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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The rich bibliography of seventy-eight references certainly serves as a useful guide to further reading with regard to the more recent research literature in the field. In addition, the author presents, with full details, the proof of Faltings’ Riemann—Roch theorem. In this context Bost obtains an arithmetic Hodge index theorem and uses this to obtain Lefschetz theorems for arithmetic surfaces.

Arakelov geometry studies a scheme X over the ring of integers Zby putting Hermitian metrics on holomorphic vector bundles over X Cthe complex points of X.

I don’t how much of these is needed to learn this stuff. 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 classical birational geometry, i.

There is this nice text by Demailly which motivates the treatment of intersection theory on the infinite fibers and probably suits you with your background.

Arakelov Geometry

The arithmetic Riemann—Roch theorem is similar except that the Todd class gets multiplied by a certain power series. Algebraic geometry Diophantine geometry.


What should I read before reading about Arakelov theory? 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 varieties.

Arakelov theory – Wikipedia

Online Price arakeloov Label: The arithmetic Riemann—Roch theorem then describes how the Chern class behaves under pushforward of vector bundles under a proper map of arithmetic varieties. Taking another look at that answer, it seems that my answer is written for people with a more algebraic background. Vamsi 1, 14 I want to learn Arakelov geometry atleast till the point I can “apply” computations of Bott-Chern forms and Analytic torsion to producing theorems of interest in Arakelov geometry.

This is where schemes and number theory come into play. Also, I understand some Geometrh. There’s many of these, but I’m not the person to tell you which one is the best to start with.

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Ariyan Javanpeykar 5, 1 22 For this one defines 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 Feometry groups.

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. A while ago I wrote my point of view on what “you should and shouldn’t read” before studying Arakelov geometry. Now, I think after reading the relevant parts in the above references, you could start reading papers about analytic torsion assuming you’re already familiar with what this is.


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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. 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. See our librarian page for additional eBook ordering options. Bruin’s master’s thesis written under the supervision of R. With this in mind the analytic part of the above book should be ok to read.

This is explained very well in Chapter 1. Print Price 3 Label: Print Price 1 Label: 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. Sign up using Facebook. Print Teometry 2 Label: Online Price 1 Label: The arithmetic Riemann—Roch theorem states.

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