Title
On unit group of commutative algebraic groups
Author(s)
AMANO, KAZUO
Citation
[岐阜大学教養部研究報告] vol.[7] p.[157]-[161]
Issue Date
1971
Rights
Version
岐阜大学教養部 (Dep. of Math. , Fac. of Gen. Educe. , Gifu
Univ.)
URL
http://repository.lib.gifu-u.ac.jp/handle/123456789/47444
※この資料の著作権は、各資料の著者・学協会・出版社等に帰属します。
157
可換代 数群 の単数群 につ いて
尼
野
岐
阜
大
一
学
教
夫
養
部
( 1971年10月30日受理)
On-unit groull of commutativealgel】raic groups
K Azuo A MAr、
,0
Dep. oJ M(1th. , Facレ oJ Gen. Educ. , Giμ Un叱
( Received Oct. 30, 1971)
lntroduction
L et A; be a locan y compa6t field under a discrete valuation. A s is well known,
the proper ties of the multiplicative structures of A;, in particular those of the unit
group ol k, play important roles in the theory of local fields.
ln the present paper,
w e shall
show
some proper ties of
unit
group
fりr
commutative h near algebraic groups which generali ze unit theorm in ん. N amely,
in
S ection l ,
w e shall introduce similar
the valuation of ん. ln Section
mapping on l inear
2 , we sha11 consider
two
algebraic groups as
special
ser ies
of
matr ices w ith el ements in ko T he ser ies have same proper ties as ! ogar ithm and
exponential
in・ local
field乱
ln S ection 3 , w e shall give unit theorem for
any
ゝ
commutative l inear algebraic groups. A nalogous r esul ts in the case of local fields
are well known 〔 5 〕 .
(
T hrough this paper, w e shall use the following notations.
ん : the locally compact fie1(L under a discrete valuation ひ,
p : the va l uat ion r i ng ol
k,
p : the maximal ideal of o, and let (p) = y ,
G : the commutative algebraic subgroup of general line乱r group G£ 0 , 幻 ,
び : the subgroup of G consisting of the elementS χこ ( xJ) such that xij eo
( 1≦f, j ≦O and ひ(det X) = j . The group び will be called the unit group
of G,
び(″
) : the subgroup of ひ consisting of the elements χ = ( 亀j )
o mod 丿 口 ≠ j ) and 恥j ≡ l mod げ 口 = j )
such that 心j ≡
f or 17= 1 , 2 , 3 , ‥‥‥
‥
158
尼
野
一
夫
§1 . Generalized valuation
rl e a
a map
m ap
g :: M
M (( n,
n , k)
k ) →
→
W e define
g
Z by
b y putting
p u tt i n g
g (( X
χ )) =
= m紬
m in
Z
g
O (( X ij
0
)
は or
a
m a t r ix
X= ( 勾 )
where 訂0 , 幻 is the full matrices algebri , and Z is the set of rational
integer s.
T hen, the map g satisfies the follow ing proper ties
P Roros171.
(1)
1
( i ) g( X十y) ≧min隔 ( X) , 副: y) ¦
( ii) g ( X y) ≧g ( X) 十g ( y)
( iii) g ( 入X) = g( X) 十べ 幻 ,
釦r m(LtTice8 χ and Y 伍 M ( 孔
八 ) , a71d μ ・r a e& meM λ 伍 k、
P nooll
( i) ,
( ii) ;
F or gy Jn matr ices X 二 ( 亀j )
and y 二 ( 馬j ) , w e have,
副:X十y) = 瀬 川べ勾 十如 珪
≧呻川min卜(あバ), べ仰 ) 目
≧min卜 ( X) , 研 y丿
g (Xy) = min卜(ぷ勾 力川
≧m回 向 n卜( 恥バ) , 削:妙 ) 口
≧UJ(。X ) 十g( y)
( iii) ; For an element 入 in だ, we have
ヽg( 入X) = 瀬芦卜( λ亀八) }
= 町阻隔( λ) 十ベ亀j ) l
= 衣 λ) 十g ( X) .
C oRoLLARY 1 . g ( X つ≧ rg ( X ) ,
yor r = 1 , 2 , … …
P RoPosITIoN 2 . The gToup U (″
) con8 1sts oJ 硫 e ete・ ent χこ ( 亀j )
OJ U 8UCh th(l t
g( χ一刀≧ 1ノ, 切heTe 1 18 伍e identi切 m(1tT謐.
P aooli
l t is tr ivial by vir tue of the definitions of the group び(″
) alld =the °ap
麗ノ.
` I
-
§2 . Logarithm and Eχponential
ln this section, w e shall consider the follow ing ser ies of matriχ χ= ( xi) in the
group G.
( L) : χ- χy2十χy3- ‥‥‥
‥・…・十( - 1) 一次ソr十‥‥‥‥
‥,
( E) : 7十X 十X y2! 十‥‥‥‥‥‥十X ソ7-! 十‥‥‥‥
‥,
F or nilpotent matr iχ,
P ROPOSITION 3 ,
(1) Cf. 〔 6〕
it is tr ivial that the ser ies ( L ) and ( E ) converge.
Th,e se石 es ( L )
conlJeTge8,
§3 .
2
汀 伍 e ma計 謐 χ sat18μ es u ( χ) > o.
可換代数群の単数群 につ卜て
P aooE.
L et 戸 ≦ r < 戸 4¯l
T hen w e have
159
for an integer (z, and J et X ゜ ( j ; )) .
g ( X ソ 杓 = g ( X つ一 球: 絢
≧rg ( X) - e叫
≧アg ( χ) - e tog声
lf g ( χ) > o, χソ r →
P RoPosITIoN 仁
∽
as r →
T he se石 es ( E )
∽ and therefore the series ( L ) converges.
conlJeTges, 汀 伍 e ・ atT謐 χ 8(1t18μ es w ( χ) > eyp - j ,
P RooF・ L et r 二 co十 c炉十 … … 十 (司八 and y ; ≦ r< だ け t w here 醜 and ci are rational
integer s,
o≦ (≒≦ p- j .
T hen w e have 心 二 r- s( r) / p- J,
w here 収 杓 = co
+ … … 十 cs
T herefore we have
ぶしX ZtX) ≧~
( X) - eレ ーれ: r) に p- j
= r卜 ( X) - eZp一万+ 9 ( 7) /p- 1.
Since s( O is・ positive, we see that X ソ 7`! →
゛
as r →
゛ ,
if g ( X ) > eZp - j
A nd hence the series ( E ) converges.
十
D EElsITloぷ) W e shall denote the ser ies ( L )
Now , we suppo叩
and ( E ) byL 昭 ( 7 十 X ) and E χp ( X ) .
that a matr iχ χ satisfies 籾( χ) > eZT) - j . T hen for r> J,
we
have
ぶしX ZrX) 一司 X) ≧ O - j ) g ( X ) - reZp- j 十卵 0 ) ZT)- 1
) e卜( 約- j随一7≧o,
and
g ( X ソ約 一司 X) ≧ O - j ) 司 X) - eら
こ
>
e {
(
T -
1 )
-
(
α こ
川
か
-
7 ≧
o .
T herefore we see that g ( E Xp X - 7) ≧ m and g ( L 昭 ( 7十 X ) ) ≧ m, when
g ( X ) ≧m
for positive integers m> eZp - j .
T HEoREM, 1 .
が g ( X ) > eZp - 1,
then lJXe haoe
E即 ( LOg( 7十X) ) こ 7十X and LOg( E聊) X ) = X,
P RooF・
T he formal identities ar e known, and all the ser ies converges by above
asser tion.
§3 . Some properties of uni.t group
F or any matr ices X 二 =( xJ) and y = ( !Jij) ,
w e can define a ∧metr iχ by putting
d( χ, Y) = Nij X-Y), where ≒ is the absolutenorm. Then themetric induces a topology in 趾 ( 馬 幻
in the usual manner (で T herefore the algebraic group G has the
induced topology such that the group び(″
) are the neighborhoods oI I .
ぐ? ぐ3
jj
C£ 〔 1〕 , C石叩. χ
l . §2. n eorem l and 3
Cf. 〔 6〕 √§3 and §4.
3
160
尼
P RoPosITIoN 5 .
P RooF・
野
一
夫
Tke g To叩 8 U c
lnd U (″
) (1Te compαd
sJ , groizp oy Gご
び(o has finite index in び and び= proj - lim UZU (o
By vir tue of Prop・ 5 ・ the lllap ひ(j) X Z 6 ( X , α) →
X
£ U (1) i s extended to the .
map び(1)X Z 。 9( X , α) →
and p- adic integer s,
W e set L Og U (゛ )= 雫(叫 T hen,
rphic to び(゛) since theorem j .
if m> e/ 7) - j , 雫(゛) is Z r isomo-
W e dence by s the rank of Z 。- modリle¥雫 師) ・ ◇
PaorosITlo1 6 . F or a ・ at石χ Z e U い, 切e h(1ue L og Z = 0,
廿 (1nd o峠 丿 廿 Z t = j;
u) heTe O is ze7ヽo ・ a tTiχ and m 18 an 振 tegeT.
P R00F・
l f Z is an unipotent matr ix,
w e have Z = E Xp ( L Og Z ) = E Xp O = jl
T herefore we suppose that Z is (1 sem i- simple matr iχ. l f Z ゛ = 7;
L Og Z = L og Z ゛= L og lこ 0 , and hence L Og Z こ 0 .
Conver sely,
we have m
for lx> eZ1) - j
and
sufficiently large integer m, w e have Z ゛ e U (″
) and hence
Z゛= EXp( m`L og Z ) = EXp O= I。
T HEoHEu.
2.
ヽ7 yxe7 e exjsX 岳 e eleme71Xs /11, j4 2 , … … , A o SUCh 伍 (1t
A = Z j 71/1;
……j ; ソoΓα71!/ ノ1 6 び( )`, 切heTe Log Z = 0 α71d α。 ( 1≦ j≦め aTe the d e・ ent8 0J
Z p●
P RooF・ 雫 (j) cnntains 畢 伽) a71d j (1) has finite index.
as p(叫 L et j 1, j 2,
are £ og ノ11, £ og ノ12,
… … , ノ1.
T her efor e 叩 (J) has some rank s
be the elements of び(1) such thi t the basis of
串 (1)
‥‥‥‥
‥, L Og A i. T hen our theorem fol low s from above me-
ntion and prop. 6 .
.
L et £ be the raりk of p over Z 。 .
T hen w e have the fol low ing.
CoRoLLARY 2 . IJ G is an u戒 potent g7ヽ
oup, 8 = X dim G
P必oE.
T he L ie algebra(4) of G is L og G, and L Og maps び onto ( ド (li°勺
for
1ノ> eZp- 7. The ideal t, has the rank £ and hence we have` our corollary.
COROLLARY 3 .
P RooF・
1L
f G iS k- Split tOTi,
S ince G is A;- sPlit,
G≧
S=
X d im
ド
Å
;
米
0
(4) Cf. 〔4 〕 , Ch l). V. §3 , 朽・op. 14
(5) Cf. 〔 5〕 ,J I . §15. e)
4
G (?
]
where A;米is the muh iph cative
可換代数群の単数群につ いて
16 1
group of A;. T herefore our corollary is the unit theorem in local fields
T HEoREM 3 、 F 07` an!j positi lXe 振 tegeT m,
th t, び 切(しA ) > 皿 j = A;
PaooE.
theTe ex18ts (l posititJe k tegeT N such
Jor a71 eleme71t Ao6 び已
L et f ≦m< p 1 :for an integer r. I L we take N > eZp一j + er. g ( £ 昭 j )
g( L og y1) > X
Since g (ム
Z/ m L og y1) = g ( £ og /1) 一祠:m) > N - er> eyp- 1,
£ 卯 ( 1Zm L og A) = 爪
is convergenc9.
Open question.
(i) Since any commutative algebraic group G have a decomposition G。χG。,
where Go and Go are the semisimple group and the unipotent group of G. W e
ask whether the rank s can be required explicitly in the group Gj .
(ii) Theorem 3 1nduces analogy to the power residue in algebraicnumber fields.
W hat means the power residue in commutative algebraic groups ?
References
〔 1 〕 E . Artin and J. Tate, Class field theory. Harvard ( 1961)
〔 2 〕 A. Borel, Groupes lin6aires algebriques. Ann. of Math. 64 ( 1956) 20- 82
〔 3 〕 A. Bore1, Linear algebraic groups。 .Benjamin ( 1969)
〔 4 〕 C. Chevalley, Th60rie des groupe,s d6 Lie. Hermann ( 1968)
〔 5 〕 H. Hasse, Zahlentheorie. Berlin, Akademie-Verlag ( 1949)
〔 6 〕 R. Hooke, Linear p-adic groups and their Lie algebra. Ann. of Math. 43 ( 1942)
64 1- 655
〔 7 〕 T. 0n0. 0n some arithmetic properties of linear algebraic groups.
Ann.
of M ath.
70 ( 1959) 26 6 - 290
。
¥
5

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