Magister thesis

khζ1, . . . , ζnik

Sλ
Mµ

R
k
k
k
Q
k k
An
¯
k=¯
kn
V(f1, . . . , fm)def ={(x1, . . . , xn)∈¯ kn|fi(x1, . . . , xn) = 0, i = 1, . . . m}
f1, . . . , fm∈k[x1, . . . , xn]
kPn
¯
k
V+(g1, . . . , gs)def ={(x0:x1:. . . :xn)∈Pn(¯ k)|gi(x0, x1, . . . , xn)=0, i = 1, . . . s}
g1, . . . , gs∈k[x0, x1, . . . , xn]
X
k K/k X(K)
An
KPn KK
X(k)X X(k)
X X(k)
f:X→Y
y∈Y Xy
¯
k f y
X y ∈Y(K)K/k Xy
K
k X X(¯ k)
f(x) = 0
f(x)2= 0

An
¯
kPn
¯
k
X=SYiYiX Yi*Yji6=j Yi
Xdim X X
n U0(U1(···(UnX
X f1, . . . , fm∈
k[x1, . . . , xn]f1, . . . , fm∈k[x0, x1, . . . , xn]
k P ∈X
rang k(dfi/∂xj)(P)k=n−dim X,
k(dfi/∂xj)(P)k¯ k
X P ∈X
k f ∈k[x0, x1, . . . , xn]
d= deg f≤n X =V+(f)
X
#X(k)≡1 mod p,
p k
N= #{(x0, x1, . . . , xn)∈kn+1, f (x0, x1, . . . , xn) = 0}.
#X(k) = N−1
q−1, q k N ≡0 mod p
F= 1 −fq−1f(x) = 0 F(x) = 0 f(x)6= 0 F(x) = 1
x= (x0, x1, . . . , xn)∈kn+1 F(x0, . . . , xn) = P
i
cixαi,0
0. . . xαi,n
n
N≡X
x∈kn+1
F(x)≡X
iX
(x0,x1,...,xn)∈kn+1
cixαi,0
0. . . xαi,n
nmod p.
N≡0 mod p
X
(x0,x1,...,xn)∈kn+1
xαi,0
0. . . xαi,n
n≡0 mod pmin
jαi,j < q −1,
αi,0+···+αi,n ≤d(q−1) <(n+1)(q−1) minjαi,j < q−1
k1
Pn
¯
kk
n