I found the famous Faltings book “Lectures on arithmetic Riemann-Roch theorem".
In the book, very analytic techniques such as Garding inequality or heat kernel are explained. I have no idea where such analytic tools must come in to prove algebraic theorem.
Because we want to calculate the Euler-Poincare number for the cohomology of vector bundle $E$ that is defined in a purely algebraic manner. So
Question 1: Why do we need to consider such analytic aspects to formulate Riemann-Roch theorem for arithmetic surfaces over Spec $\mathbb{Z}$?
I also found in Lang's book that to formulate Arithmetic Riemann-Roch theorem for arithmetic surface over Spec $\mathbb{Z}$, one might have to take care of Arakelov metric for arithmetic surface over infinite place $R$.
Question 2: What does it mean to “choose'' a nice metric such as Arakelov-metric for base-changed arithmetic surface over real place? Does it change the structure of the given arithmetic surface? Or is it incomplete to formulate Faltings-Riemann Roch Theorem that we merely consider the algebraic defining equation of arithmetic surface over Spec $\mathbb{Z}$?
Best Answer
I am by no means an expert here, so this is just a "long comment" regarding the question why analytic tools come in. In algebraic geometry intersection theory over complex numbers is a powerful tool to study curves and varieties in general. However, for number theory and arithmetic geometry one needs also other fields, different from the complex numbers. But for other fields, many difficulties arise, and new tools are really necessary - an important example here is Arakelov theory. Arakelov introduced an intersection pairing on arithmetic surfaces. G. Faltings’ arithmetic analogues of Riemann-Roch theorem and adjunction formula from classical intersection theory on surfaces are two outstanding examples for the successes of Arakelov theory. As for archimedian fields, similar things can be done for non-archimedian fields. Harmonic analysis on metrized graphs, for example, is used to study Arakelov-Green functions and related continuous Laplacian operators. Various arithmetic results are obtained after these studies. For example, the proof of "effective Bogomolov conjecture" over function fields of characteristic zero.
In general, to prove an algebraic theorem, often a "new technique" has to be used. There are several examples for this in algebra (and other fields).