Skip to main content Accessibility help
×
Hostname: page-component-7c8c6479df-24hb2 Total loading time: 0 Render date: 2024-03-28T05:15:23.331Z Has data issue: false hasContentIssue false

From the Birch and Swinnerton-Dyer Conjecture to non-commutative Iwasawa theory via the Equivariant Tamagawa Number Conjecture - a survey

Published online by Cambridge University Press:  20 April 2010

Otmar Venjakob
Affiliation:
Universität Heidelberg Mathematisches Institut Im Neuenheimer Feld 288 69120 Heidelberg, Germany. otmar@mathi.uni-heidelberg.de
David Burns
Affiliation:
King's College London
Kevin Buzzard
Affiliation:
Imperial College of Science, Technology and Medicine, London
Jan Nekovář
Affiliation:
Université de Paris VI (Pierre et Marie Curie)
Get access

Summary

Introduction

This paper aims to give a survey on Fukaya and Kato's article [23] which establishes the relation between the Equivariant Tamagawa Number Conjecture (ETNC) of Burns and Flach [9] and the noncommutative Iwasawa Main Conjecture (MC) (with p-adic L-function) as formulated by Coates, Fukaya, Kato, Sujatha and the author [14]. Moreover, we compare their approach with that of Huber and Kings [24] who formulate an Iwasawa Main Conjecture (without p-adic L-functions). We do not discuss these conjectures in full generality here, in fact we are mainly interested in the case of an abelian variety defined over ℚ. Nevertheless we formulate the conjectures for general motives over ℚ as far as possible. We follow closely the approach of Fukaya and Kato but our notation is sometimes inspired by [9, 24]. In particular, this article does not contain any new result, but hopefully serves as introduction to the original articles. See [47] for a more down to earth introduction to the GL2 Main Conjecture for an elliptic curve without complex multiplication. There we had pointed out that the Iwasawa main conjecture for an elliptic curve is morally the same as the (refined) Birch and Swinnerton Dyer (BSD) Conjecture for a whole tower of number fields. The work of Fukaya and Kato makes this statement precise as we are going to explain in these notes. For the convenience of the reader we have given some of the proofs here which had been left as an exercise in [23] whenever we had the feeling that the presentation of the material becomes more transparent thereby.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×