We consider $\mathbb{Z}_p^{\mathbb{N}}$-extensions $\mathcal{F}$ 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 $\mathcal{F}$. 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 $\mathbb{Z}_p^d$-extensions). The final section provides more details for the cyclotomic $\mathbb{Z}_p^{\mathbb{N}}$-extension arising from the torsion of the Carlitz module: in particular, we relate cyclotomic units with Bernoulli-Carlitz numbers by a Coates-Wiles homomorphism.

Aspects of Iwasawa theory over function fields / Bandini, Andrea; F., Bars; I., Longhi. - (In corso di stampa). (Intervento presentato al convegno t-motives: Hodge structures, transcendence and other motivic aspects tenutosi a Banff International Research Station (BIRS), Banff, Canada nel 28-09/02-10 2009).

Aspects of Iwasawa theory over function fields

BANDINI, Andrea;
In corso di stampa

Abstract

We consider $\mathbb{Z}_p^{\mathbb{N}}$-extensions $\mathcal{F}$ 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 $\mathcal{F}$. 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 $\mathbb{Z}_p^d$-extensions). The final section provides more details for the cyclotomic $\mathbb{Z}_p^{\mathbb{N}}$-extension arising from the torsion of the Carlitz module: in particular, we relate cyclotomic units with Bernoulli-Carlitz numbers by a Coates-Wiles homomorphism.
In corso di stampa
Aspects of Iwasawa theory over function fields / Bandini, Andrea; F., Bars; I., Longhi. - (In corso di stampa). (Intervento presentato al convegno t-motives: Hodge structures, transcendence and other motivic aspects tenutosi a Banff International Research Station (BIRS), Banff, Canada nel 28-09/02-10 2009).
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11381/2524473
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