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!


1 Answer 1


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.

  • $\begingroup$ Can I ask how you actually found this? I'd like to be able to do it myself next time $\endgroup$ Aug 10 at 22:14
  • $\begingroup$ I searched for the title and year to find the journal and page numbers (which you had omitted) and then looked for the Comptes Rendus online archive for math papers from en.wikipedia.org/wiki/…. In the "Online Open Archives" section, after "Séries A et B, Sciences Mathématiques et Sciences Physiques (1975-1980)" is a link to papers in the relevant years. At that link I tried various search terms from the article title and author name, but I can't reconstruct exactly what I did. You need to do such a search yourself enough times. $\endgroup$
    – KConrad
    Aug 11 at 14:30

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