有限阿贝尔群中的$p^\ell$ -扭点与组合恒等式
$p^\ell$-Torsion Points In Finite Abelian Groups And Combinatorial Identities
摘要：
本文的主要目的是计算某类有限阿贝尔群中$p^\ell$ -扭元个数的全部矩。这些矩中的平均值是为类组的Cohen-Lenstra估计和它们对Tate-Shafarevich组的适应而定义的。特别地，我们证明了Tate-Shafarevich群的启发式模型与Poonen和Rains最近关于椭圆曲线的$p$ -Selmer群阶矩的猜想是相容的.为了我们的目的，我们被引导到定义某些多项式索引的整数分区，并研究它们的组合方式。此外，从我们的概率模型中，我们得到组合身份，其中一些似乎是新的，其他有关的对称函数的理论。因此，在某种意义上，我们的方法为这些恒等式提供了某种自然的代数背景。
英文摘要：
The main aim of this article is to compute all the moments of the number of $p^\ell$-torsion elements in some type of nite abelian groups. The averages involved in these moments are those de ned for the Cohen-Lenstra heuristics for class groups and their adaptation for Tate-Shafarevich groups. In particular, we prove that the heuristic model for Tate-Shafarevich groups is compatible with the recent conjecture of Poonen and Rains about the moments of the orders of $p$-Selmer groups of elliptic curves. For our purpose, we are led to de ne certain polynomials indexed by integer partitions and to study them in a combinatorial way. Moreover, from our probabilistic model, we derive combinatorial identities, some of which appearing to be new, the others being related to the theory of symmetric functions. In some sense, our method therefore gives for these identities a somehow natural algebraic context.
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