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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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Arakelov theory

Peter Arndt 8, 3 41 The arithmetic Riemann—Roch theorem is geomtry except that the Todd class gets multiplied by a certain power series. After explaining classical results goemetry 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.

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. Dear Vamsi, A while ago I wrote my point of view on what “you should and shouldn’t read” before studying Arakelov geometry. I don’t how much of these is needed to learn this stuff. This is explained very well in Chapter 1. In addition, the author presents, with full details, the proof of Faltings’ Riemann—Roch theorem.


The rich bibliography of seventy-eight references certainly arskelov as a useful guide to further reading with regard to the more recent research literature in the field.

soft question – Learning Arakelov geometry – MathOverflow

I also want to know if there are any applications of Analytic torsion outside Arakelov geometry. This page was last edited on 28 Mayat Home Questions Tags Users Unanswered. 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. The exposition stands out of its high degree of clarity, completeness, rigor and topicality, which also makes the volume an gelmetry textbook on the subject for seasoned graduate students and young researchers in arithmetic algebraic geometry.

[] New Approach to Arakelov Geometry

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 Arqkelov. There’s many of these, but I’m not the person to tell you which one is the best to start with.

Thanks for the answer. Prerequisites for reading this book are the basic results of algebraic geometry and the language of schemes. Online Price 2 Label: The arithmetic Riemann—Roch theorem states.


Compared to the earlier books on Arakelov geometry, the current monograph is much more up-to-date, detailed, comprehensive, and self-contained. Taking another look at that answer, it seems that my answer is written for people with a more algebraic background.

I know almost nothing of schemes or of number theory. I would say Fulton’s book is not necessary since you anyway do intersection theory via K-theory.

Also, I understand some PDE. I just don’t know any of them. aramelov

Post as a guest Name. Sign up or log in Sign up using Google. Graduate students interested in Diophantine and Arakelov geometry.

Mathematics > Algebraic Geometry

With this in mind the analytic part of the above book should be ok to read. Sign up using Facebook. See What should I read before reading about Arakelov theory? Print Price 3 Label: Email Required, but never shown.