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Some time ago I started a stub characteristic variety to record few references, mainly in D-module context. Regarding that the related notion of a characteristic ideal also appears in the treatment of Iwasawa polynomial and Alexander polynomial which Urs wants to understand from the point of view of connections and differential refinements of cohomology, maybe we should do some effort to make some pages which will connect various notions of characteristic ideals and their loci across various subjects. I just recorded
at characteristic ideal for the version in the context of Iwasawa theory.
Of course, I am not quite sure about the connection with the PDE notion. Cauchy had introduced the concept of characteristics to partial differential equations. Maybe we could read through sec. 3.6 in the Michael Bachtold’s thesis, Fold-type Solution Singularities and Characteristic Varieties of Nonlinear PDEs, pdf or he could help us as a member of $n$Forum.
Unfortunately I don’t know anything about Iwasawa theory. From a glimpse at the first paragraph of the articles you linked I don’t see a relation to the characteristic variety in D-modules and PDEs.
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