Английская Википедия:Arthur–Selberg trace formula

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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 Шаблон:Math on the discrete part Шаблон:Math of Шаблон:Math in terms of geometric data, where Шаблон:Math is a reductive algebraic group defined over a global field Шаблон:Math and Шаблон:Math 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 Шаблон:Harv 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 case when Шаблон:Math 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 Шаблон:Math of a locally compact group Шаблон:Math with Шаблон:Math compact. The group Шаблон:Math acts on the space of functions on Шаблон:Math by the right regular representation Шаблон:Math, and this extends to an action of the group ring of Шаблон:Math, considered as the ring of functions Шаблон:Math on Шаблон:Math. The character of this representation is given by a generalization of the Frobenius formula as follows. The action of a function Шаблон:Math on a function Шаблон:Math on Шаблон:Math is given by

<math>\displaystyle R(f)(\phi)(x) = \int_G f(y)\phi(xy) \,dy = \int_{\Gamma\backslash G}\sum_{\gamma\in \Gamma}f(x^{-1}\gamma y)\phi(y)\,dy. </math>

In other words, Шаблон:Math is an integral operator on Шаблон:Math (the space of functions on Шаблон:Math) with kernel

<math>\displaystyle K_f(x,y) = \sum_{\gamma\in \Gamma}f(x^{-1}\gamma y).</math>

Therefore, the trace of Шаблон:Math is given by

<math>\displaystyle \operatorname{Tr}(R(f)) = \int_{\Gamma\backslash G}K_f(x,x) \,dx.</math>

The kernel K can be written as

<math>K_f(x,y) = \sum_{o\in O}K_o(x,y) </math>

where Шаблон:Math is the set of conjugacy classes in Шаблон:Math, and

<math>K_o(x,y)= \sum_{\gamma\in o}f(x^{-1}\gamma y) = \sum_{\delta\in \Gamma_\gamma\backslash \Gamma}f(x^{-1}\delta^{-1}\gamma\delta y)</math>

where Шаблон:Math is an element of the conjugacy class Шаблон:Math, and Шаблон:Math is its centralizer in Шаблон:Math.

On the other hand, the trace is also given by

<math>\displaystyle \operatorname{Tr}(R(f)) = \sum_{\pi} m(\pi)\operatorname{Tr}(R(f)|\pi)</math>

where m(π) is the multiplicity of the irreducible unitary representation Шаблон:Math of Шаблон:Math in Шаблон:Math

Examples

Difficulties in the non-compact case

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

  • The representation on Шаблон:Math contains not only discrete components, but also continuous components.
  • The kernel is no longer integrable over the diagonal, and the operators Шаблон:Math 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 Шаблон:Math of a real Lie group Шаблон:Math (usually Шаблон:Math). In higher rank it is more convenient to replace the Lie group with an adelic group Шаблон:Math. One reason for this that the discrete group can be taken as the group of points Шаблон:Math for Шаблон:Math 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 Шаблон:Harv asserts the equality of two distributions on Шаблон:Math:

<math>\sum_{o\in O}J_o^T = \sum_{\chi\in X}J_\chi^T.</math>

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 Шаблон:Math of Шаблон:Math, while the right hand side is the spectral side of the trace formula and is a sum over certain representations of subgroups of Шаблон:Math.

Distributions

Шаблон:Empty section

Geometric terms

Шаблон:Empty section

Spectral terms

Шаблон:Empty section

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. Шаблон:Harvtxt found a modification in which the terms are invariant.

The invariant trace formula states

<math> \sum_M\frac{|W_0^M|}{|W_0^G|} \sum_{\gamma\in (M(Q))}a^M(\gamma)I_M(\gamma,f)

= \sum_M\frac{|W_0^M|}{|W_0^G|} \int_{\Pi(M)}a^M(\pi)I_M(\pi,f) \, d\pi</math>

where

Stable trace formula

Шаблон:Harvtxt 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 Шаблон:Harvtxt.

Two elements of a group Шаблон:Math are called stably conjugate if they are conjugate over the algebraic closure of the field Шаблон:Math. 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 Шаблон:Math in terms of stable distributions. However these stable distributions are not distributions on the group Шаблон:Math, but are distributions on a family of quasisplit groups called the endoscopic groups of Шаблон:Math. Unstable orbital integrals on the group Шаблон:Math correspond to stable orbital integrals on its endoscopic groups Шаблон:Math.

Simple trace formula

There are several simple forms of the trace formula, which restrict the compactly supported test functions f in some way Шаблон:Harv. 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

<math>\int_{n\in N(A)}f(xny) \, dn=0</math>

for any unipotent radical Шаблон:Math of a proper parabolic subgroup (defined over Шаблон:Math) and any x, y in Шаблон:Math, then the operator Шаблон:Math has image in the space of cusp forms so is compact.

Applications

Шаблон:Harvtxt used the Selberg trace formula to prove the Jacquet–Langlands correspondence between automorphic forms on Шаблон:Math 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.

Шаблон:Harvtxt used the Arthur–Selberg trace formula to prove the Weil conjecture on Tamagawa numbers.

Шаблон:Harvtxt described how the trace formula is used in his proof of the Langlands conjecture for general linear groups over function fields.

See also

References

External links

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