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arXiv · 1412.5957

Iwasawa Main Conjecture for the Carlitz cyclotomic extension and applications

Bruno Anglès·Andrea Bandini·Francesc Bars·Ignazio Longhi
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Abstract

We prove an Iwasawa Main Conjecture for the class group of the $\mathfrak{p}$-cyclotomic extension $\mathcal{F}$ of the function field $\mathbb{F}_q(\theta)$ ($\mathfrak{p}$ is a prime of $\mathbb{F}_q[\theta]\,$), showing that its Fitting ideal is generated by a Stickelberger element. We use this and a link between the Stickelberger element and a $\mathfrak{p}$-adic $L$-function to prove a close analog of the Ferrero-Washington theorem for $\mathcal{F}$ and to provide informations on the $\mathfrak{p}$-adic valuations of the Bernoulli-Goss numbers $\beta(j)$ (i.e., on the values of the Goss $\zeta$-function at negative integers).

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Bruno Anglès, Andrea Bandini, Francesc Bars, Ignazio Longhi. 2014-12-18. Iwasawa Main Conjecture for the Carlitz cyclotomic extension and applications. https://arxiv.org/abs/1412.5957

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