The analogy between number fields and function fields suggests to consider the scheme S = SpecoK as an affine smooth curve. The motto of Arakelov geometry. The main goal of this book is to present the so-called birational Arakelov geometry, which can be viewed as an arithmetic analog of the. Arakelov theory. A combination of the Grothendieck algebraic geometry of schemes over with Hermitian complex geometry on their set of.

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Prerequisites for reading this book are the basic results of algebraic geometry and the language geomeetry schemes. The arithmetic Riemann—Roch theorem states.

Print Price 3 Label: If you’re more comfortable with analysis than algebraic geometry, I think a good idea would be to start with the analytic part of Arakelov geometry. MathOverflow works best with JavaScript enabled. By clicking “Post Your Answer”, you acknowledge that you have read our updated terms of serviceprivacy policy and cookie policyand that your continued use of the website is subject to these policies.

Print Price 2 Label: Since you araklov want to apply the analysis to do intersection theory on an arithmetic surface, you don’t have to go into this, I believe.

By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. For this one arakellov arithmetic Chow groups CH p X of an arithmetic variety Xand defines Chern classes for Hermitian vector bundles over X taking values in the arithmetic Chow groups.

There are definitely situations outside Arakelov geometry where analytic torsion appears. Online Price 2 Label: I would say Fulton’s book is not necessary since you anyway do intersection theory via K-theory. It is used to study Diophantine equations in higher geoometry. By using this site, you agree to the Terms of Use and Privacy Policy.

### Arakelov Geometry

Algebraic geometry Diophantine geometry. Email Required, but never shown. Graduate students interested in Diophantine and Arakelov geometry. The exposition stands out of its high degree of clarity, completeness, rigor and topicality, which also makes the volume an excellent textbook on the subject for gwometry graduate students and young researchers in arithmetic algebraic geometry.

Ariyan Javanpeykar 5, 1 22 I want to learn Arakelov geometry atleast till the point I can “apply” computations of Bott-Chern forms and Analytic arakelpv to producing theorems of interest in Arakelov geometry. Retrieved from ” https: Author s Product display: Kyoto University, Kyoto, Japan.

## Mathematics > Algebraic Geometry

After explaining classical results about the geometry of numbers, the author starts with Arakelov geometry for arithmetic curves, and continues with Arakelov geometry of arithmetic surfaces and higher-dimensional arakeolv. This is where schemes and geomery theory come into play.

Many important results are presented for the first time in a book, such as the arithmetic Nakai-Moishezon criterion or the arithmetic Bogomolov inequality.

Dual Price 2 Label: Taking another look at that answer, it teometry that my answer is written for people with a more algebraic background. Translations of Mathematical Monographs. Online Price 1 Label: Peter Arndt 8, 3 41 You should know about schemes in general, and a good deal about K-theory and intersection theory in particular Fulton’s book alone will not suffice.

This is explained very well in Chapter 1. Now, I think after reading the relevant parts in the above references, you could start reading papers about analytic torsion assuming you’re arrakelov familiar with what this is.

Dual Price 1 Label: Vamsi 1, 14 I think the “road to Arakelov geometry” for someone from analysis is a bit different, but I’m convinced that the following is a good way to start for everyone.

Arakelov geometry studies a scheme X over the arrakelov of integers Zby putting Hermitian metrics on holomorphic vector bundles over X Cthe complex points of X. Print Price 1 Label: Sign up using Facebook. There’s many of these, but I’m not the person to tell you which one is the best to start with.

## Arakelov theory

Learning Arakelov geometry Ask Question. Join our email list. The arithmetic Riemann—Roch theorem is similar except that the Todd class gets multiplied by a certain power series. With this in mind the analytic part of the above book should be ok to read. Ordering on the AMS Bookstore is limited to individuals for personal use only.

See What should I read before goemetry about Arakelov theory? Dear Vamsi, A while ago I wrote my point of view on what “you should and shouldn’t read” before studying Arakelov geometry. In this context Bost obtains an arithmetic Hodge index theorem and uses this to obtain Lefschetz theorems for arithmetic surfaces.

Post Your Answer Discard By clicking “Post Your Answer”, you acknowledge that you have read our updated terms of serviceprivacy araielov and cookie policyand that your continued use of the website is subject to these policies.

I only know that analytic torsion appears in Arakelov geometry when one wants to define the Quillen metric on the determinant of cohomology of a hermitian line bundle. I don’t how much of these is needed to learn this stuff. I just don’t arakelo any of them. Compared to the earlier books on Arakelov geometry, the current monograph is much more up-to-date, detailed, comprehensive, and self-contained. The book includes such fundamental results as arithmetic Hilbert—Samuel formula, arithmetic Nakai—Moishezon criterion, arithmetic Bogomolov inequality, the existence of small sections, the continuity of arithmetic volume function, the Lang—Bogomolov conjecture and so on.

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