# What are non-abelian $L$-functions?

I have heard people discussing the utility of $L$-functions, claiming that since they are essentially cohomological entities, they are "abelian" and therefore lack force.

From looking around on the web, I see that this idea has a base of believers, and that there is some notion of non-abelian $L$-functions.

### Question

What is the definition of non-abelian $L$-functions? Does it have to do with replacing cohomology with homotopy in some way? How does it relate to the original definition of $L$-function (in particular, what is the analogue of the characteristic polynomial?)? What is the context in which it arises?

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Allow me a bit of snark: "abelian" L-functions may lack force, but "non-abelian" L-functions may lack a definition. More seriously, I personally have never heard to such a thing, aside from e.g. arxiv.org/abs/math/0412008, which does not seem to be what you want. But I imagine that some people do speculate on what such a thing could be (or, rather, what sort of homotopical invariants could replace L-functions). Since these speculations haven't filtered down to us hoi polloi, I am skeptical they have found practical use. – B R Oct 23 '11 at 22:46
Since I have "The collected papers of Emil Artin" to hand, allow me to quote the first paragraph of "Über eine neue Art von L-Reihen"; "Für die Untersuchung beliebiger, auch nicht Abelscher algebraischer Zahlkörper benötigt man eine Reihe neuer analytischer Funktionen, die mit FROBENIUSschen Gruppencharakteren gebildet sind und im Abelschen Falle mit den gewöhnlichen L-Reihen zusammenfallen. Ihrer Untersuchung sind die folgenden Zeilen gewidmet". Sehr schön! – Barinder Banwait Oct 23 '11 at 22:59
One kind of non-abelian L-function that promises some applications comes up in non-abelian Iwasawa theory, where one can find some speculation and some computation. You could check, for example, the paper of Coates, Fukaya, Kato, Sujatha, Venjakob, IHES publications, 101. – Minhyong Kim Oct 23 '11 at 23:40
Link to the Coates, Fukaya, Kato, Sujatha, Venjakob paper: archive.numdam.org/ARCHIVE/PMIHES/PMIHES_2005__101_/… – B R Oct 24 '11 at 0:53

"I have heard people discussing the utility of L-functions, claiming that since they are essentially cohomological entities, they are "abelian" and therefore lack force." This sounds like twaddle to me. Artin L-series are non-abelian; the L-series of representations of the Weil group are non-abelian; automorphic L-series are nonabelian (unless they are on GL1); the L-series of motives are usually nonabelian. Perhaps you could clarify your question.

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I know very little about this, so my ability to elaborate is limited. I agree that L-functions can deal with non-abelian extensions. But it is also true that their definition goes through cohomology. I imagine that the "non-abelian" part of "non-abelian L-functions" has to do with their construction, and not with their applications. As for in what sense they would be stronger than the usual L-functions, this is part of my question. – James D. Taylor Oct 23 '11 at 21:46