Torsion of abelian varieties, Weil classes and cyclotomic extensions
Torsion of abelian varieties, Weil classes and cyclotomic extensions
复制标题
阿贝尔簇的扭转、Weil 类和分圆扩张
DOI：
10.1017/s0305004198003235
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发表时间：
1997
影响因子：
0.8
通讯作者：
Y. Zarhin
中科院分区：
文献类型：
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作者：
Y. Zarhin
设K <$C是Q上生成的域，K（a）<$C是K的代数闭包，G（K）=Gal（K（a）/K）是其Galois群.对于每个正整数m，我们将K（a）的子域记为K（μm），该子域是通过将所有m次单位根邻接到K而获得的。对于每个素数[lscr ]，我们将K（[lscr ]）写为K（a）的子域，通过将所有[lscr ]-单位幂根邻接到K而获得。我们把K（a）的子域记为K（c），它是通过把K（a）中所有单位根都与K相连而得到的。设K（ab）<$K（a）是K的极大阿贝尔扩张.域K（ab）包含K（c）;如果K=Q，则Q（ab）=Q（c）（克罗内克-韦伯定理）。我们写χ[lscr ][ratio ]G（K）→Z*[lscr ]为定义伽罗瓦作用于所有[lscr ]-幂单位根的分圆特征标。对于定义单位根上伽罗瓦作用的分圆特征标，我们记为χ[lscr]= χ[lscr ] mod [lscr ][ratio ]G（K）→Z*[lscr ]→（Z/[lscr ]Z）*。字符χ[lscr ]将Gal（K（[lscr ]）/K）标识为Z*[lscr ]=Gal（Q（[lscr ]）/Q）的子群。设μ（Z[lscr ]）是Z*[lscr ]中所有单位根的有限循环群μ（Z[lscr ]）.如果[lscr ]是奇数，则其阶等于[lscr ]−1;如果[lscr ]=2，则其阶等于2。令Q（[lscr ]）'是Q（[lscr]）中μ（Z[lscr]）-不变量的子域。显然，Gal（Q（[lscr ]）/Q（[lscr ]）′）=μ（Z[lscr ]）和Gal（Q（[lscr ]）′/Q）= Z*[lscr ]/μ（Z[lscr ]）同构于Z[lscr ]。
Let K⊂C be a field finitely generated over Q, K(a)⊂C the algebraic closure of K and G(K)=Gal (K(a)/K) its Galois group. For each positive integer m we write K(μm) for the subfield of K(a) obtained by adjoining to K all mth roots of unity. For each prime [lscr ] we write K([lscr ]) for the subfield of K(a) obtained by adjoining to K all [lscr ]-power roots of unity. We write K(c) for the subfield of K(a) obtained by adjoining to K all roots of unity in K(a). Let K(ab)⊂K(a) be the maximal abelian extension of K. The field K(ab) contains K(c); if K=Q then Q(ab)=Q(c) (the Kronecker-Weber theorem). We write χ[lscr ][ratio ]G(K)→Z*[lscr ] for the cyclotomic character defining the Galois action on all [lscr ]-power roots of unity. We write χ[lscr ]= χ[lscr ] mod [lscr ][ratio ]G(K) →Z*[lscr ]→(Z/[lscr ]Z)* for the cyclotomic character defining the Galois action on the [lscr ]th roots of unity. The character χ[lscr ] identifies Gal (K([lscr ])/K) with a subgroup of Z*[lscr ]=Gal (Q([lscr ])/Q). Let μ(Z[lscr ]) be the finite cyclic group μ(Z[lscr ]) of all roots of unity in Z*[lscr ]. Its order is equal to [lscr ]−1 if [lscr ] is odd and 2 if [lscr ]=2. Let Q([lscr ])′ be the subfield of μ(Z[lscr ])-invariants in Q([lscr ]). Clearly, Gal (Q([lscr ])/Q([lscr ])′)=μ(Z[lscr ]) and Gal (Q([lscr ])′/Q)= Z*[lscr ]/μ(Z[lscr ]) is isomorphic to Z[lscr ].