CHAPTER 2

Preliminaries

2.1

For v ∈ Σfin, let dv be the local differential exponent of Fv over Qp, where p is

the characteristic of ov/pv:

pv

−dv

= {x ∈ Fv| trFv/Qp (xov) ⊂ Zp }.

Then the global different dF/Q is the ideal of oF such that dF/Q ov = pvv d for all

v ∈ Σfin; the discriminant DF of F/Q is defined to be the absolute norm N(dF/Q).

Then, S(dF/Q) coincides with the set of ramified places of F/Q.

Let ψ be the additive character of the adele ring AQ of Q with archimedean

component x → exp(2π

√

−1x), x ∈ R. Then, ψF = ψ ◦ trF/Q is a non-trivial

additive character of the adele ring A of F , which is decomposed to a product of

additive characters ψF,v of Fv over all places v of F . We note that ψF,v|pv −dv = 1,

ψF,v|pv

−dv−1

= 1.

The R-algebra F ⊗Q R is denoted by F∞, which is a direct product of Fv

∼

= R

over all v ∈ Σ∞.

2.2

Let G be the F -algebraic group GL(2). For any F -subgroup M of G and for

a place v of F , the Fv-points of M is denoted by Mv. The points of finite ade-

les and the F∞-points of M are denoted by Mfin and M∞, respectively. Then,

MA = Mfin M∞. The points of finite adeles Gfin of G is realized as a restricted

direct product of the local groups Gv = GL(2,Fv) with respect to the maxi-

mal compact subgroups Kv = GL(2, ov) over all v ∈ Σfin. The direct product

Kfin =

v∈Σfin

Kv is a maximal compact subgroup of Gfin. The Lie group G∞

is isomorphic to

v∈Σ∞

GL(2,Fv). For each v ∈ Σ∞, let Kv be the image of

O(2, R) by the isomorphism GL(2, R)

∼

= Gv. Then, K∞ =

v∈Σ∞

Kv is a maximal

compact subgroup of G∞ and K = KfinK∞ is a maximal compact subgroup of

GA = Gfin G∞.

Let Z be the center of G, which coincides with the scalar matrices in G. Let

H be the F -split torus of G consisting of all the diagonal matrices and N the F -

subgroup of G consisting of all the upper triangular unipotent matrices. Then B =

HN is a Borel subgroup of G consisting of all the upper triangular matrices in G.

For any place v of F , we have the Iwasawa decomposition Gv = BvKv = Hv Nv Kv.

2.3. Haar measures

For any place v, we take the self-dual Haar measure dxv of the additive group

Fv with respect to the duality defined by ψF,v. If v ∈ Σfin, then vol(ov) = qv

−dv /2

.

If v ∈ Σ∞, then dxv is the Lebesgue measure of Fv = R. Fix a multiplicative

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