On Principal Values on P-Adic Manifolds

R. Langlands and D. Shelstad

In the paper [L] a project for proving the existence of transfer factors for forms of SL(3), especially for the unitary groups studied by Rogawski, was begun, and it was promised that it would be completed by the present authors. Their paper is still in the course of being written, but the present essay can serve as an introduction to it. It deals with SL(2) which has, of course, already been dealt with systematically [L-L], the existence of the transfer factors being easily verified. Thus it offiers no new results, but develops, in a simple context, some useful methods for computing the principal value integrals introduced in [L].

We describe explicitly the Igusa fibering, form and integrand associated to orbital integrals on forms of SL(2), taking the occasion to clarify the relation of this fibering to the Springer-Grothendieck resolution (cf. §3). The Igusa data established, there are two problems: (i) to show that certain principal values are zero, (ii) to compare principal values on two twisted forms of the same variety. To deal with the first we have, in §1, computed directly some very simple principal values on P^{1}, and shown that principal values behave like ordinary integrals under standard geometric operations such as fibering and blowing-up. The second problem is dealt with in a similar way, by using Igusa’s methods to establish, in a simple case, a kind of comparison principle (Lemma 4.B).

The endoscopic groups for a form of SL(2) are either tori or SL(2). For tori the solution of the first problem (Lemma 4.A) leads immediately to the existence of transfer factors, and the hypotheses of [L_{1}, pp. 102, 149] are trivially satisfied. If G is anisotropic over F and the endoscopic group is SL(2) the solution of the second problem (Lemma 4.B with κ ≡ 1) and the characterization of stable orbital integrals (cf. [V]) yields the existence of transfer factors as well as the local hypothesis of [L_{1}, p. 102]. The analogous results at archimedean places are known in general (cf. [L_{1}, Lemma 6.17]). The global hypothesis [L_{1}, p. 149] follows from [L_{1}, Lemma 7.22].

Principal values on p-adic manifolds 2

The principal values which arise for forms of SL(2) are computed without difficulty, but we expressly avoid such calculations. The aim of the project begun in [L], and continued here, is to develop methods for proving the existence of transfer factors which appeal only to geometric techniques of some generality and thus have some prospect of applying to all groups. One encouraging sign is the smoothness with which they mesh with the notion of κ-orbital integral. They can be easily applied to the study of the germ at regular unipotent elements. A further test, perhaps not easy to carry out, would be the semi-regular elements, already studied for GL(n) by Repka [R].

Throughout this paper F will be a nonarchimedean local field of characteristic zero, with residue field of q elements; | · |_{F} = | · | will denote the valuation on F and ϖ a prime element; F̄ will be an algebraic closure of F.

§1. Remarks.

The following lemmas concern the simplest of the principal value integrals which arise in §1 of [L].

Let N = N(m_{1}, …, m_{n}) be the box
(1.1)  |u_{j}| ≤ q^{−m_{j}}  (1 ≤ j ≤ n)

in F^{n}. Consider the (multi-valued) differential form

(1.2)  ν_{(c_{1},...,c_{n})} ∏_{j=1}^{n} u_{j}^{c_{j}} \frac{du_{1}}{u_{1}} ∧ … ∧ \frac{du_{n}}{u_{n}} ,

where c_{1}, …, c_{n} are rational numbers. Let θ_{1}, …, θ_{n} be quasicharacters on F^{×}. Writing

(1.3)  θ_{j} = θ_{j}| · |^{t_{j}}

with θ_{j} unitary and g_{j} a real number, we assume

(1.4)  t_{j} + c_{j} ≠ 0 if θ_{j} ≡ 1  (1 ≤ j ≤ n).

Set

(1.5)  h_{(θ_{1},...,θ_{n})}(u_{1}, …, u_{n}) = ∏_{j=1}^{n} θ_{j}(u_{j}).

Then (1.4) allows us to define the principal value integral

(1.6)  ∮_{N} h_{(θ_{1},...,θ_{n})}|ν_{(c_{1},...,c_{n})}|

following [L, Lemma 1.3]. Thus consider for Re(s_{j}) ≫ 0 (1 ≤ j ≤ n)

∫_{N} ∏_{j=1}^{n} |u_{j}|^{s_{j}}h_{(θ_{1},...,θ_{n})}|ν_{(c_{1},...,c_{n})}| = ∏_{j=1}^{n} ∫_{|u_{j}|≤q^{−m_{j}}} θ_{j}(u_{j})|u_{j}|^{s_{j}+t_{j}+c_{j}} \frac{du_{j}}{|u_{j}|}

= ∏_{j=1}^{n} ∑_{n−m_{j}}^{∞} ( ∫_{|u_{j}|=q^{−n}} θ_{j}(u_{j})\frac{du_{j}}{|u_{j}|}) q^{−(s_{j}+t_{j}+c_{j})n} (1.7)

= ϵ(1 − \frac{1}{q})^{n} ∏_{j=1}^{n} \frac{(θ_{j}(ϖ)q^{−(s_{j}+t_{j}+c_{j})})^{m_{j}}}{1 − θ_{j}(ϖ)q^{−(s_{j}+t_{j}+c_{j})}} ,

where

ϵ = \{ \begin{matrix}1 if each θ_{j} is unramified \\ 0 otherwise. \end{matrix}

The analytic continuation of this function is, thanks to (1.4), analytic at s_{1} = … = s_{n} = 0; (1.6) is the value at s_{1} = … = s_{n} = 0. Thus:

Lemma 1.A.

∮_{N(m_{1},...,m_{n})} h_{(θ_{1},...,θ_{n})}|ν_{(c_{1},...,c_{n})}| = ϵ(1 − \frac{1}{q})^{n} ∏_{j=1}^{n} \frac{(θ_{j}(ϖ)q^{−(t_{j}+c_{j})})^{m_{j}}}{1 − θ_{j}(ϖ)q^{−(t_{j}+c_{j})}} .

To define now ∮_{X} h|ν| we assume:

(1.8) X is an F -manifold, h is a C-valued function supported on a compact open subset of X, ν is a differential form on X; and

(1.9) the support of h is the disjoint union of neighborhoods U with the following properties: (1.10) there are local coordinates u_{1}, …, u_{n} on X such that U is given by (1.1) for some

m_{1}, …, m_{n},

(1.11) on U, ν = αν_{(c_{1},...,c_{n})} with |α| constant, and h = γh_{(θ_{1},...,θ_{n})} with γ constant, where c_{1}, …, c_{n} and θ_{1}, …, θ_{n} satisfy (1.4).

Then

∮_{X} h|ν| =^{def.} ∑_{U}γ|α| ∮_{N(m_{1},...,m_{n})} h_{(θ_{1},...,θ_{n})}|ν_{(c_{1},...,c_{n})}|.

Our definition is that for the case r = s = 1 in the proof of Proposition 1.2 in [L]. The integral is independent of the choice for {U, u_{1}, …, u_{n}} ([L, Proposition 1.2]). Note especially that the conditions (1.9) – (1.11) are local, i.e., they are satisfied if we can find around each point in the support of h a neighborhood U satisfying (1.10) and (1.11). We will allow α to take values in a finite Galois extension L of F; in that case, |α| = |Nm_{F}^{L}α|^{1/[L:F]}.

The following remark will simplify a later argument. Given X, h and ν as in (1.8), a patch U as in (1.9), and integers M_{j}(1 ≤ j ≤ r) such that M_{j} ≥ m_{j}, let Ū = Ū(M_{1}, …, M_{r}) be the subset

|u_{j}| = q^{−M_{j}}(1 ≤ j ≤ r), |u_{j}| ≤ q^{−m_{j}} (r + 1 ≤ j ≤ n)

of U. Then:

Lemma 1.B.

∮_{U^{}}¯ h|ν| exists and equals the value at s_{1} = … = s_{n} = 0 of

γ|α| ∫_{U^{}}¯∏_{j=1}^{n} |u_{j}|^{s_{j}}h_{(θ_{1},...,θ_{n})}|ν_{(c_{1},...,c_{n})}| .

Moreover, if the support of h is the disjoint union of a collection S of such neighborhoods then

∮_{X} h|ν| = ∑_{U^{}¯ ∈S} ∮_{U^{}}¯ h|ν| .

Proof: The first assertion follows from the definitions, and the second from the independence of ∮_{X} h|ν| from the choice of decomposition for the support of h.

We consider an example. Let U_{0}, …, U_{n} be homogeneous coordinates on P^{n}. Suppose that θ_{0}, …, θ_{n} are quasicharacters on F^{×} such that ∏_{j=1}^{n} θ_{j} ≡ 1, and that c_{0}, …, c_{n} are rational numbers

such that ∑_{j=0}^{n} c_{j} = 0. Assume

(1.12) t_{j} + c_{j} ≠ 0 if θ_{j} ≡ 1(0 ≤ j ≤ n), where θ_{j} = θ_{j}| · |^{t_{j}}. Let ν be the form on P^{n} given on U_{k} ≠ 0 by

(1.13)  (−1)^{k} ∏_{j=0}^{n} U_{j}^{c_{j}} \frac{dU_{0}}{U_{0}} ∧ … ∧ \frac{d̂U_{k}}{U_{k}} ∧ … ∧ \frac{dU_{n}}{U_{n}} ,

where indicates deletion. Let h be the function

(1.14)  h(U_{0}, …, U_{n}) = ∏_{j=0}^{n} θ_{j}(U_{j}).

Then (1.12) ensures that ∮_{P^{n}(F)} h|ν| is well-defined.

Lemma 1.C.

∮_{P^{n}(F)} h|ν| = 0.

Proof for n = 1: Set θ = θ_{0}, t = t_{0}, c = c_{0}, u = U_{0}, on U_{1} = 1 and u = U_{1} on U_{0} = 1. Then

∮_{P^{1}(F)} h|ν| = ∮_{|u|≤1} θ(u)|u|^{t+c} \frac{du}{|u|} + ∮_{|u|≤q^{−1}} θ^{−1}(u)|u|^{−(t+c)} \frac{du}{|u|}

= ϵ(1 − \frac{1}{q}) (\frac{1}{1 − θ(ϖ)q^{−(t+c)}} + \frac{θ(ϖ)^{−1}q^{t+c}}{1 − θ(ϖ)^{−1}q^{t+c}} ) (1.15)

= 0.

The proof for n > 1 will be by reduction to the case n = 1; it follows Lemma 1.F.

Consider X, h and ν as in (1.8) – (1.11) and a neighborhood U as in (1.9). To compute ∮_{U} h|ν| we may change coordinates and assume that m_{1} = … = m_{n} = 1. This will be done for the next lemma.

Suppose that we blow up X at u_{1} = … = u_{n} = 0 to obtain the F -manifold X̄ and projection π : X̄ → X. Let Ū = π^{−1}(U), h̄ = h ∘ π and ν̄ = π^{∗}(ν).

Lemma 1.D. Assume that ∑_{j=1}^{n}(t_{j} + c_{j}) ≠ 0 if ∏_{j=1}^{n} θ_{j} ≡ 1. Then ∮_{U^{}}¯h̄|ν̄| exists and equals ∫_{U} h|ν|.

Proof: Near u_{1} = … = u_{n} = 0, X̄ is given by u_{i}U_{j} = U_{i}u_{j}(i, j = j, …, n), where U_{1}, …, U_{n} are homogeneous coordinates on P^{n−1}(F).

On U_{i} = 1 we have the coordinates U_{1}, …, U_{i−1}, z = u_{i}, U_{i+1}, …, U_{n}. Then u_{j} = zU_{j}(j ≠ i). Thus

ν̄ = αz^{(c_{1}+...+c_{n})} ∏_{j≠i} U_{j}^{c_{j}} \frac{dU_{1}}{U_{1}} ∧ … ∧ \frac{dz}{z} ∧ … \frac{dU_{n}}{U_{n}}

and

\begin{aligned}h̄ & = γ ∏_{j=1}^{n} θ_{j}(z) ∏_{j≠i} θ_{j}(U_{j}) \\ & = γ|z|^{(t_{1}+...+t_{n})} ∏_{j≠i} |U_{j}|^{t_{j}} ∏_{j=1}^{n} θ_{j}(z) ∏_{j≠i} θ_{j}(U_{j}) \end{aligned}

on this patch.

Let x ∈ U have coordinates u_{1}, …, u_{n}. Set S(x) = {i : |u_{i}| = max_{1≤j≤n} |u_{j}|}. For S ⊆ {1, 2, …, n}, let U_{S} = {x ∈ U : S(x) = S}. Then U is the disjoint union of the U_{S}. Let Ū_{S} = π^{−1}(U_{S}). Fix i ∈ S. Then Ū_{S} is contained in U_{i} = 1; it consists of the points in U_{i} = 1 with |z| ≤ 1, |U_{j}| = 1 (j ∈ S, j ≠ i) and |U_{k}| < 1(k ∉ S). The assumption in the statement of the lemma ensures that ∮_{U^{}_{¯ S}} h̄|ν̄| and ∮_{U^{}}¯h̄|ν̄| are well-defined.

On the other hand, ∮_{U} h|ν| is the value at s_{1} = … = s_{n} = 0 of

\begin{matrix}∫_{U} ∏_{j=1}^{n} |u_{j}|^{s_{j}}h|ν| = ∑_{S} ∫_{U_{S}} ∏_{j=1}^{n} |u_{j}|^{s_{j}}h|ν| \\ = ∑_{S} ∫_{U^{}_{¯ S}} |z|^{s_{1}+...+s_{n}} ∏_{j≠i} |U_{j}|^{s_{j}} h̄|ν̄|, \end{matrix}

where for each S ⊆ {1, 2, …, n} we have fixed i ∈ S. Then by Lemma 1.B,

∮_{U} h|ν| = ∑_{S} ∮_{U^{}_{¯ S}} h̄|ν̄| = ∮_{U^{}}¯h̄|ν̄|,

and we are done.

Corollary 1.E. Under the assumption of Lemma 1.D, ∮_{X̄} h̄|ν̄| is well-defined and equals ∮_{X} h|ν|.

Lemma 1.F. Suppose that ϕ : X → X′ is a smooth (submersive) map of F -manifolds. Suppose that X, h and ν satisfy (1.9) – (1.11) with the further constraint on the coordinates u_{1}, …, u_{n}:

(1.16) there are coordinates v_{1}, …, v_{r} on ϕ(U) such that u_{j} = v_{j} ∘ ϕ, j = 1, …, r. Let ν′ be a differential form on X′ given on ϕ(U) by

(1.17)  α′ ∏_{j=1}^{r} v_{f}^{d_{j}} \frac{dv_{1}}{v_{1}} ∧ … ∧ \frac{dv_{r}}{v_{r}} ,

where |α′| is constant and d_{j} is rational, 1 ≤ j ≤ r. Then for x′ ∈ X′(F) the principal value integral

H(x′) = ∮ h\frac{|ν|}{|ϕ^{∗}(ν′)|},

taken over the fiber above x′ in X, is well-defined outside a locally finite family of divisors. Moreover, ∮_{X′} H|ν′| is well-defined and

∮_{X} h|ν| = ∮_{X′} H|ν′|.

Proof: We may assume that the support of h is contained in a neighborhood U as in the statement of the lemma. Let x′ ∈ ϕ(U) have coordinates v_{1}, …, v_{r}. The fiber integral

H(x′) = γ|α||α′|^{−1} ∏_{j=1}^{r} θ_{j}(u_{j})|u_{j}|^{c_{j}−d_{j}} ∮ ∏_{j=r+1}^{n} θ_{j}(u_{j})|u_{j}|^{c_{j}} \frac{du_{1+1}}{|u_{r+1}|} ⋯ \frac{du_{n}}{|u_{n}|}

is well-defined provided none of v_{1}, …, v_{r} vanish at x′. Then

∮ H|ν′| = γ|α| ∮ ∏_{j=r+1}^{n} θ_{j}(u_{j})|u_{j}|^{c_{j}} \frac{du_{r+1}}{|u_{r+1}|} ⋯ \frac{du_{n}}{|u_{n}|} ∮ ∏_{j=1}^{r} θ_{j}(u_{j})|u_{j}|^{c_{j}} \frac{du_{1}}{|u_{1}|} ⋯ \frac{du_{r}}{|u_{r}|}.

is well-defined and coincides with ∮ h|ν|. Thus the lemma is proved. Proof of Lemma 1.C: Let p be the point in P^{n}(F) where U_{0} = U_{1} = ⋯ = U_{n−1} = 0. Suppose that we blow up P^{n} at p to obtain the smooth variety Q over F. The local conditions of Lemma 1.D are met since

∑_{j=0}^{n−1}(t_{j} + c_{j}) = −(t_{n} + c_{n}) and ∏_{j=0}^{n−1} θ_{j} = θ_{n}^{−1}

(cf. (1.12)). Corollary 1.E then implies that ∮_{P^{n}(F)} h|ν| = ∮_{Q(F)} h̄|ν̄|. We define a smooth map ϕ^{0} : P^{n} − {p} ⟶ P^{n−1} by mapping the point with homogeneous coordinates U_{0}, …, U_{n} in P^{n} to the point with homogeneous coordinates U_{0}, …, U_{n−1} in P^{n−1}. There is a smooth extension ϕ : Q ⟶ P^{n−1} of ϕ^{0}, with fiber P^{1}. An easy calculation verifies that the conditions of Lemma 1.F are met and that the integral H(x′) over the fiber above x′ ∈ P^{n−1}(F) takes the form (1.15). But then H ≡ 0. We conclude that ∮_{Q(F)} h̄|ν̄| = 0, and the lemma is proved.

Finally, there are two remarks which will be useful for the proof of Lemma 4.B. We state them only in the generality needed for that lemma. Remark 1.G.

Let L ⊂ F̄ be a quadratic extension of F. Denote the natural action of the nontrivial element σ of Gal(F/F) by a bar. We define a twisted form S of P^{1} by requiring that σ act on the homogeneous coordinates U_{0}, U_{1} by U_{0} ⟶ U_{1}, U_{1} ⟶ U_{0}. Then S(F) is contained in the affine patch U_{1} ≠ 0 and is given by uū = 1 if we require U_{1} = 1 and set U_{0} = u. The form ν on P^{1}(L) = S(L) given by (1.13) with c_{1} = c_{2} = 0 is preserved by the Galois action of S; |ν| = \frac{du}{|u|} is a Haar measure on S(F). Thus, for any character θ on \{u ∈ L^{×} : uū = 1\}, ∫_{S(F)} θ(u)\frac{du}{|u|} exists as an ordinary integral and is zero unless θ is trivial. Remark 1.H.

Again L will be a quadratic extension of F. We regard P^{1}(L) as the F -rational points on a twisted form R of P^{1} × P^{1} as follows: R(L) = P^{1}(L) × P^{1}(L) and σ acts by (p, q) ⟶ (q̄, p̄), so that R(F) = \{(p, p̄) : p ∈ P^{1}(L)\}. Define a form on R(L) by ν = \frac{du∧dv}{uv}, where u (respectively, v) denotes the coordinate U_{0} on U_{1} = 1 in the first (respectively, second) copy of P^{1}(L). At a point of R(F) on (U_{1} = 1) × (U_{1} = 1) we have v = ū. Let h be given at such a point by θ(uū)|uū|^{t}, where θ is a character on F^{×} and t is a real number such that t ≠ 0 if θ^{2} ≡ 1. Observe that, in general, h and ν do not satisfy the conditions of (1.9) - (1.11). We may, however, blow up R at u = v = 0 to obtain a variety R̄ over F and projection π : R̄ ⟶ R. Set h̄ = h ∘ π and ν̄ = π^{∗}(ν). Let N̄ be the inverse image in R̄(F) of the neighborhood |u|_{L} ≤ 1 of u = v = 0 in R(F). Then a calculation with coordinates shows that

(1.18)  ∮_{N̄} h̄|ν̄| is well-defined

(here t ≠ 0 if θ^{2} ≡ 1 is needed) and

(1.19)  ∮_{N} h̄|ν̄| = ∮_{|u|_{L}≤1} θ ∘ Nm(u)|u|_{L}^{t} \frac{d_{L}u}{|u|_{L}} ,

where the subscript L indicates that we are computing on the L-manifold |u|_{L} ≤ 1 in P^{1}(L). Observe that if θ is trivial on Nm_{F}^{L}L^{×} and t = −1 then

∮_{R̄(F)−N̄} h̄|ν̄| = ∮_{R(F)−N} h|ν|

is well-defined and equals

∮_{|u|_{L}<1} d_{L}u = ∮_{|u|_{L}>1} \frac{d_{L}u}{|u|_{L}^{2}} .

Thus, in this case, we have

(1.20)  ∮_{R̄(F)} h̄|ν̄| = ∮_{P^{1}(L)} \frac{d_{L}u}{|u|_{L}^{2}} = 0 (Lemma 1.C) .

§2. Igusa Theory.

Recall the setting of [L, §1]: Y is a smooth variety over F; ϕ : Y ⟶ C is an Igusa fibering of Y over a smooth curve C over F; ω is an Igusa form on Y; and f an Igusa integrand (the definitions will be reviewed presently). There is a distinguished point c_{0} on C(F) and ϕ is smooth except on the special fiber ϕ^{−1}(c_{0}). Choose an F -coordinate λ around c_{0} on C; assume λ(c_{0}) = 0. Then Igusa’s theory establishes the existence of an asymptotic expansion

∑_{(θ,β,r)} θ(λ)|λ|^{β−1}(− log_{q} |λ|)^{r−1}F_{r}(θ, β, f)

near λ = 0 for the integral

F (λ) = ∫ f \frac{|ω|}{|ϕ^{∗}(dλ)|}

over the fiber in Y (F) above the point in C with coordinate λ. Here θ denotes a character on F^{×}, β a real number and r a positive integer. The coefficients F_{r}(θ, β, f) are the principal value integrals of [L, Proposition 1.2]. Under an assumption we will make (2.9), only r = 1 occurs and F_{1}(θ, β, f) is an integral of the type considered in the last section. In this paragraph we will relax the constraints on the form ω and integrand f. The fiber integal F (λ) may then exist only as a principal value integral, but it will still have an asymptotic expansion. The coefficients are again given by [L, Proposition 1.2], i.e., by (related) principal value integrals.

For the rest of this section we require the following of Y, C, ϕ, ω and f:

(2.1) Y is a smooth variety over F, C is a smooth curve over F with distinguished point c_{0} ∈ C(F), ϕ : Y ⟶ C is an F -morphism smooth except over c_{0}, ω is a differential form of maximal degree on Y, f is a C-valued function supported on a compact open subset of Y (F); and

(2.2) if y_{0} ∈ Y (F) lies over the coordinate patch for λ, a fixed local F -coordinate around c_{0} on C, then there exist local F -coordinates μ_{1}, …, μ_{n} around y_{0} on Y such that:

(2.3) if y_{0} ∈ ϕ^{−1}(c_{0}) then ϕ is given near y_{0} by λ = αμ_{1}^{a_{1}} … μ_{n}^{a_{n}}, where α is regular and invertible at y_{0} and a_{1}, …, a_{n} are nonnegative integers; if y_{0} ∉ ϕ^{−1}(c_{0}) and λ_{0} is the coordinate of ϕ(y_{0}) then μ_{1} = λ − λ_{0};

(2.4) ω is given near y_{0} by

W ∏_{j=1}^{n} μ_{j}^{b_{j}} \frac{dμ_{1}}{μ_{1}} ∧ … ∧ \frac{dμ_{n}}{μ_{n}} ,

where W is regular and invertible at y_{0} and b_{1}, …, b_{n} are rational numbers; if y_{0} ∉ ϕ^{−1}(c_{0}) then b_{1} = 1;

(2.5) f is given on points of Y (F) near y_{0} by γK_{1}(μ_{1}) … K_{n}(μ_{n}), where γ is locally constant around y_{0} and K_{1}, …, K_{n} are quasicharacters on F^{×} such that:

(2.6) if μ_{j} = 0 is the branch of a divisor E in ϕ^{−1}(c_{0}) through y_{0} then K_{j} depends only on E; if y_{0} ∉ ϕ^{−1}(c_{0}) then K_{j} ≡ 1; and

(2.7) if K_{j} = κ_{j}| · |^{t_{j}} with κ_{j} unitary and g_{j} real then either t_{j} + b_{j} ≠ 0 or k_{j} ≢ 1, 1 ≤ j ≤ n. For Lemma 2.A, (2.7) need only be satisfied for y_{0} ∉ ϕ^{−1}(c_{0}). Remark. These are the conditions of [L, §1] for ϕ : Y ⟶ C to be an Igusa fibering; ω is an Igusa form if b_{1}, …, b_{n} are positive integers, i.e., ω has no singularities, and the zeros ω lie on the special fiber, i.e., b_{j} = 0 unless μ_{j} = 0 is the branch of a divisor in ϕ^{−1}(c_{0}): f is an Igusa integrand if K_{1}, … K_{n} are unitary and K_{j} ≡ 1 unless μ_{j} = 0 is the branch of a divisor in ϕ^{−1}(c_{0}).

Let 𝔈 be the set of all divisors in ϕ^{−1}(c_{0}) meeting the support of f. If μ_{j} = 0 is the branch of E ∈ 𝔈 through y_{0} then a_{j} = a(E), the multiplicity of E in ϕ^{−1}(c_{0}). We then also set b_{j} = b(E), K_{j} = K(E), κ_{j} = κ(E) and t_{j} = t(E), as our assumptions allow.

Let

F (λ) = ∮ f \frac{|ω|}{|ϕ^{∗}(dλ)|},

the integral being taken over the fiber in Y (F) above the point on C(F) with coordinate λ ≠ 0. Then F (λ) is a well-defined principal value integral of the type studied in the last section. To check this we may assume that f is supported on a neighborhood |μ_{j}| ≤ ϵ_{j}, 1 ≤ j ≤ n, in a coordinate patch (2.2) around y_{0} ∉ ϕ^{−1}(c_{0}). We may also assume |W | and γ constant. Then

(2.8)  F (λ) = γ|W | ∮_{{|u_{j}|<ϵ_{j} \atop (j>1)}} ∏_{j>1} K_{j}(μ_{j})|μ_{j}|^{b_{j}} \frac{dμ_{2}}{|μ_{2}|} ⋯ \frac{dμ_{n}}{|μ_{n}|}.

and we are done.

The data for the asymptotic expansion of F (λ) will be a slight modification of that of [L, Proposition 1.1]. Consider pairs (θ, β), where θ is a character on F^{×} and β is a real number. Let 𝔈(θ, β) be the set of those E ∈ 𝔈, i.e., of those divisors E in ϕ^{−1}(c_{0}) meeting the support of f, such that k(E) = θ^{a(E)} and

β(E) =^{def.} \frac{b(E) + t(E)}{a(E)} = β .

Let e(θ, β) be the maximum number of branches of divisors in 𝔈(θ, β) meeting at a point. For the purposes of this paper it will be sufficient to consider the case:

(2.9)  e(θ, β) ≤ 1 .

Then:

Lemma 2.A. For |λ| sufficiently small,

F (λ) = ∑_{(θ,β)} θ(λ)|λ|^{β−1}F_{1}(θ, β, f)

where F_{1}(θ, β, f) is the constant of [L, Proposition 1.2].

If e(θ, β) = 0 then F_{1}(θ, β, f) = 0. Otherwise, let E be a divisor in 𝔈(θ, β). Suppose that y_{0} ∈ E(F). Choose coordinates μ_{1}, …, μ_{n} as in (2.2) and assume that μ_{1} = 0 is a branch of E through y_{0}. Following [L, Proposition 1.2] we define h and ν near y_{0} by

h = h(μ_{2}, …, μ_{n}) = \frac{γ(0, μ_{2}, …, μ_{n})}{θ^{β}(α(0, μ_{2}, …, μ_{n}))} ∏_{j=2}^{n} K_{j}(μ_{j})θ(μ_{j}^{−a_{j}})|μ_{j}|^{−βa_{j}} ,

where θ^{β} = θ| · |^{β}, and

ν = W (0, μ_{2}, …, μ_{n}) ∏_{j=2}^{n} μ_{j}^{b_{j}} \frac{dμ_{2}}{μ_{2}} ∧ … ∧ \frac{dμ_{n}}{μ_{n}} .

Then

(2.10)  F_{1}(θ, β, f) = ∑_{E} ∮_{E(F)} h|ν|,

These integrals to be calculated by the methods of §1. Proof of Lemma 2.A:

We may assume that f is supported on a coordinate patch (2.2) around y_{0} ∈ ϕ^{−1}(c_{0}). Then f and ω, and hence F (λ), come with the parameters t = (t_{1}, …, t_{n}) and b = (b_{1}, …, b_{n}). We write F (λ) = F (λ, t, b).

If t_{j} + b_{j} ≥ 1 (1 ≤ j ≤ n) then arguments of [L, Propositions 1.1 and 1.2] carry through without modification, for F (λ, t, b) is an ordinary integral. Thus the lemma is proved in this case.

We now relax this condition on t and b. Let t′ = (t′_{1}, …, t′_{n}) ∈ R^{n}. It is convenient to assume that

\frac{t′}{a_{i}} = \frac{t′_{j}}{a_{j}} if a_{i}, a_{j} ≠ 0.

Suppose that E ∈ 𝔈 has data (θ, β) with respect to (t, b), i.e., with respect to f and ω. If μ_{j} = 0 is a branch of E through y_{0} then E has data θ′ = θ and

β′ = \frac{b_{j} + t_{j} + t′_{j}}{a_{j}} = β + \frac{t′_{j}}{a_{j}}

with respect to (t + t′, b). Thus (2.9) is satisfied by (t + t′, b). If t′_{j} ≫ 0, 1 ≤ j ≤ n, then t_{j} + t′_{j} + b_{j} ≥ 1, 1 ≤ j ≤ n, and there exists ϵ > 0 independent of t′ such that

(2.11)  F (λ, t + t′, b) = ∑_{(θ′,β′)} θ′(λ)|λ|^{β′−1}F_{1}(θ′, β′, f)

for |λ| < ϵ. One verifies easily that F (λ, t, b) is the value at t′ = 0 of F (λ, t + t′, b). At t′ = 0 the right side of (2.11) has the value

∑_{(θ,β)} θ(λ)|λ|^{β−1}F_{1}^{0}(θ, β′, f)

where F_{1}^{0}(θ, β′, f) is the value of F_{1}(θ, β′, f) at t′ = 0. This is readily seen to be F_{1}(θ, β, f), and the lemma is proved.

The following remarks will not be needed in this paper.

Let ϵ > 0, I(ϵ) = {c ∈ C(F) : |λ| = |λ(c)| ≤ ϵ} and Y (ϵ) be the inverse image of I(ϵ) in Y (F). The asymptotic expansion for F (λ) allows us to define the principal value integral ∮_{I(ϵ)} F (λ)dλ as the value of ∫_{I(ϵ)} F (λ)|λ|^{s}dλ at s = 0 provided F_{1}(1, 0, f) = 0, i.e., provided there is no contribution from the pair θ ≡ 1, β = 0 to the expansion. On the other hand our initial assumptions ensure that ∮_{Y (ϵ)} f|ω| is well-defined (in the sense of §1).

Lemma 2.B. Assume F_{1}(1, 0, f) = 0. Then

∮_{Y (ϵ)} f|ω| = ∮_{I(ϵ)} F (λ)dλ .

Proof: We may assume that f is supported on a neighborhood in a coordinate patch (2.2) around y_{0} ∈ ϕ^{−1}(c_{0}). Suppose that μ_{1} = 0, …, μ_{r} = 0 are branches of divisors in ϕ^{−1}(c_{0}), and that μ_{r+1} = 0, …, μ_{n} = 0 are not. Then for Res ≫ 0 ∮_{Y(ϵ)} f|λ|^{s}|ω| is the value at s_{r+1} = … = s_{n} = 0

of ∫_{Y (ϵ)} f|λ|^{s}|μ_{r+1}|^{s_{r+1}} … |μ_{n}|^{s_{n}}|ω|. Since λ = αμ_{1}^{a_{1}} … μ_{r}^{a_{r}}, where |α| ≠ 0 on the support of f, it

follows that lim_{ϵ′→0} ∮_{Y (ϵ′)} f|λ|^{s}|ω| = 0. Since ϕ is smooth away from the special fiber we have (cf. Lemma 1.F) that for ϵ′ < ϵ and Res ≫ 0

∮_{Y (ϵ)−Y (ϵ′)} f|λ|^{s}|ω| = ∫_{I(ϵ)−I(ϵ′)} F (λ)|λ|^{s}dλ .

The asymptotic expansion for F (λ) implies that

lim_{ϵ′→0} ∫_{I(ϵ′)} F (λ)|λ|^{s}dλ = 0 for Res ≫ 0 .

Thus

∮_{Y (ϵ)} f|λ|^{s}|ω| = ∫_{I(ϵ)} F (λ)|λ|^{s}dλ .

Since the value of the left side at s = 0 is ∮_{Y (ϵ)} f|ω| the lemma is proved.

Lemma 2.C. If F_{1}(θ, β, f) = 0 for all β ≤ 0 then

lim_{ϵ→0} ∮_{Y (ϵ)} f|ω| = 0 .

Proof: Under this assumption the asymptotic expansion involves only positive exponents β. Then lim_{ϵ→0} ∮_{I(ϵ)} F (λ)dλ = 0. Hence, by the last lemma, lim_{ϵ→0} ∮_{Y (ϵ)} f|ω| = 0.

§3. Some Igusa Data.

Following [L, §§2-5] we now construct a smooth variety Y over F, an Igusa fibering ϕ : Y ⟶ C of Y over a curve C, a differential form ω of maximal degree on Y, and an integrand f_{κ} (notation of [L, §2]) on Y (F).

Fix an inner form G of SL(2) and a maximal torus T over F in G. Let c_{0} be a point in the center of G. For the curve C we take T with the other central point removed.

The construction of Y starts with the variety S of stars ([L, §2]). Here S is just ℬ × ℬ, ℬ denoting the variety of Borel subgroups of G. Let B_{∞} ∈ ℬ. Then S(B_{∞}) is ℬ−{B_{∞}}×ℬ−{B_{∞}}. Let B_{0} ∈ ℬ − {B_{∞}}. Then S(B_{∞}, B_{0}) consists of the pairs (B_{+}, B_{−}) in S(B_{∞}) with B_{+} = B_{0}. If N(·) indicates unipotent radical and B^{g} the Borel subgroup g^{−1}Bg, g ∈ ℬ, we have

S(B_{∞}) = {(B_{0}^{n_{1}}, B_{0}^{nn_{1}}); n, n_{1} ∈ N(B_{∞})}
≃ S(B_{∞}, B_{0}) × N(B_{∞})
≃ N(B_{∞}) × N(B_{∞}) .

Coordinates for S(B_{∞}) are evident, but the demands for Galois action require that a little care be taken in the choice.

First, and for the rest of the paper, we fix data as in [L, §2]: G^{∗} = SL(2), B^{∗} is the upper triangular subgroup of G^{∗}, B_{∗} the lower triangular subgroup, T^{∗} the diagonal subgroup; ψ : G ⟶ G^{∗} is an inner twist such that ψ : T ⟶ G^{∗} is defined over F, T^{∗} denotes ψ(T), η^{∗} : G^{∗} ⟶ G^{∗} is a diagonalization of T^{∗} and, finally, η denotes η^{∗} ∘ ψ.

By means of ψ we identify G with G^{∗} as a group over F̄, and hence ℬ with ℬ^{∗} and S with ℬ^{∗} × ℬ^{∗}, where ℬ^{∗} is the variety of Borel subgroups of SL(2). View ℬ^{∗} as the variety P^{1} of lines through the origin in A^{2} via

(B^{∗})^{g} ↔ [0, 1] · g .

Write a for [a, 1].

Returning to B_{∞} and B_{0}, now elements of ℬ^{∗}, we choose h ∈ G^{∗} such that

(3.1)  (B_{0})^{h} = B^{∗} and (B_{∞})^{h} = B_{∗} .

Then h allows us to identify S(B_{∞}) with S(B_{∗}). If n = [{1 \atop x} {0 \atop 1}] and n_{1} = [{1 \atop y} {0 \atop 1}] we have

(3.2)  S(B_{∗}) ∋ ((B^{∗})^{n_{1}}, (B^{∗})^{nn_{1}}) ↔ (y, x + y) ∈ P^{1} × P^{1} .

Thus h provides coordinates, informally denoted x and y, on S(B_{∞}).

The variety S_{1}(B_{∞}) of [L, §3] is naturally identified with S(B_{∞}); S_{1} is then obtained by gluing together the S(B_{∞}), B_{∞} ∈ ℬ^{∗}, according to the rules of [L, (3.7) and (3.8)]. But these are the rules for the natural gluing of open subsets of ℬ^{∗} × ℬ^{∗} = P^{1} × P^{1}, and so S_{1} = S = P^{1} × P^{1} (cf. [L, Lemma 3.10(a)]).

To describe the Galois action on S and at the same time maintain our identification of G(F̄) with G^{∗}(F̄) and of S with ℬ^{∗} × ℬ^{∗} we equip G^{∗}(F̄) with the Galois action σ_{G} = ψ ∘ σ ∘ ψ^{−1}, σ ∈ Gal(F̄/F). Recall that the identification ψ : T ⟶ T^{∗} is over F. Let L ⊂ F̄ be a quadratic extension of F. Write 𝒯_{L} for the set of tori in G defined over F (i.e., in G^{∗} and preserved by σ_{G}, σ ∈ Gal(F̄/F)) which are anisotropic over F and split over L. We allow also L = F, then meaning by 𝒯_{L} the set of F -split tori in G.

The action of σ ∈ Gal(F̄/F) on S = ℬ^{∗} × ℬ^{∗} will be denoted σ_{(G,T)}. From [L, §2 and §4] we get

(3.3)  σ_{(G,T)}((B_{+}, B_{−})) = (σ_{G}(B_{−}), σ_{G}(B_{+}))

if T ∈ 𝒯_{L}, L ≠ F, and σ is nontrivial on L, and

(3.4)  σ_{(G,T)}((B_{+}, B_{−})) = (σ_{G}(B_{+}), σ_{G}(B_{−}))

otherwise.

The following elaborate remark will be helpful later on.

(3.5) If σ_{G}(B^{∗}) = B^{∗} and σ_{G}(B_{∗}) = B_{∗}, as we may assume if G is split over F, then S is covered by patches S(B_{∞}), where σ_{G}(B_{∞}) = B_{∞} and σ_{G}(B_{0}) = B_{0}, σ ∈ Gal(F̄/F), for some B_{0} ≠ B_{∞}. For example, S = S(B_{∗}) ∪ S(B^{∗}). Each such patch S(B_{∞}) is preserved by Gal(F̄/F). The element h of (3.1) can be chosen so that σ_{G}(h)h^{−1} is central, σ ∈ Gal(F̄/F). Then the identification of S(B_{∞}) with S(B_{∗}) provided by h respects Galois action.

(3.6) Suppose that L is a quadratic extension of F. Assume, as we may if T ∈ 𝒯_{L}, that σ_{G}(B^{∗}) = B_{∗} for σ nontrivial on L and σ_{G}(B^{∗}) = B^{∗}, σ_{G}(B_{∗}) = B_{∗} otherwise. Then S is covered by coordinate patches S(B_{∞}) where for some B_{0} ≠ B_{∞}, σ_{G}(B_{∞}) = B_{0} for σ nontrivial on L and σ_{G}(B_{∞}) = B_{∞}, σ_{G}(B_{0}) = B_{0} otherwise. Again S = S(B_{∗}) ∪ S(B^{∗}) will do. Now, however, σ_{(G,T)} preserves only S(B_{∞}) ∩ S(B_{0}) = S(B_{∞}) − {(B_{0}, B_{0})} if σ is nontrivial on L. The element h of (3.1) may be chosen so that σ_{G}(h)h^{−1} is central, σ ∈ Gal(F̄/F). Then the identification of S(B_{∞}) ∩ S(B_{0}) with S(B_{∗}) ∩ S(B^{∗}) provided by h respects Galois action.

Returning to the construction of Y we find it convenient to make yet another identification, that of T and T^{∗} with T^{∗} using the diagonalization η^{∗}. We equip T^{∗}(F̄) with the action σ_{T} = σ_{T^{∗}} = η^{∗} ∘ σ ∘ (η^{∗})^{−1}|_{T}^{∗}, σ ∈ Gal(F̄/F), and regard C as a curve in T^{∗} preserved by this action. Note that η^{∗}(c_{0}) = c_{0}.

A star s = (B_{+}, B_{−}) is regular in the sense of [L, §2] if and only if B_{+} ≠ B_{−}. For the variety X_{1} of [L, §4] we take the closure in G^{∗} × S of {(g, s = (B_{+}, B_{−})) : g, s regular, g ∈ B_{+} ∩ B_{−}}; X_{1} is defined over F for the Galois action given by σ_{G} × σ_{(G,T)}, σ ∈ Gal(F̄/F). There are maps defined over F:

\begin{matrix}X_{1} ⟶^{π} G^{∗} \\ ↓ϕ_{1} \\ T & ∗ \end{matrix}

where σ ∈ Gal(F̄/F) acts on G^{∗} by σ_{G} and on T^{∗} by σ_{T}. The horizontal arrow is projection on the first component. To define ϕ_{1}, note that X_{1} is contained in {(g, s = (B_{+}, B_{−})) ∈ G^{∗} × S : g ∈ B_{+} ∩ B_{−}}. Thus if (g, s) ∈ X_{1} we may choose h ∈ G^{∗} such that B_{+}^{h} = B^{∗}. Then ϕ_{1}((g, s)) is the image of h^{−1}gh ∈ B^{∗} under the projection B^{∗} = T^{∗}N(B^{∗}) ⟶ T^{∗}.

The variety Y will be the intersection of ϕ_{1}^{−1}(C) with the closure in X_{1} of ϕ_{1}^{−1}(C − {c_{0}}). By restriction we have:

\begin{matrix}Y ⟶^{π} G^{∗} \\ ↓ϕ \\ C \end{matrix}

Let M be the Springer-Grothendieck variety {(g, b) : g ∈ B} ⊂ G^{∗} × ℬ^{∗}, with the usual maps:

\begin{matrix}M ⟶^{π_{M}} G^{∗} \\ ϕ_{M} ↓ \\ T & ∗ \end{matrix}

Define ξ : Y ⟶ M by (g, (B_{+}, B_{−})) ⟶ (g, B_{+}). Then ϕ = ϕ_{M} ∘ ξ and π = π_{M} ∘ ξ. If M′ is M with the fibers over the central points removed then ϕ^{−1}(C − {c_{0}}) ⟶^{ξ} M′ is an isomorphism of varieties over F̄. In particular, ϕ^{−1}(C − {c_{0}}) is smooth.

To examine the special fiber ϕ^{−1}(c_{0}) we introduce coordinates as in [L, §3]. Let Y ⊂ G^{∗}×S ⟶ S be projection on the second factor. Let Y (B_{∞}) be the inverse image of S(B_{∞}), B_{∞} ∈ ℬ^{∗}. Identify S(B_{∞}) with S(B_{∗}) by means of some h as in (3.1). We may then work with the coordinates x, y of (3.2) on S(B_{∗}) and with Y (B_{∗}).

Let λ be a local F -coordinate around c_{0} in C. Recall that C ⊂ T^{∗} and that σ ∈ Gal(F̄/F) acts by σ_{T}. Assume that λ = 0 at c_{0}. If α is the root of T^{∗} in B^{∗} then we may write 1 − α^{−1} as λb(λ) near c_{0}, with b regular and invertible near λ = 0. Suppose that (g, s) ∈ Y (B_{∗}). As in (3.2) write s as ((B^{∗})^{n_{1}}, (B^{∗})^{nn_{1}}), with n = ( \begin{matrix}1 & 0 \\ x & 1 \end{matrix} ) and n_{1} = ( \begin{matrix}1 & 0 \\ y & 1 \end{matrix} ). Note that x is the coordinate z(W_{+}, α) from [L, §3]. Write

(3.7)  g = n_{1}^{−1}t ( \begin{matrix}1 & u \\ 0 & 1 \end{matrix} ) n_{1}, with t ∈ T^{∗}, u ∈ F̄ .

Assume x ≠ 0. Then g ∈ (B^{∗})^{nn_{1}} is equivalent to

1 − α(t)^{−1} = xu

or, if (g, s) is near ϕ^{−1}(c_{0}) and we pull back λ to Y, to

(3.8)  λb(λ) = xu .

As a consequence, u, x and y serve as coordinates on Y (B_{∗}), and Y (B_{∗}) is smooth. Then each Y (B_{∞}) is smooth, B_{∞} ∈ ℬ^{∗}. Hence Y is a smooth variety.

Near ϕ^{−1}(c_{0}) on Y (B_{∗}), ϕ is given by

(3.9)  λ = Axu

with A regular and invertible near λ = 0. Thus u = 0 is the branch of a divisor E_{1} of ϕ^{−1}(c_{0}). This branch consists of the pairs (c_{0}, s), s ∈ S(B_{∗}), and so E_{1} must be {c_{0}} × S = {c_{0}} × P^{1} × P^{1}; E_{1} maps under π : Y ⟶ G to {c_{0}}. On the other hand, x = 0 is the branch {(g, (B, B)) : B ≠ B_{∗}, g ∈ G, c_{0}g unipotent}. For convenience we call g c_{0}-unipotent if c_{0}g is unipotent. Then π maps E_{2} − E_{1} isomorphically to the orbit of regular c_{0}-unipotent elements in G. Note that the two divisors E_{1} and E_{2} cover ϕ^{−1}(c_{0}), and that E_{2} has no F -rational points unless G is split over F.

The relation of Y to the Springer-Grothendieck variety M is now evident. Under Y ⟶^{ξ} M the divisor E_{2} is mapped isomorphically to the fiber over c_{0}; Y is obtained from M with the fiber over −c_{0} removed by blowing up along the subvariety {c_{0}} × ℬ^{∗} = ξ(E_{1}) of the fiber over c_{0}.

To verify that ϕ : Y ⟶ C is an Igusa fibering it remains only to check that one of E_{1}, E_{2} is defined over F. For then both divisors are defined over F and we may apply (3.9) and Hilbert’s Theorem 90 (for the field of functions regular and invertible near a point) to replace around each F -rational point y_{0} ∈ ϕ^{−1}(c_{0}) the coordinates u, x and y with F -coordinates μ_{1}, μ_{2}, μ_{3} such that:

(3.10) μ_{i} = 0 is a branch of E_{i} if y_{0} lies on E_{i}(i = 1, 2).

(3.11) λ is given near y_{0} by λ = αμ_{1}^{a_{1}}μ_{2}^{a_{2}}, where α is regular and invertible at y_{0}, and a_{i} = 1 if y_{0} lies on E_{i} and a_{i} = 0 otherwise (i = 1, 2). Since E_{1} = {c_{0}} × S is clearly defined over F, we are done.

The indices a(·) of §2 are:

(3.12)  a(E_{1}) = a(E_{2}) = 1 if G is split over F ,
a(E_{1}) = 1  otherwise.

The next step is to define an Igusa form ω. Let ω_{T} be the (right) invariant form on T^{∗} equal to dλ at c_{0}. Let a ∈ F̄ be such that ω̄ = aω_{T} is defined over F for the Galois action on T^{∗} as F -split torus. Let H ∈ Lie(T^{∗}) be such that ω̄(H) = 1. Choose X_{+} ∈ Lie(N(B^{∗})), X_{−} ∈ Lie(N(B_{∗})) and right invariant 1-forms ω, ω_{+}, ω_{−} on G defined over F so that ⟨ω_{0}, ω_{+}, ω_{−}⟩ is dual to ⟨H, X_{+}, X_{−}⟩. Then ω_{G} = ω_{0} ∧ ω_{+} ∧ ω_{−} is a (right) invariant form of maximal degree on G defined over F. The form ω_{M} on M associated to ω_{G} (more precisely, to ν_{1} = ω_{0}, ν_{2} = ω_{+}, ω_{1} = ω_{−}) in [L, Lemma 2.8] is π_{M}^{∗}(ω_{G}). We set ω_{Y} = ξ^{∗}(ω_{M}) = π^{∗}(ω_{G}) and ω = a^{−1}ω_{Y}.

The form ω is regular; it is nonvanishing off the special fiber. The discussion of [L, §2] implies that locally ω = W′ω′, where W′ is a regular invertible function and ω′ is defined over F. This ensures that the measure |ω| is well-defined.

Suppose that y_{0} ∈ Y (B_{∗}) is near but not on ϕ^{−1}(c_{0}). We may as well take X_{+} = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} )

and X_{−} = ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ). Then it may be shown that ω is given near y_{0} by ϕ^{∗}(dλ) ∧ du ∧ dy = W (λ)d(xu) ∧ du ∧ dy = W (λ)u dx ∧ du ∧ dy, where W is regular and invertible near λ = 0, with W (0) = (ab(0))^{−1} (cf. (3.8)). From this it follows that

(3.13)  ω = W (λ)u^{2}x\frac{dx}{x} ∧ \frac{du}{u} ∧ dy

around a point of Y (B_{∗}) ∩ ϕ^{−1}(c_{0}).

Note that ω may be expressed in terms of the coordinates μ_{1}, μ_{2}, μ_{3} of (3.10), but that the coordinates u and x will do just as well to compute the indices b(·) of §2:

(3.14)  b(E_{1}) = 2, b(E_{2}) = 1 if G is split over F
b(E_{1}) = 1  otherwise.

It remains to define the Igusa integrand. Let κ be a character on 𝒟(T) = 𝒟(T, F), the definition of which will be recalled in (3.15). Recall that T (F̄)\𝔄(T, F) is the set of F -rational points in T (F̄)\G(F̄) = (T \G)(F̄). If γ ∈ T (F)−{c_{0}} then π : Y ⟶ G induces an F -isomorphism from the fiber ϕ_{γ}^{−1} over γ in Y to T \G (cf. [L, Lemma 2.1]). We have therefore:

(3.15)  ϕ_{γ}^{−1}(F) ⟶ T (F̄)\𝔄(T, F) ⟶ 𝒟(T, F) = T (F̄)\𝔄(T, F)/G(F)
≃ H^{1}(Gal(F̄/F), T(F̄)) ,

allowing us to regard κ as a function m_{κ} on ϕ_{γ}^{−1}(F). If κ is trivial then m_{κ} ≡ 1. Suppose then that T ∈ 𝒯_{L}, L ≠ F, and κ is nontrivial. We will need an explicit formula for m_{κ} near an F -rational point y_{0} on the special fiber. Proposition 5.1 of [L] shows that m_{κ} depends locally only on the coordinate x, at least if G is split over F, but for the formula we will need an F -coordinate.

Suppose that S(B_{∞}) is a coordinate patch as in (3.5). We may as well take σ_{G} = σ_{G^{∗}}, σ ∈ Gal(F̄/F), or G = SL(2). Identify S(B_{∞}) with S(B_{∗}) using h as in (3.5). Recall that this identification respects the Galois action on S. The formulas (3.2), (3.3) and (3.4) imply that the coordinates x, y on S(B_{∗}) satisfy σ(y) = x + y if σ is nontrivial on L and that σ(y) = y, σ(x + y) = x + y otherwise. Then σ(x) = σ(x + y − y) = y − (x + y) = −x for σ nontrivial on L. Fix τ ∈ L − F such that τ^{2} ∈ F. Then μ = τx is an F -coordinate (and will serve as μ_{2} in (3.10)). Let (g, s) ∈ ϕ_{γ}^{−1}(F) lie in Y (B_{∞}), which we have identified with Y (B_{∗}). The coordinate μ then being F -valued, we have that

σ ⟶ ( \begin{matrix}μ & 0 \\ 0 & μ^{−1} \end{matrix} ) if σ|_{L} ≢ 1, σ ⟶ 1 otherwise ,

represents an element of H^{1}(T) that we denote μ_{σ}. Let ϵ_{σ} denote the image of (g, s) under (3.15). Then Proposition 5.2 of [L] implies that there is an element t_{σ} of H^{1}(T) independent of (g, s) such that

(3.16)  ϵ_{σ} = μ_{σ}t_{σ}

(see the Appendix to this section). Thus

m_{κ}((g, s)) = κ(ϵ_{σ}) = κ(μ_{σ})κ(t_{σ}) = κ(μ)κ(t_{σ}) ,

where κ now also denotes the quadratic character on F^{×} attached to L/F.

By requiring that S(B_{∞}) be as in (3.5) we have excluded the case G anisotropic over F. This is of no consequence, for then if (g, s) ∈ Y (F) the star s = (B_{+}, B_{−}) must be regular (σ_{G}(B_{+} ∩ B_{−}) = B_{+} ∩ B_{−} implies B_{+} ∩ B_{−} is a torus so that B_{+} ≠ B_{−}). From Lemma 2.10 of [L] we conclude that m_{κ} is locally constant on Y (F).

Finally, fix f ∈ C_{c}^{∞}(G(F)). The Igusa integrand will be:

\begin{aligned}f_{κ}(g, s)) & = m_{κ}((g, s))(f ∘ π)(g, s) \\ & = m_{κ}((g, s))f(g), (g, s) ∈ Y (F) − ϕ^{−1}(c_{0}) . \end{aligned}

The characters κ(·) of §2 are:

(3.17)  \{ \begin{matrix}κ(E_{1}) ≡ 1 and κ(E_{2}) = κ & if G is split over F \\ κ(E_{1}) ≡ 1 & otherwise. \end{matrix}

Appendix

Here we note the explicit calculation of ϵ_{σ} in (3.16) and another local expression for m_{κ} which applies to anisotropic groups as well.

For (3.16) recall that we have assumed that G is SL(2) (and ψ ≡ 1, T = T^{∗}). We refrain from identifying T (F̄) with T^{∗}(F̄). Then ϵ_{σ} is the class of σ ⟶ σ(h_{1})h_{1}^{−1}, where h_{1} ∈ G(L) satisfies:

h_{1}gh_{1}^{−1} ∈ T, B_{+}^{h_{1}} = η^{−1}(B^{∗}) and B_{−}^{h_{1}} = η^{−1}(B_{∗})

if s = (B_{+}, B_{−}). Write η as t ⟶ h_{2}th_{2}^{−1}, h_{2} ∈ G(L). For h_{1}, we can take h_{2}^{−1}h_{3} if h_{3} ∈ G(L) satisfies:

h_{3}gh_{3}^{−1} ∈ T^{∗}, B_{+}^{h_{3}} = B^{∗} and B_{−}^{h_{3}} = B_{∗} .

On Y (B_{∗}) we have

g = ( \begin{matrix}1 & 0 \\ −y & 1 \end{matrix} ) t ( \begin{matrix}1 & u \\ 0 & 1 \end{matrix} ) ( \begin{matrix}1 & 0 \\ y & 1 \end{matrix} ) ,

with t ∈ T^{∗} and ux = 1 − α(t)^{−1}. It is easily checked that

h_{3} = ( \begin{matrix}1 & 1/x \\ 0 & 1 \end{matrix} ) ( \begin{matrix}1 & 0 \\ y & 1 \end{matrix} )

will do;

σ(h_{3})h_{3}^{−1} = ( \begin{matrix}0 & −1/x \\ x & 0 \end{matrix} ) = ( \begin{matrix}μ^{−1} & 0 \\ 0 & μ \end{matrix} ) ( \begin{matrix}0 & −1/τ \\ τ & 0 \end{matrix} )

for σ nontrivial on L. Then

\begin{aligned}σ(h_{1})h_{1}^{−1} & = σ(h_{2}^{−1}) ( \begin{matrix}μ^{−1} & 0 \\ 0 & μ \end{matrix} ) ( \begin{matrix}0 & −1/τ \\ τ & 0 \end{matrix} ) h_{2} \\ & = η^{−1} ( \begin{matrix}μ & 0 \\ 0 & μ^{−1} \end{matrix} ) σ(h_{2}^{−1}) ( \begin{matrix}0 & −1/τ \\ τ & 0 \end{matrix} ) h_{2} , \end{aligned}

so that (3.16) holds with τ_{σ} the class of

σ ⟶ σ(h_{2}^{−1}) ( \begin{matrix}0 & −1/τ \\ τ & 0 \end{matrix} ) h_{2} .

Suppose now that L is a quadratic extension of F and that

(3.18)  σ_{G} = \{ \begin{matrix}ad ( \begin{matrix}0 & 1 \\ ζ & 0 \end{matrix} ) ∘ σ_{G^{∗}} & if σ|_{L} ≢ 1 , \\ σ_{G^{∗}} & otherwise , \end{matrix}

where ζ ∈ F^{×}. Note that G is split over F if and only if ζ ∈ Nm_{F}^{L}L^{×}. Assume that T ∈ 𝒯_{L}. Now σ_{G} satisfies the conditions of (3.6). The coordinates x and y on S(B_{∗}) satisfy σ(x+y) = ζ/y, σ(y) = ζ/(x + y) if σ|_{L} ≢ 1 and σ(x + y) = x + y, σ(y) = y otherwise. Then x/(x + y) = 1 − y/(x + y) is defined over F. It serves as a coordinate around an F -rational point (g, s) of Y (B_{∗}) near ϕ^{−1}(c_{0}). A calculation as in the last paragraph shows that:

(3.19)  m_{κ}((g, s)) = κ(x/(x + y)) ,

where κ now denotes the quadratic character of F^{×} attached to L/F if κ is nontrivial, and the trivial character otherwise.

§4. Application

Continuing from the last section, we have f ∈ C_{c}^{∞}(G(F)), Haar measures |ω_{G}| on G(F) and |ω_{T} | on T (F), and a character κ on 𝒟(T). For γ regular in T (F), form the κ-orbital integral

\begin{aligned}Φ^{κ}(γ, f) & = Φ_{T}^{κ} (γ, f, |ω_{T} |, |ω_{G}|) \\ & = ∑_{δ} κ(δ) ∫_{T^{h}(F)\G(F)} f(g^{−1}h^{−1}γhg)\frac{|ω_{G}|}{|ω_{T} |^{h}} , \end{aligned}

where h ∈ 𝔄(T, F) represents δ ∈ 𝒟(T, F), and then the normalized integral

F^{κ}(γ, f) = |1 − α(γ^{−1})|Φ^{κ}(γ, f).

Recall that T has been identified with T^{∗} by means of η; α is the root of T^{∗} in B^{∗}.

Assume that γ lies in C(F) near c_{0} and has coordinate λ. The Igusa data of the last section were chosen so that

F^{κ}(γ, f) = ∫_{ϕ_{γ}^{−1}(F)} f_{κ} \frac{|ω|}{|dλ|}.

Thus for |λ| sufficiently small we have that

(4.1)  F^{κ}(γ, f) = |λ|Λ_{1} + κ(λ)Λ_{2}

where, in the notation of Lemma 2A,

Λ_{1} = F_{1}(1, 2, f) (4.2) and (4.3) Λ_{2} = \{ \begin{matrix}F_{1}(κ, 1, f) & if G is split over F \\ 0 & otherwise. \end{matrix} On the right side of (4.1) we have, as in §3, regarded κ as a character on F^{×}, trivial if κ is trivial on 𝒟(T) and the character on F^{×} attached to the quadratic splitting field L of T otherwise.

The term Λ_{1} is the contribution from the divisor E_{1} which maps to {c_{0}} under π: Y → G, while Λ_{2} is the contribution from E_{2}; under π, E_{2} maps (isomorphically) to the conjugacy class of regular c_{0}-unipotent elements in G. Thus (4.1) assumes a familiar form, but with the coefficients now expressed as principal value intregrals.

If T is split over F then blowing up the Springer-Grothendieck variety M to obtain Y is unnecessary, and as a consequence we have introduced the spurious term Λ_{1}. It is quickly dismissed, for if T is split over F then E_{1} = {c_{0}} × S is F -isomorphic to P^{1} × P^{1} (cf. (3.4)) and by (2.10) Λ_{1} is given, up to a constant, by

∮_{P^{1}(F)×P^{1}(F)} \frac{da db}{|a − b|^{2}}

where a, b each denote the coordinate U_{0} on U_{1} = 1 in P^{1}. We apply Lemma 1.F to this integral and the fibering ϕ: P^{1} × P^{1} → P^{1} given by projection on the first component. The fiber integral H(x′), x′ ∈ P^{1}(F), is seen to be an integal over P^{1}(F) of the form (1.15). Thus it is zero. We conclude then from Lemma 1.F that Λ_{1} = 0.

Lemma 4.A. If κ is nontrivial then Λ_{1} = 0.

Proof: If G is anisotropic over F then this is immediate from the definition of κ-orbital integral (see also the remark following the proof of Lemma 4.B). Suppose then that G is split over F. We may as well assume that G = G^{∗} = SL(2). Since κ is nontrivial T ∈ 𝒯_{L}, some L ≠ F; E_{1} is the variety R = Res_{F}^{L}P^{1} of Remark 1.H, i.e., E_{1}(L) = P^{1}(L) × P^{1}(L) and E_{1}(F) = {(p, p̄): p ∈ P^{1}(L)}, where the bar denotes the action of the nontrivial element of Gal(L/F) (cf. (3.3), (3.4)). Then by (2.10), (3.2), (3.5) and (3.16) λ_{1} is, up to a constant,

(4.4)  ∮_{R(F)} \frac{κ (\frac{b−a}{τ}) da db}{|b − a|^{2}}

where a, b each denote the coordinate U_{0} on U_{1} = 1 in P^{1}(L). The element τ of L − F was fixed for (3.16); \frac{b−a}{τ} lies in F^{×} if b = ā ≠ 0.

Abbreviate the point U_{0} = a, U_{1} = 1 in P^{1} by a, and U_{0} ≠ 0, U_{1} = 0 by ∞. We define a smooth morphism ϕ: R − \{(∞, ∞)\} → P^{1} by (a, b) → \frac{b−a}{τ} and (a, ∞), (∞, b) → ∞; ϕ^{0} is defined over F. We blow up R at (∞, ∞) to obtain the variety R̂ over F. The fiber over (∞, ∞) in R̂ meets the proper inverse image of the divisor a = b at a single point p_{0}. Blow up R̂ at this point, which is F -rational, to obtain the variety ˆ R̂ over F. A calculation with coordinates shows that ϕ^{0} extends to an Igusa fibering ϕ: ˆ R̂ → P^{1} with distinguished point c_{0} = ∞; moreover, the fiber over ∞ is the union of three divisors, each occurring with multiplicity one. Only one of the divisors has F -rational points. We conclude then that on ˆ R̂(F) the map ϕ is smooth.

To compute (4.4) by lifting to ˆ R̂(F) we must check the conditions of Lemma 1.D at (∞, ∞) on R(F) and at p_{0} on R̂(F). We find that θ_{1} = κ, c_{1} = −1 and θ_{2} ≡ 1, c_{2} = 1 for a suitable choice of local coordinates around (∞, ∞) on R(F). Thus θ_{1}θ_{2} = κ and c_{1} + c_{2} = 0, and so to lift to R̂(F) it is crucial that κ be nontrivial. On R̂(F) at p_{0} we find θ_{1} = κ, c_{1} = 0 and θ_{2} = κ, c_{2} = −1. The conditions of Lemma 1.D are met and we may apply Corollary 1.E to rewrite (4.4) as a principal value integral I over ˆ R̂(F). We compute I by applying Lemma 1.F to the fibering ϕ. For any p ∈ P^{1}(F) − {0, ∞} the corresponding fiber integral is seen immediately to be a constant times ∮_{P^{1}(F)} da = 0 (cf. (1.15)). Thus Lemma 1.F implies that I = 0, and Lemma 4.A is proved.

Suppose now that T ∈ 𝒯_{L}, L ≠ F; κ may be either character on 𝒟(T). Recall that the maximal torus T^{∗} = ψ(T) in G^{∗} is defined over F, as is the map ψ: T → T^{∗}. Let κ^{∗} be the character on 𝒟(T^{∗}) associated to κ by ψ. We indicate by Λ_{1}^{∗} the contribution (4.2) for the data G^{∗}, T^{∗}, κ^{∗} and f^{∗} ∈ C_{c}^{∞}(G^{∗}(F)). It may be written as M_{1}^{∗}f^{∗}(c_{0}). Similarly the contribution (4.2) for the data G, T, κ and f can be written as M_{1}f(c_{0}).

Lemma 4.B.

M_{1} = ε(κ, G)M_{1}^{∗}

where ε(κ, G) = 1 if κ is nontrivial and ε(1, G) = \{ \begin{matrix}1 & if G is split over F \\ −1 & otherwise. \end{matrix}

Proof: We may assume that G satisfies (3.18), i.e.,

σ_{G} = \{ \begin{matrix}ad ( \begin{matrix}0 & 1 \\ ζ & 0 \end{matrix} ) ∘ σ_{G^{∗}} & if σ _{L} ≢ 1 \\ σ_{G^{∗}} & otherwise, \end{matrix}

where ζ ∈ F^{×}; G is split over F if and only if ζ ∈ Nm_{F}^{L}L^{×}. We write M_{1}(ζ) for the term M_{1}. Then from (2.10), (3.2), (3.6) and (3.19) we find that M_{1}(ζ) is given up to a constant independent of G (i.e., of ζ) by

(4.5)  ∮_{Q_{ζ}(F)} \frac{κ (1 − \frac{1}{b}) da db}{|b − a|^{2}}

where Q_{ζ} is the form of P^{1} × P^{1} on which 1 ≠ σ ∈ Gal(L/F) acts on the homogeneous coordinates U_{0}, U_{1} (on the first copy of P^{1}) and V_{0}, V_{1} (on the second copy) by U_{0} → ζV_{1}, U_{1} → V_{0}, V_{0} → ζU_{1}, V_{1} → U_{0}. Also a denotes U_{0} on U_{1} = 1 and b = V_{0} on V_{1} = 1.

We define now a smooth variety Y and an Igusa fibering ϕ: Y → A^{1} with distinguished point zero on A^{1} such that if ζ is the coordinate on A^{1} then (4.5) is the fiber integral F (ζ), ζ ≠ 0. The asymptotic expansion for F (ζ) at ζ = 0 will be seen to have the one term, that corresponding to θ = θ_{L}κ, where θ_{L} is the quadratic character of F^{×} attached to L/F, and β = 1. Then in the notation of Lemma 2.A we have

(4.6)  F (ζ) = θ_{L}(ζ)κ(ζ)F_{1}(θ_{L}κ, 1,^{∗})

for |ζ| sufficiently small, where ^{∗} is the Igusa integrand yet to be defined. This will prove the lemma.

We start with a variety Y_{1} ⊂ (P^{1})^{4} × A^{1}. let a, b, a_{1}, b_{1} each denote the coordinate U_{0} on U_{1} = 1 in P^{1}, and ζ be the coordinate on A^{1}. On (U_{1} = 1)^{4} × A^{1}, Y_{1} is given by ab_{1} = a_{1}b = ζ. Let a′, b′, a′_{1}, b′_{1} each denote U_{1} on U_{0} = 1. On (U_{0} = 1) × (U_{1} = 1)^{3} × A^{1}, Y_{1} is given by a_{1}b = ζ and b_{1} = a′ζ; on (U_{0} = 1)^{2} × (U_{1} = 1)^{2} × A^{1} by b_{1} = a′ζ, a_{1} = b′ζ, and so on. We define Y_{1} over F by twisting the natural F -structure by σ(a) = a_{1}, σ(b) = b_{1}, σ(a′) = a′_{1}, σ(b′) = b′_{1}, σ being the nontrivial element of Gal(L/F).

The variety Y_{1} is smooth except at the point y_{1} given by a = a_{1} = b = b_{1} = 0. Let ϕ_{1} be the projection of Y_{1} ⊂ (P^{1})^{4} × A^{1} onto A^{1}; ϕ_{1}: Y_{1} − {y_{1}} → A^{1} is an Igusa fibering with distinguished point zero.

For ζ ≠ 0 the projection of Y_{1} onto the product of the first and second copies of P^{1} yields an F -isomorphism of the fiber ϕ_{1}^{−1}(ζ) with Q_{ζ}. Let Y′_{1} = Y_{1} − ϕ_{1}^{−1}(0). We define a form ω_{1} on Y′_{1} by \frac{da∧db∧dζ}{(b−a)^{2}} and an integrand f_{1} = κ (1 − \frac{a}{b}) on Y′_{1}(F). The fiber integral

(4.7)  ∮_{ϕ_{1}^{−1}(ζ)(F)} f_{1} \frac{|ω_{1}|}{|dζ|}

is (4.5).

We now attend to the fiber in Y_{1} − {y_{1}} over ζ = 0. It is the union of divisors E′_{1}, …, E′_{4}. Their branches on ((U_{1} = 1)^{4} × A^{1}) ∩ (Y_{1} − {y_{1}}) are:

\begin{matrix}(E′_{1}) b = 0, b_{1} = 0 \\ (E′_{3}) a = 0, b = 0 \end{matrix}  \begin{matrix}(E′_{2}) a = 0, a_{1} = 0 \\ (E′_{4}) a_{1} = 0, b_{1} = 0. \end{matrix}

Note that E′_{1}, E′_{4} have no F -rational points; E′_{1}, E′_{2} are each defined over F.

The point y_{1} on Y_{1} is F -rational. Blow up Y_{1} at this point to obtain the variety Y over F and projection π: Y → Y_{1}. Set ϕ = ϕ_{1} ∘ π, ω = π^{∗}(ω_{1}) and f_{Y} = f_{1} ∘ π. Then Y, C = A^{1}, c_{0} = 0, ϕ, ω and f_{Y} satisfy the conditions of (2.1)–(2.7) (i.e., are “generalized” Igusa data), as well as (2.9). The proof is routine. We will include as much of it as will be needed to write down the asymptotic expansion for the fiber integral which, by construction, coincides with the integral (4.7).

The fiber ϕ^{−1}(0) is the union of five divisors E_{0}, E_{1}, …, E_{4}, where E_{i} is the proper inverse image of E′_{i} (i = 1, …, 4). Let u_{1} = a, u_{2} = b, u_{3} = a_{1}, u_{4} = b_{1}; let U_{1} = A, U_{2} = B, U_{3} = A_{1}, U_{4} = B_{1} be homogeneous coordinates on P^{3}. Then Y is given near π^{−1}(y_{1}) by u_{i}U_{j} = u_{j}U_{i} (i, j = 1, …, r). The divisor E_{0} is given by a = b = a_{1} = b_{1} = 0 and AB_{1} = BA_{1} (homogeneous coordinates); on E_{0} ∩ E_{1} we have B = 0, B_{1} = 0; on E_{0} ∩ E_{2}, A = 0 and A_{1} = 0, and so on. The divisors E_{0}, E_{1} and E_{2} are each defined over F, while E_{3} and E_{4} have no F -rational points and so may be ignored for the asymptotic expansion. Also E_{1} ∩ E_{2} is empty and E_{i} ∩ E_{j} (i = 1, 2, and j = 3, 4) consists of a single point on E_{0} which is not F -rational.

.^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} E_{0} E_{1} E_{2} E_{3}  E_{4}

The variety E_{0} is a form of P^{1} × P^{1}, the natural projections being given by (A, B, A_{1}, B_{1}) → \frac{B}{A} = \frac{B_{1}}{A_{1}} , \frac{A_{1}}{A} = \frac{B_{1}}{B} (where we allow the value ∞ and ignore quotients of the form \frac{0}{0}). For these to be defined over F, the first P^{1} has to be provided with its natural F -structure and the second with the structure of Remark 1.G. The variety E_{1} is the blow-up R̄ of the twisted form R of P^{1} × P^{1} described in Remark 1.H (cf. also (4.4)). To see this, we note that the projection of Y_{1} ⊂ (P^{1})^{4}×A^{1} onto the product of the first and third copies of P^{1} yields an F -isomorphism of E′_{1} with R − {r_{0}}, where r_{0} is given by a = a_{1} = 0. Then E_{1} is the blow-up R̄ of R at r_{0}, and E_{0} ∩ E_{1} is the inverse image of r_{0} in R̄. The divisor E_{2} is described similarly.

Suppose that y_{0} ∈ E_{0}(F) and that A ≠ 0 at y_{0}. We may assume A = 1. Then t = a_{1}, A_{1}, B serve as coordinates on Y near y_{0}; a_{1} = tA_{1} and b = tB. For 1 ≠ σ ∈ Gal(L/F) we have σ(B) = B, A_{1}σ(A_{1}) = 1 and σ(t) = tA_{1}. To obtain F -coordinates we may take B, r and s with t = st_{0}, t_{0} ≠ 0, σ(t_{0})/t_{0} = A_{1} and t_{0} = 1 + τr, where τ ∈ L − F and τ^{2} ∈ F. Also, t = 0 is a branch of E_{0}, A_{1} ≠ 0 since A_{1}σ(A_{1}) = 1, and B = 0 is a branch of E_{1}. Finally, ζ = a_{1}b = t^{2}A_{1}B,

ω = \frac{da ∧ db ∧ dζ}{(b − a)^{2}} = tB \frac{dt ∧ dB ∧ dA_{1}}{(B − 1)^{2}}

and f_{Y} = κ(1 − 1/B) = κ(B)κ(B − 1). We conclude that

β(E_{0}) = \frac{b(E_{0})}{a(E_{0})} = \frac{2}{2} = 1, κ(E_{0}) ≡ 1.

Similarly,

β(E_{1}) = \frac{2}{1} = 2, κ(E_{1}) = κ

and

β(E_{2}) = \frac{2}{1} = 2, κ(E_{2}) ≡ 1.

This implies that (2.9) is satisfied (i.e., e(θ, β) ≤ 1 for all (θ, β)) since E_{1} and E_{2} do not intersect. The asymptotic expansion for the fiber integral, i.e., for the integral (4.5), is then

(4.8)  ∑_{θ} θ(ζ)F_{1}(θ, 1, f_{Y}) + |ζ|κ(ζ)F_{1}(κ, 2, f_{Y}) + |ζ|F_{1}(1, 2, f_{Y})

if κ is nontrivial, or

(4.9)  ∑_{θ} θ(ζ)F_{1}(θ, 1, f_{Y}) + |ζ|F_{1}(1, 2, f_{Y})

if κ is trivial. The summation is over characters θ of F^{×} for which θ^{2} = 1.

The integrals F_{1}(κ, 2, f_{Y}) and F_{1}(1, 2, f_{Y}) of (4.8) and the two integrals contributing to

F_{1}(1, 2, f_{Y}) in (4.9) (cf. (2.10)) are each of the form (1.20) and hence vanish.

Since ζ = s^{2}(t_{0}^{2}A_{1}B) and t_{0}^{2}A_{1}B = t_{0}σ(t_{0})B the formula (2.10) yields

(4.10)  F_{1}(θ, 1, f_{Y}) = ∮_{E_{0}(F)} \frac{κ(B)}{θ(B)} \frac{κ(B − 1)}{|B − 1|^{2}} \frac{1}{θ̂(A_{1})} \frac{dA_{1}}{|A_{1}|} dB,

where θ̂(A_{1}) = θ(t_{0}σ(t_{0})). Recall that A_{1} ranges over {x ∈ L^{×}: Nm_{F}^{L}x = 1}. The character θ̂ is

trivial if and only if θ ≡ κ or θ ≡ κθ_{L}. In the case θ̂ ≢ 1,

∫ \frac{dA_{1}}{θ̂(A_{1})|A_{1}|} = 0 (cf. Remark 1.G)

and so (4.10) vanishes. In the case θ ≡ κ,

∮_{P^{1}(F)} \frac{κ(B − 1)}{|B − 1|^{2}} dB = ∮_{P^{1}(F)} \frac{κ(B)}{|B|^{2}} dB = 0 (Lemma 1.C)

and again (4.10) is zero. Thus only θ ≡ κθ_{L} may give a nonzero contribution. The expansions

(4.8) and (4.9) therefore take the form (4.6), and Lemma 4.B is proved.

Lemma 4.B and Lemma 4.A in the case G split over F imply Lemma 4.A for G anisotropic

over F.

References

[L-L] J.-P. Labesse and R. Langlands, L-indistinguishability for SL(2), Canad. J. Math., vol. 31 (1979), pp. 726–785.

[L] R. Langlands, Orbital integrals on forms of SL(3), I, Amer. J. Math., vol. 105 (1983), to appear.

[L_{1}] , Les débuts d’une formule des traces stable, Publ. Math. Univ. Paris VIII, vol. 13 (1983).

[R] J. Repka, Shalika’s germs for p-adic GL(n) II: the subregular term, preprint.

[V] M.-F. Vignéras, Caractérisation des intégrales orbitales sur un groupe réductif p-adique, J. Fac. Sci. Univ. Tokyo, vol. 28 (1982), pp. 945–961.