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
DOI: https://doi.org/10.1090/S1056-3911-2011-00539-0
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
Email: mkakde@math.princeton.edu

DOI: https://doi.org/10.1090/S1056-3911-2011-00539-0
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

American Mathematical Society