Title: Finding the optimal fraction for wildly ramified finite abelian $p$-extensions in characteristic $p$
Abstract: Let $p$ be a fixed prime, $G$ a fixed finite abelian group of order $p^n$ and let the type of $G$ be denoted by the corresponding partition $\lambda$ of $n$. In dependence of $\lambda$, we define the map $f_{\lambda}: D(\lambda) \rightarrow \mathbb{Q}, \ d = (m_1, \ldots, m_n) \mapsto \frac{1 + \sum\limits_{i=1}^{n} \left( m_i - \lfloor \frac{m_i}{p} \rfloor \right)}{\sum\limits_{i=1}^{n} (p^i - p^{i-1})(m_i+1)}$ where the domain $D(\lambda) \subseteq \mathbb{N}^n$ can be characterised by finitely many inequalities of the entries $m_i$ depending on $\lambda$. \\We are interested in the following questions concerning the map $f_{\lambda}$: Does the function $f_{\lambda}$ have a maximum $M(\lambda)$? What is the maximum value? For which and for how many $d \in D(\lambda)$ is it attained?
In this talk, we discuss the important special cases of elementary-abelian groups $\lambda = (1, \ldots, 1)$ and cyclic groups $\lambda = (n)$ and provide a full solution for the previous questions. Based on these two cases, we outline the key ideas for solving the problem for any $\lambda$.