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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Peter Arndt 8, 3 41 Libraries and resellers, please contact cust-serv ams. This is a timely monograph that should appeal to researchers in this important area of mathematics. Arakelov Geometry Share this page.
Prerequisites for reading this book are the basic results of algebraic geometry and the language of schemes. Print Price 3 Label: The arithmetic Riemann—Roch theorem is similar except that the Todd afakelov gets multiplied by a certain power series. 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 feometry is.
There’s many of these, but I’m not the person to tell you which one is the best to start with. I would say Fulton’s book is not necessary since you anyway do intersection theory via K-theory.
Author s Product display: Algebraic geometry Diophantine geometry. If not, I guess I would have to learn the scheme stuff Thanks for the answer. Online Price 1 Label: Print Price 2 Geomery There are definitely situations outside Arakelov geometry where analytic torsion appears.
The arithmetic Riemann—Roch theorem then describes how the Chern class behaves under pushforward of vector bundles under a proper map of arithmetic varieties.
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 seasoned graduate students and young researchers in arithmetic algebraic geometry. Many important results are presented for the first time in a book, such as the arithmetic Nakai-Moishezon criterion or the arithmetic Bogomolov inequality.
Views Read Edit View history. This page was last edited on 28 Mayat Taking another look at that answer, it seems that my answer is written for people with a more algebraic background. Graduate students interested in Diophantine and Arakelov geometry. Email Required, but never shown. In mathematicsArakelov theory or Arakelov geometry is an approach to Diophantine geometrynamed for Suren Arakelov. Join our email list.
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. Gfometry as a guest Name. Sign up using Facebook.
Also, I understand some PDE. Vamsi 1, 14 Learning Arakelov geometry Ask Question. I know almost nothing of schemes or of number theory. With this in mind the analytic part of the above book should be ok to read.
I also want to know if there are any applications of Analytic torsion outside Arakelov geometry. Retrieved from ” https: 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 arzkelov arithmetic Chow groups.
The arithmetic Riemann—Roch theorem states. Sign up or log in Sign up using Google. Sign up using Email and Password. Ordering on the AMS Bookstore is limited to individuals for personal use only.
Arakelov geometry in nLab
In this context Bost obtains an arithmetic Hodge index theorem and uses this to obtain Lefschetz theorems for arithmetic surfaces. Bruin’s arakflov thesis written under the supervision of R. I just don’t know any of them. 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.
This is where schemes and number theory come into play.
In addition, the author presents, with full details, the proof of Faltings’ Riemann—Roch theorem. 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.
See What should I read before reading about Arakelov theory? It is used to study Diophantine equations in higher dimensions.