Families of extensions of local fields (reviewed)

Let $K$ be a $p$-adic field whose residue field has cardinality $q$ and let $I=[s_1,\dots,s_w]_{\epsilon}^f$ be a Herbrand invariant. The family $I/K$ of extensions associated to this data is the set of all $K$-isomorphism classes of extensions $L/K$ of residue field degree $f$ and ramification index $e=\epsilon p^w$ whose Swan slopes are $s_1,\dots,s_w$ (see Monge [MR:3256847, arXiv:1109.4617, 10.1142/S1793042114500511]). If $K=\Q_p$ we say that $I/\Q_p$ is an absolute family; otherwise say $I/K$ is a relative family.

Associated to the family $I/K$ is a generic polynomial $f_I(x)$. Let $K_f$ be the unramified extension of $K$ of degree $f$. Define a function $\sigma(i,j)=ej+i$ and let $\mathcal{R}$ be a set of coset representatives for $\lambda = \mathcal{O}_{K_f}/\mathcal{M}_{K_f}$ which includes 0 and 1. The generic polynomial for $I/K$ has the form \[ f_I(x)=d_0\pi_K+\sum_{(i,j)\in\mathcal{A}}a_{\sigma(i,j)}\pi_K^ix^j+ \sum_{(i,j)\in\mathcal{B}}b_{\sigma(i,j)}\pi_K^ix^j+ \sum_{(i,j)\in\mathcal{C}}c_{\sigma(i,j)}\pi_K^ix^j+x^e, \] where $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ are finite sets. More precisely, the elements of $\mathcal{C}$ correspond to a subset of the set $\{s_1,\dots,s_w\}$ of Swan slopes, and the elements of $\mathcal{A}$ correspond to certain indices of inseparability. The Eisenstein diagram gives a visual description of the process for producing the generic polynomial.

One obtains Eisenstein polynomials over $\mathcal{O}_{K_f}$ known as semicanonical polynomials by specializing the parameters in $f_I(x)$. More precisely, we may choose $a_{\sigma(i,j)}\in\mathcal{R}\smallsetminus\{0\}$ and $b_{\sigma(i,j)}\in\mathcal{R}$ arbitrarily. For each $(i,j)\in\mathcal{C}$, $c_{\sigma(i,j)}$ is chosen from a subset $S_{(i,j)}$ of $\mathcal{R}$ which corresponds to coset representatives for a certain quotient of $\lambda$. We choose $d_0$ from a subset of $\mathcal{R}\smallsetminus\{0\}$ corresponding to coset representatives for $\lambda^\times / (\lambda^\times)^\epsilon$; in particular if $\gcd(\epsilon, q^f-1) = 1$ then we set $d_0=1$.

Each element of $I/K$ is represented by an extension $L$ of $K_f$ which is generated by a root of one of these semicanonical polynomials. The ambiguity of the family is an upper bound on the number of semicanonical polynomials corresponding to each isomorphism class of extensions in $I/K$.