Herbrand invariant (reviewed)

A Herbrand invariant $I$ is the collection of ramification data associated to a finite separable extension of $p$-adic fields. Let $L/K$ be an extension of $p$-adic fields with residue characteristic $p$. Let $f$ be the residue field degree and $\epsilon p^w$ the ramification index of $L/K$, with $p\nmid\epsilon$. The Herbrand invariant of $L/K$ can be expressed in terms of the Swan slopes of $L/K$ as $I=[s_1,\dots,s_w]_{\epsilon}^f$; in terms of the rams of $L/K$ as $I=(r_1,\dots,r_w)_{\epsilon}^f$; or in terms of the means of $L/K$ as $I=\langle m_1,\dots,m_w\rangle_{\epsilon}^f$.

As a convention, we put $s_0=r_0=m_0=0$. The three representations of the Herbrand invariant of $L/K$ are related by

• $\displaystyle s_k=\frac{p m_k-m_{k-1}}{p-1}$,
• $\displaystyle m_k=\sum_{j=1}^k\frac{p-1}{p^{k+1-j}}s_j$,
• $\displaystyle r_k= \epsilon p^k \frac{m_k- m_{k-1}}{p-1}$,
• $\displaystyle m_k=\sum_{j=1}^k \frac{p-1}{\epsilon p^j}r_j$

for $1\le k\le w$.

Alternatively, one can start with a Herbrand invariant $I$ and then identify fields $K$ such that there exist extensions $L/K$ whose Herbrand invariant is $I$. Let $I=(r_1,\dots,r_w)_{\epsilon}^f$, with $r_1,\dots,r_w$ a weakly increasing sequence of positive rationals and $\epsilon,f\in\N$. Let $b_1<\dots