Asymptotic behavior of modular representations over abelian $p$-groups

In this paper, we prove some results on the asymptotic behavior arising in modular representation theory over abelian $p$-groups. First, we embed the representation ring of a cyclic $p$-group into a real algebra of functions. Second, we calculate the asymptotic order of the dimension of the core of $n$-th tensor power of a direct sum of syzygies and cosyzygies of the trivial module, which is of the form $C{\gamma }^n{n}^{\alpha }$. This result leads to a negative answer to a question by Benson and Symonds, that is, the dimension of the core of $M^{\otimes n}$ for certain $\mathrm{\Omega }$-algebraic module $M$ is not eventually recursive. Third, we give a systematic way of computing the core series of $\mathrm{\Omega }$-algebraic modules. Finally, we show the existence of a transcendental core series, which comes from iterated syzygy modules of the trivial representation.