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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