Let $F$ be a finite field extension of the $p$-adic numbers $\mathbb{Q}\_p$, whose residue field has $q$ elements. Let $\mathfrak{p}$ be the prime ideal of $F$. Given a finite field extension $K/F$, write $\operatorname{Disc}\_\mathfrak{p}(K)$ for $q^{v\_\mathfrak{p}(\operatorname{Disc}(K/F))}$. Then **Serre's mass formula** states that for all $n\geq 1$, we have


$$
\sum\_{[K:F] = n \text{ totally ramified}} \frac{1}{\#\operatorname{Aut}(K/F)}\cdot\frac{1}{\operatorname{\operatorname{Disc}\_{\mathfrak{p}}(K)}} = \frac{1}{q^{n-1}}.
$$


This result first appeared in Serre's 1978 paper **Une “ formule de masse ” pour les extensions totalement ramifiees de degre donne d’un corps local**. As far as I can tell, this paper is only available in the book **Oeuvres/Collected Papers. 3 volumes.** Given that I am unable to obtain this book, I am looking for a reference on the internet that contains either Serre's original paper or an exposition of his proof (ideally both proofs from the paper).


I would also (delightedly) accept an answer that directly explains the proof (i.e. no reference); such an answer would in my opinion be a great addition to the learning resource that is MO, seeing as the proof is so hard to find on the internet!



Looking for proof of Serre's mass formula
The article is not only available in Serre's Collected Papers: it first appeared in a journal, after all. Here's a scan from the Comptes Rendus archives: <https://gallica.bnf.fr/ark:/12148/bpt6k6234149b/f323.item>.


