representation-theoretic and arguably geometric

Automorphic-representative and geometric

Functoriality

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.

Explanation

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.