The Tate conjecture for almost ordinary Abelian varieties over finite fields
The Tate conjecture for almost ordinary Abelian varieties over finite fields
复制标题
有限域上几乎普通阿贝尔簇的泰特猜想
DOI：
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发表时间：
1993
期刊：
影响因子：
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通讯作者：
Y. Zarhin
中科院分区：
文献类型：
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作者：
H. Lenstra;Y. Zarhin
设K是由q个元素组成的特征p的有限域。设K(o)表示K和G(K)的代数闭包：= G&\(K(o)/K}表示K的伽罗瓦群。设Υ为K上的光滑射影变，设Υ（α）：= Υ χ Κ（α）。设£是一个不同于p的有理素数。伽罗瓦群G(K)作用于（扭曲的）£进上同群F 2m (Y(a),Q i)(m)。[Tl] Täte推测在伽罗瓦作用下的子空间是由Y上的共维m代数环的上同调类张成的。这一猜想已在满足一定数值条件的费马超曲面[Sh， Y]、椭圆型Ä‘3曲面[AS]、有限高的A’3曲面[N， NO]和普通KZ曲面的幂[Z3]等情况下得到了证明。现在，设Y = X是一个p维的阿贝尔变换。在这种情况下，Täte [T2]证明了m = 1时的猜想。请注意，在Täte模块Vt(X)上，对X的äs偏对称多线性形式的输入上同调群的众所周知的解释允许我们识别伽罗瓦不变子空间H(X(a),Qe)(m)-^>与所有偏对称的2m线性形式E在Vt(X)上的空间，这样
Let K be a finite field of characteristic p consisting of q elements. Let K (o) denote the algebraic closure of K and G(K) := G&\(K(o)/K} the Galois group of K. Let Υ be a smooth projective variety over K and set Υ(α) := Υ χ Κ(α). Let £ be a rational prime different from p. The Galois group G(K) acts on the (twisted) £-adic etale cohomology groups F 2 m (Y(a) ,Q i )(m). In [Tl] Täte conjectured that the subspace fixed under the Galois action is spanned by the cohomology classes of co-dimension m algebraic cycles on Y. This conjecture has been proved in certain cases, e.g., Fermat hypersurfaces satisfying certain numerical conditions [Sh, Y], elliptic Ä'3 surfaces [AS], A'3 surfaces of finite height [N, NO] and powers of ordinary KZ surfaces [Z3]. Now, let Y = X be a p-dimensional Abelian variety. In this case, Täte [T2] has proved his conjecture for m = 1. Notice that the well known Interpretation of ί-adic etale cohomology groups of X äs skew-symmetric multilinear forms on the Täte module Vt(X) allows us to identify the Galois invariant subspace H(X(a),Qe)(m)-^> with the space of all skewsymmetric 2m-linear forms E on Vt(X) such that