I'm looking for a good reference for learning about Hasse-invariants ($p$-ranks) for curves of arbitrary genus over a field characteristic $p$.

All the usual suspects I've searched (Milne's Étale cohomology, Mumford's red book, Hartshorne, Silverman) either omit the topic entirely, give a definition only for elliptic curves, or assume that the reader is already familiar with the Hasse-Witt matrix. I didn't see anything in the index of SGA 1 either, but perhaps somebody more familiar with SGA (or with a searchable copy) might be able to find something.

Any suggestions? Thanks.


1 Answer 1


Have you looked at Yuri Manin's paper The Hasse–Witt matrix of an algebraic curve?

I remember also looking for the paper of Pierre Cartier but I couldn't find it. It's listed as a reference in many places, so perhaps you'll find something there (if you find it, please let me know!).

  • $\begingroup$ Thanks. I did a mathscinet search for the Cartier paper but none of the titles that came up looked like they had what I'm looking for. I recall seeing the Manin paper in some citations, and just got my hands on a copy, but my Russian is unfortunately not up to the task :( $\endgroup$ Nov 18, 2013 at 20:17
  • $\begingroup$ Hi @Brett: There is an English translation of this paper in the volume Selected Papers of Yu I Manin (World Scientific Series in 20th Century Mathematics). $\endgroup$ Nov 19, 2013 at 18:55
  • $\begingroup$ Indeed there is, and I now have a copy in my hands! (Why didn't this show up on Google or mathscinet?) Thanks so much. $\endgroup$ Nov 19, 2013 at 19:15
  • $\begingroup$ Dear @Brett: the pleasure is all mine! It seems you'll be able to live a little longer without having to learn Russian. Regards, $\endgroup$ Nov 19, 2013 at 19:17

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