Averages of groups involving \(p^{\ell}\)-rank and combinatorial identities
A finite abelian group \(G\) is called a group of type \(S\) if it is endowed with a non-degenerate, bilinear, alternating pairing \(\beta: G \times G \rightarrow \mathbb Q/ \mathbb Z\). If \(G\) is a group of type \(S\), \(|\Aut^S(G)|\), the number of automorphisms of \(G\) that respect the pairing \(\beta\) does not depend on the choice of the pairing \(\beta\). Note that Tate-Shafarevich groups are groups of type \(S\). Let \(h\) be a complex-valued function defined on isomorphism classes of finite abelian groups \(H\) of order \(n\). Define \[ \zeta(h,s)=\sum_{n\geq 1}\frac{1}{n^s}\sum_{H(n)}\frac{h(H)}{|\Aut(H)|} \] and \[ M_u(h)=\lim_{s\rightarrow u}\frac{\zeta(h,s)}{\zeta(1,s)}. \] The value \(M_u(h)\) is called the \(u\)-\textit{average} of the function \(h\). By taking \(h\) to be the function \(T_l(H)\), which adjoins to a group \(H\), \(|H[p^l]|=|\{x\in H:\;p^lh=0\}|\), the number of the \(p^l\)-torsion elements, the author proves that the \(u\)-average of \(T_l\) is \[ M_u(T_l)=1+\frac{1}{p^u}+\dots +\frac{1}{p^{ul}} \] for a finite abelian group and \[ M_u(T_l)=1+\frac{1}{p^{2u-1}}+\dots +\frac{1}{p^{(2u-1)l}} \] for a group of type \(S\). The author also finds probabilities for both finite abelian groups and groups of type \(S\) to have given \(p^l\)-rank (which is the number of cyclic components of the group whose order is divisible by \(p^l\)).
scientific article; zbMATH DE number 5853445
scientific article