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a a b
a
b
L
K n :K→N
K(x) = 1 x6= 0
(0) = 0 Kn(x, y)7→
w (y−x) w (x) = Pn
i=1 (xi)x
f:C→Kn
C Kn
∀x∈C, wf(x)= w (x).

(ei)1≤i≤nKn
Kn
Kn(π, λ1, . . . , λn)
S(n)n(K×)nei7→ λieπ(i)
U λi
U K×
Kn
Kn
Kn
AnA
U K×G K×/U
x∈Knx U
CU(x): G→Nr∈G
cr(x)x rU
f:C→Kn
∀x∈C, CU(x) = CU(f(x)),
U U
KnU
U
U
K:K→C
w
Knw (x) = Pn
i=1 (xi) (x, y)7→
w (y−x)Kn
f:C→Kn

∀x∈C, w (x)=w f(x).
K×
U( ) = {λ∈K×| ∀x∈K, (λx) = (x)}.
Kn
U Kn
G K×/U U =U( )
f∗g f g G
t7→ f∗g(t) = X
xy=t
f(x)g(y) = X
x∈G
f(x)f(tx−1).
g:G→Cg
CGCGC
x∈Knhx:G→Cr
cr(x)λ∈K×λx
hx
w (λx) = X
r∈G
cr(λx) (r) = X
r∈G
cr/λ(x) (r) = hx∗(λ)
K U
G K×/U
CG
f x f
hx∗=hf(x)∗.
hx=hf(x)CU(x) = CU(f(x))
∆ =
. . . (rs−1). . .
r,s∈G
=Y
χ∈b Gb(χ)

b(χ) = Ps∈G(s)χ(s)χ
KF``
`
(t) = 
t, 0≤t≤`/2;
`−t, `/2< t < `.
U{−1,+1}
GF`×/{±1}n:= (`−1)/2
06=t∈F`
t U =2
`= 7
`∆6= 0
b(1) = Ps∈G(s) =
Pn
k=1 k=1
2(n+ 1)n.
b(χ)6= 0 χ
`
`= 1 + 2p ` = 1 + 4p p
`= 1 + 6p
L
∆
χF`χ(−1) =
1`−1
2
G=F`×/{−1,+1}
f:G→Cχ

b
f(χ) = X
x∈G
f(x)χ(x) = X
k< `
2
f(k)χ(k).
τtt G [ f◦τ(χ) =
¯χ(t)b f(χ) 0 ≤x < `/2G
x < `/4 2x < `
2(2x) = 2 (x)`/4< x < `/2
`/2<2x < ` (2x) = `−2 (x)(2x)−2 (x)
(2x) + 2 (x)−`G
(2x)2−4 (x)2=(2x)−2 (x)`.
(¯χ(2) −4)b(χ) = ( ¯χ(2) −2)b(χ)`.
r2r≡ ±1 mod `
(2r+ 1)l−1
2r∆ = ``−1 2∆
b(χ) = 0
B1(χ)B2(χ)
B1(χ) = 1
`
`
X
k=1
kχ(k), B2(χ) = 1
`
`
X
k=1
(k2−lk)χ(k).
χ
2b 1(χ) = 2 X
k< `
2
χ(k) =
`
X
k=1
χ(k) = 0.
B1(χ) = X
k< `
2
kχ(k) + X
k< `
2
(`−k)χ(k) = `X
k< `
2
χ(k)=2b 1(χ)=0
b(χ)=0 b(χ)=0