Let 𝔸[n] be the group of n-torsion points of a commutative algebraic group 𝔸 defined over a number field F. For a prime 𝔭 of F, we let N𝔭(𝔸[n]) be the number of 𝔽𝔭-solutions of the system of polynomial equations defining 𝔸[n] when reduced modulo 𝔭. Here, 𝔽𝔭 is the residue field at 𝔭. Let πF(x) denote the number of primes 𝔭 of F such that N(𝔭) ≤ x. We then, for algebraic groups of dimension one, compute the kth moment limit Mk(𝔸/F,n) =limx→∞ 1 πF(x)∑N(𝔭)≤xN𝔭k(𝔸[n]) by appealing to the Chebotarev density theorem. We further interpret this limit as the number of orbits of the action of the absolute Galois group of F on k copies of 𝔸[n] by an application of Burnside’s Lemma. These concrete examples suggest a possible approach for determining the number of orbits of a group acting on k copies of a set.

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关于模素数的扭点矩及其应用

让𝔸[n]成为一组n-交换代数群的扭点𝔸在数字字段上定义F. 对于一个素数𝔭的F，我们让ñ𝔭(𝔸[n])是数量𝔽𝔭-定义多项式方程组的解𝔸[n]当减少模𝔭. 这里，𝔽𝔭是残差场𝔭. 让πF(X)表示素数的数量𝔭的F这样ñ(𝔭) ≤ X. 然后，对于一维的代数群，我们计算ķ矩极限 米ķ(𝔸/F,n) =林X→∞ 1 πF(X)∑ñ(𝔭)≤Xñ𝔭ķ(𝔸[n]) 通过诉诸 Chebotarev 密度定理。我们进一步将此限制解释为绝对伽罗瓦群的作用的轨道数F在ķ的副本𝔸[n]通过伯恩赛德引理的应用。这些具体的例子提出了一种可能的方法来确定一个群体的轨道数量ķ一套的副本。