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Summary: STABILITY AND ENDOSCOPY: INFORMAL MOTIVATION
James Arthur
University of Toronto
The purpose of this note is described in the title. It is an elementary introduction
to some of the basic ideas of stability and endoscopy. We shall not discuss the
techniques of the theory, which among other things entail a sophisticated use of
Galois cohomology. Our aim is rather to persuade a reader that the theory was
created in response to some very natural problems in harmonic analysis. The article
is intended for people who are starting (or even just thinking of starting) to learn
the subject.
Langlands was actually lead to the theory of endoscopy by questions in algebraic
geometry, particularly Shimura varieties [17, §1]. However, he quickly realized
that the questions had remarkable implications for harmonic analysis. It is in this
context that we will discuss the basic ideas.
We begin with a simple form of the trace formula. Suppose that G is a reductive
algebraic group defined over a number field F. The ad`eles A of F are a locally
compact ring in which F embeds as a discrete subring, and the group of F-rational
points G(F) embeds as a discrete subgroup of the locally compact group G(A)
of ad`elic points. We shall be concerned with the case that G is anisotropic, or
equivalently, that the quotient space G(F)\G(A) is compact. It is then known that
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