“Arthur–Selberg trace formula”的意思、由来-开放百科全书

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Arthur–Selberg trace formula

释义

Notation
The compact case
Examples Difficulties in the non-compact case
The trace formula in the non-compact case
Distributions Geometric terms Spectral terms
The invariant trace formula
Stable trace formula
Simple trace formula
Applications
References
External links

In mathematics, the Arthur–Selberg trace formula is a generalization of the Selberg trace formula from the group SL₂ to arbitrary reductive groups over global fields, developed by James Arthur in a long series of papers from 1974 to 2003. It describes the character of the representation of G(A) on the discrete part L{{su|b=0|p=2}}(G(F)∖G(A)) of L²(G(F)∖G(A)) in terms of geometric data, where G is a reductive algebraic group defined over a global field F and A is the ring of adeles of F.

There are several different versions of the trace formula. The first version was the unrefined trace formula, whose terms depend on truncation operators and have the disadvantage that they are not invariant. Arthur later found the invariant trace formula and the stable trace formula which are more suitable for applications. The simple trace formula {{harv|Flicker|Kazhdan|1988}} is less general but easier to prove. The local trace formula is an analogue over local fields.

Jacquet's relative trace formula is a generalization where one integrates the kernel function over non-diagonal subgroups.

Notation

F is a global field, such as the field of rational numbers.
A is the ring of adeles of F.
G is a reductive algebraic group defined over F.

The compact case

In the (rare) case when G(F)∖G(A) is compact the representation splits as a direct sum of irreducible representations, and the trace formula is similar to the Frobenius formula for the character of the representation induced from the trivial representation of a subgroup of finite index.

In the compact case, which is essentially due to Selberg, the groups G(F) and G(A) can be replaced by any

discrete subgroup Γof a locally compact group G with Γ\\G compact. The group G acts on the space of functions on

Γ∖G by the right regular representation R, and this extends to an action of the group ring of G, considered as the ring of functions f on G. The character of this representation is given by a generalization of the Frobenius formula as follows.

The action of a function f on a function φ on Γ∖G is given by

In other words, R(f) is an integral operator on L²(Γ∖G) (the space of functions on Γ∖G) with kernel

Therefore, the trace of R(f) is given by

The kernel K can be written as

where O is the set of conjugacy classes in Γ, and

where γ is an element of the conjugacy class o, and Γ_γ is its centralizer in Γ.

On the other hand, the trace is also given by

where m(π) is the multiplicity of the irreducible unitary representation π of G in L²(Γ∖G).

Examples

If Γ and G are both finite, the trace formula is equivalent to the Frobenius formula for the character of an induced representation.
If G is the group R of real numbers and Γ the subgroup Z of integers, then the trace formula becomes the Poisson summation formula.

Difficulties in the non-compact case

In most cases of the Arthur–Selberg trace formula, the quotient G(F)∖G(A) is not compact, which causes the following (closely related) problems:

The representation on L²(G(F)∖G(A)) contains not only discrete components, but also continuous components.
The kernel is no longer integrable over the diagonal, and the operators R(f) are no longer of trace class.

Arthur dealt with these problems by truncating the kernel at cusps in such a way that the truncated kernel is integrable over the diagonal. This truncation process causes many problems; for example, the truncated terms are no longer invariant under conjugation. By manipulating the terms further, Arthur was able to produce an invariant trace formula whose terms are invariant.

The original Selberg trace formula studied a discrete subgroup Γ of a real Lie group G(R) (usually SL₂(R)).

In higher rank it is more convenient to replace the Lie group with an adelic group G(A). One reason for this that the discrete group can be taken as the group of points G(F) for F a (global) field, which is easier to work with

than discrete subgroups of Lie groups. It also makes Hecke operators easier to work with.

The trace formula in the non-compact case

One version of the trace formula {{harv|Arthur|1983}} asserts the equality of two distributions on G(A):

The left hand side is the geometric side of the trace formula, and is a sum over equivalence classes in the group of rational points G(F) of G, while the right hand side is the spectral side of the trace formula and is a sum over certain representations of subgroups of G(A).

Distributions

Geometric terms

Spectral terms

The invariant trace formula

The version of the trace formula above is not particularly easy to use in practice, one of the problems being that the terms in it are not invariant under conjugation. {{harvtxt|Arthur|1981}} found a modification in which the terms are invariant.

The invariant trace formula states

where

f is a test function on G(A)
M ranges over a finite set of rational Levi subgroups of G
(M(Q)) is the set of conjugacy classes of M(Q)
Π(M) is the set of irreducible unitary representations of M(A)
a^M(γ) is related to the volume of M(Q,γ)\\M(A,γ)
a^M(π) is related to the multiplicity of the irreducible representation π in L²(M(Q)\\M(A))
is related to
is related to trace
W₀(M) is the Weyl group of M.

Stable trace formula

{{harvtxt|Langlands|1983}} suggested the possibility a stable refinement of the trace formula that can be used to compare the trace formula for two different groups. Such a stable trace formula was found and proved by {{harvtxt|Arthur|2002}}.

Two elements of a group G(F) are called stably conjugate if they are conjugate over

the algebraic closure of the field F. The point is that when one compares elements in two different groups, related for example by inner twisting, one does not usually get a good correspondence between conjugacy classes, but only between stable conjugacy classes. So to compare the geometric terms in the trace formulas for two different groups, one would like the terms to be not just invariant under conjugacy, but also to be well behaved on stable conjugacy classes; these are called stable distributions.

The stable trace formula writes the terms in the trace formula of a group G in terms of stable distributions. However these stable distributions are not distributions on the group G, but are distributions on a family of quasisplit groups called the endoscopic groups of G. Unstable orbital integrals on the group G correspond to stable orbital integrals on its endoscopic groups H.

Simple trace formula

There are several simple forms of the trace formula, which restrict the compactly supported test functions f in some way {{harv|Flicker|Kazhdan|1988}}. The advantage of this is that the trace formula and its proof become much easier, and the disadvantage is that the resulting formula is less powerful.

For example, if the functions f are cuspidal, which means that

for any unipotent radical N of a proper parabolic subgroup (defined over F) and any x, y in G(A), then the operator R(f) has image in the space of cusp forms so is compact.

Applications

{{Harvtxt|Jacquet|Langlands|1970}} used the Selberg trace formula to prove the Jacquet–Langlands correspondence between automorphic forms on GL₂ and its twisted forms. The Arthur–Selberg trace formula can be used to study similar correspondences on higher rank groups. It can also be used to prove several other special cases of Langlands functoriality, such as base change, for

some groups.

{{harvtxt|Kottwitz|1988}} used the Arthur–Selberg trace formula to prove the Weil conjecture on Tamagawa numbers.{{harvtxt|Lafforgue|2002}} described how the trace formula is used in his proof of the Langlands conjecture for general linear groups over function fields.

References

{{Citation | last1=Arthur | first1=James | title=The trace formula in invariant form | jstor=1971376 | mr=625344 | year=1981 | journal=Annals of Mathematics |series=Second Series | volume=114 | issue=1 | pages=1–74 | doi=10.2307/1971376 }}
{{Citation | last1=Arthur | first1=James | title=Conference on automorphic theory (Dijon, 1981) | chapter-url=http://www2.maths.ox.ac.uk/cmi/library/cw/arthur/pdf/tfreductive.pdf | publisher=Univ. Paris VII | location=Paris | series=Publ. Math. Univ. Paris VII | mr=723181 | year=1983 | volume=15 | chapter=The trace formula for reductive groups | pages=1–41 | doi=10.1007/978-1-4684-6730-7_1| isbn=978-0-8176-3135-2 | citeseerx=10.1.1.207.4897 }}
{{Citation|last1=Arthur |first1=James |title=A stable trace formula. I. General expansions |url=http://www.claymath.org/cw/arthur/pdf/54.pdf |mr=1954821 |year=2002 |journal=Journal of the Institute of Mathematics of Jussieu. JIMJ. Journal de l'Institute de Mathématiques de Jussieu |volume=1 |issue=2 |pages=175–277 |doi=10.1017/S1474-748002000051 |deadurl=yes |archiveurl=https://web.archive.org/web/20080509063705/http://www.claymath.org/cw/arthur/pdf/54.pdf |archivedate=2008-05-09 }}
{{Citation|last1=Arthur |first1=James |title=Harmonic analysis, the trace formula, and Shimura varieties |chapter-url=http://www.claymath.org/cw/arthur/pdf/62.pdf |publisher=American Mathematical Society |location=Providence, R.I. |series=Clay Math. Proc. |mr=2192011 |year=2005 |volume=4 |chapter=An introduction to the trace formula |pages=1–263 |deadurl=yes |archiveurl=https://web.archive.org/web/20080509055029/http://www.claymath.org/cw/arthur/pdf/62.pdf |archivedate=2008-05-09 }}
{{Citation | author2-link=David Kazhdan | last1=Flicker | first1=Yuval Z. | last2=Kazhdan | first2=David A. | title=A simple trace formula | doi=10.1007/BF02796122 | year=1988 | journal=Journal d'Analyse Mathématique | volume=50 | pages=189–200}}
{{Citation | last1=Gelbart | first1=Stephen | author1-link = Stephen Gelbart | title=Lectures on the Arthur-Selberg trace formula | publisher=American Mathematical Society | location=Providence, R.I. | series=University Lecture Series | isbn=978-0-8218-0571-8 | mr=1410260 | year=1996 | volume=9|arxiv=math.RT/9505206 | doi=10.1090/ulect/009}}
{{Citation | last1=Jacquet | first1=H. | last2=Langlands | first2=Robert P. | title=Automorphic forms on GL(2) | url=http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/JL.html#book | publisher=Springer-Verlag | location=Berlin, New York | series=Lecture Notes in Mathematics, Vol. 114 | doi=10.1007/BFb0058988 | mr=0401654 | year=1970 | volume=114 | isbn=978-3-540-04903-6}}
{{Citation | last1=Konno | first1=Takuya | title=A survey on the Arthur-Selberg trace formula | url=http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1173-20.pdf | mr=1840082 | year=2000 | journal=Surikaisekikenkyusho Kõkyuroku | issue=1173 | pages=243–288}}
{{citation|mr=0942522|last= Kottwitz|first= Robert E. |title=Tamagawa numbers |journal= Ann. of Math. |series= 2 |volume= 127 |year=1988|issue= 3|pages=629–646|doi=10.2307/2007007|jstor= 2007007}}
{{Citation | last1=Labesse | first1=Jean-Pierre | title=La formule des traces d'Arthur-Selberg | mr=837215 | year=1986 | journal=Astérisque | issue=133 | pages=73–88}}
{{Citation | last1=Langlands | first1=Robert P. | title=The trace formula and its applications: an introduction to the work of James Arthur | doi=10.4153/CMB-2001-020-8 | mr=1827854 | year=2001 | journal=Canadian Mathematical Bulletin | issn=0008-4395 | volume=44 | issue=2 | pages=160–209}}
{{Citation | authorlink=Laurent Lafforgue | last1=Lafforgue | first1=Laurent | title=Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002) | publisher=Higher Ed. Press | location=Beijing | mr=1989194 | year=2002 | chapter=Chtoucas de Drinfeld, formule des traces d'Arthur-Selberg et correspondance de Langlands | pages=383–400}}
{{Citation | last1=Langlands | first1=Robert P. | title=Les débuts d'une formule des traces stable | url=http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/endoscopy.html#debuts | publisher=Université de Paris VII U.E.R. de Mathématiques | location=Paris | series=Publications Mathématiques de l'Université Paris VII [Mathematical Publications of the University of Paris VII] | mr=697567 | year=1983 | volume=13}}
{{Citation | doi=10.4153/CMB-2001-020-8 | authorlink=Robert Langlands | last1=Langlands | first1=Robert P. | title=The trace formula and its applications: an introduction to the work of James Arthur | url=http://journals.cms.math.ca/cgi-bin/vault/view/langlands8065 | mr=1827854 | year=2001 | journal=Canadian Mathematical Bulletin | volume=44 | issue=2 | pages=160–209}}
{{Citation | last1=Shokranian | first1=Salahoddin | title=The Selberg-Arthur trace formula | publisher=Springer-Verlag | location=Berlin, New York | series=Lecture Notes in Mathematics | isbn=978-3-540-55021-1 | doi=10.1007/BFb0092305 | mr=1176101 | year=1992 | volume=1503}}

External links

Works of James Arthur at the Clay institute
Archive of Collected Works of James Arthur at the University of Toronto Department of Mathematics

2 : Automorphic forms|Theorems in harmonic analysis

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