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Is there a version of Herbrand-Ribet or Mazur-Wiles (relating divisibility of class groups to special values of L-functions) for functions fields (over finite fields)?

Probably the proofs would have to be very different since we don't have a nice tool like modular forms to construct representations and extensions with nice properties out of (or do we?).

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(I was hoping somebody else would answer this, because function fields are not really my area and I hoped I would learn something from the answer; but nobody seems to be biting, so...)

Iwasawa theory over function fields definitely exists, and in many ways it's easier than number-field Iwasawa theory -- there are more nice tools available, such as the Grothendieck--Lefschetz trace formula, which aren't there in the number field setting.

For instance, here is a paper of Goss and Sinnott from the 1980s which (among many other results) proves an analogue of Herbrand--Ribet for the class groups of function field extensions arising from Drinfeld modules.

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  • $\begingroup$ Your link appears to behind warwick login wall. Is the paper "Class-groups of Functions Fields"? $\endgroup$
    – Asvin
    Commented Mar 10, 2018 at 17:19
  • $\begingroup$ @Asvin Yes, that's it. Link now fixed. $\endgroup$
    – David Loeffler
    Commented Mar 10, 2018 at 19:19

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