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Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Proof of the main conjecture of noncommutative Iwasawa theory for totally real number fields in certain cases

Author: Mahesh Kakde
Journal: J. Algebraic Geom. 20 (2011), 631-683
Published electronically: April 5, 2011
MathSciNet review: 2819672
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Abstract | References | Additional Information

Abstract: Fix an odd prime $p$. Let $G$ be a compact $p$-adic Lie group containing a closed, normal, pro-$p$ subgroup $H$ which is abelian and such that $G/H$ is isomorphic to the additive group of $p$-adic integers $\mathbb {Z}_p$. First we assume that $H$ is finite and compute the Whitehead group of the Iwasawa algebra, $\lambda (G)$, of $G$. We also prove some results about certain localisation of $\lambda (G)$ needed in Iwasawa theory. Let $F$ be a totally real number field and let $F_{\infty }$ be an admissible $p$-adic Lie extension of $F$ with Galois group $G$. The computation of the Whitehead groups are used to show that the Main Conjecture for the extension $F_{\infty }/F$ can be deduced from certain congruences between abelian $p$-adic zeta functions of Deligne and Ribet. We prove these congruences with certain assumptions on $G$. This gives a proof of the Main Conjecture in many interesting cases such as $\mathbb {Z}_p\rtimes \mathbb {Z}_p$-extensions.

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Additional Information

Mahesh Kakde
Affiliation: Trinity College, Trinity Street, Cambridge CB2, United Kingdom 01223 338 400
Address at time of publication: Princeton University, Fine Hall, Washington Road, Princeton, New Jersey 08540

Received by editor(s): December 9, 2008
Received by editor(s) in revised form: April 20, 2009
Published electronically: April 5, 2011
Additional Notes: This work is a part of the author’s Ph.D. thesis. The author thanks Trinity College, Cambridge for the Ph.D. studentship