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 geometry in nLab
Also, I understand some PDE. 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.
I know almost nothing of schemes or of number theory. I don’t how much of these is needed to learn this stuff. A while ago I wrote my point of view on what “you should and shouldn’t read” before studying Arakelov geometry. What gometry I read before reading about Arakelov theory? Taking another look at that answer, it seems that my answer is written for people with a more algebraic background.
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.
Mathematics > Algebraic Geometry
If you’re more comfortable with analysis than algebraic geomegry, I think a good idea would be to start with the analytic part of Arakelov geometry. This is explained very well in Chapter 1.
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.
This is where schemes and number theory come into play. 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.
There’s many of these, but I’m geomery the person to tell you which one is the best to start with. 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.
 New Approach to Arakelov Geometry
Vamsi 1, 14 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. I would say Fulton’s book is not necessary since you anyway arakeolv intersection theory via K-theory. Dear Vamsi, A while ago I wrote my point of view on what “you should and shouldn’t read” before studying Arakelov geometry.
See What should I read before reading about Arakelov theory? Bruin’s master’s thesis written adakelov the supervision of R. Ariyan Javanpeykar 5, 1 22 Thanks for the answer. I also want to know if there are any applications of Analytic torsion outside Arakelov geometry.
If not, I guess I would have to learn arakelpv scheme stuff There are definitely situations outside Arakelov geometry where analytic torsion appears.
I just don’t know any of them. 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. Peter Arndt 8, 3 41 Adakelov up or log in Sign up using Google.