STNB 2016 (30th edition)

Iwasawa theory and STNB, a personal view.


Francesc Bars Cortina


Iwasawa theory is the study of objects of arithmetic interest over infinite towers of global fields.

Iwasawa explored this study for the p-th cyclotomic extension of number fields, obtaining a general theory of cyclotomic Iwasawa modules, formulating some sort of cyclotomic Iwasawa main conjecture. This was latter generalized considering $\mathbb{Z}_p^d$-extensions by Greenberg and many others (In STNB2014 we worked on a proof for some specific object concerning its cyclotomic Iwasawa main conjecture).

Kato in 1993 observed a strong relation between Iwasawa main conjecture with the $p$-part of Tamagawa number conjecture on $p$-valuation of special values of $L$-functions for a motive, therefore for attack BSD a good Iwasawa main conjecture for non-abelian infinite towers will be useful, with this inspiration a lot of big mathematicians develope lasts years non-commutative Iwasawa theory for example for extensions of the form $GL_2(\mathbb{Z}_p)$. Kato and Fukaya formulated a general non-commutative Iwasawa main conjecture for what knowadays is called $\Lambda$-rings, some sort of Iwasawa algebras. (In a talk in STNB2008 we introduced a little bit the non-commutative formulation).

By explicit class field theory, on globals fields of positive characteristic, appears natural to consider abelian profinite groups (in the infinite tower of global fields) which are not $\Lambda$-rings by adding the torsion of rank 1 Drinfeld modules
which is the ''cyclotomic''-tower in the positive characteristic, and one could obtain a non-commutative and non-abelian Iwasawa algebras if one introduces torsion of higher rank Drinfeld modules (considering the work of Pink and his school). (In STNB2001 we worked on Drinfeld modules and Hayes explicit class field theory, in particular the construction of the ''cyclotomic''-tower).

First, in the talk we want to remember the classical commutative Iwasawa theory over a global field of positive characteristic, and present the recent results on ``Carlitz-cyclotomic''-commutative Iwasawa theory (insights on non-noetherian Iwasawa theory began since STNB2010). After, we will try to present few of the results on non-commutative Iwasawa theory of Witte and Burns for $\Lambda$-rings (workshops inside STNB2010).



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