Aspects of Iwasawa theory over function fields

Abstract

We consider Z N p -extensions F of a global function field F and study various aspects of Iwasawa theory with emphasis on the two main themes already (and still) developed in the number fields case as well. When dealing with the Selmer group of an abelian variety A defined over F , we provide all the ingredients to formulate an Iwasawa Main Conjecture relating the Fitting ideal and the p-adic L-function associated to A and F. We do the same, with characteristic ideals and p-adic L-functions, in the case of class groups (using known results on characteristic ideals and Stickelberger elements for Z d p -extensions). The final section provides more details for the cyclotomic Z N p -extension arising from the torsion of the Carlitz module: in particular, we relate cyclotomic units with Bernoulli-Carlitz numbers by a Coates-Wiles homomorphism.

Iwasawa main conjecture for the Carlitz cyclotomic extension and applications

“…. Then (5) implies that the restriction of Fr q to F is exactly κ −1 (π q ). Theorem 3.15 implies that the series L p (X, y, ω i p ) converges on the closed unit disc.…”

“…The theorem above allows us to compute the Fitting ideal of C(χ) as the inverse limit of the Fitting ideals appearing in the (natural) filtration of F given by the fields F n . A different approach to the same problem is provided in [5,Section 5] where the authors use a filtration of Z d p -extensions (a more general approach and the fact that the limit is independent from the filtration are shown in [6]). In that paper the statement of the Main Conjecture involves characteristic ideals but (for Iwasawa modules) they coincide with Fitting ideals whenever the Fitting is principal (see, for example, [4, Lemma 5.10]).…”