Is there any book explaining in detail the book "Basic Number Theory" by Andre Weil as Dirichlet did to "Disquisitiones Arithmeticae"?
This is because I have read the two books mentioned above and I hope there will be one.
[Math] ny book explaining in detail the book “Basic Number Theory” by André Weil as Dirichlet did to “Disquisitiones Arithmeticae” by Gauss
nt.number-theoryreading-listreference-requestsoft-question
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First of all, Kevin is being quite modest in his comment above: his paper
Buzzard, Kevin. Integral models of certain Shimura curves. Duke Math. J. 87 (1997), no. 3, 591--612.
contains many basic results on integral models of Shimura curves over totally real fields, and is widely cited by workers in the field: 22 citations on MathSciNet. The most recent is a paper of mine:
Clark, Pete L. On the Hasse principle for Shimura curves. Israel J. Math. 171 (2009), 349--365.
http://alpha.math.uga.edu/~pete/plclarkarxiv7.pdf
Section 3 of this paper spends 2-3 pages summarizing results on the structure of the canonical integral model of a Shimura curve over $\mathbb{Q}$ (with applications to the existence of local points). From the introduction to this paper:
"This result [something about local points] follows readily enough from a description of their [certain Shimura curves over Q] integral canonical models. Unfortunately I know of no unique, complete reference for this material. I have myself written first (my 2003 Harvard thesis) and second (notes from a 2005 ISM course in Montreal) approximations of such a work, and in so doing I have come to respect the difficulty of this expository problem."
I wrote that about three years ago, and I still feel that way today. Here are the documents:
is my thesis. "Chapter 0" is an exposition on Shimura curves: it is about 50 pages long.
- For my (incomplete) lecture notes from 2005, go to
http://alpha.math.uga.edu/~pete/expositions2012.html
and scroll down to "Shimura Curves". There are 12 files there, totalling 106 pages [perhaps I should also compile them into a single file]. On the other hand, the title of the course was Shimura Varieties, and although I don't so much as attempt to give the definition of a general Shimura variety, some of the discussion includes other PEL-type Shimura varieties like Hilbert and Siegel moduli space. These notes do not entirely supercede my thesis: each contains some material that the other omits.
When I applied for an NSF grant 3 years ago, I mentioned that if I got the grant, as part of my larger impact I would write a book on Shimura curves. Three years later I have written up some new material (as yet unreleased) but am wishing that I had not said that so directly: I would need at least a full semester off to make real progress (partly, of course, to better understand much of the material).
Let me explain the scope of the problem as follows: there does not even exist a single, reasonably comprehensive reference on the arithmetic geometry of the classical modular curves (i.e., $X_0(N)$ and such). This would-be bible of modular curves ought to contain most of the material from Shimura's book (260 pages) and the book of Katz and Mazur Arithmetic Moduli of Elliptic Curves (514 pages). These two books don't mess around and have little overlap, so you get a lower bound of, say, 700 pages that way.
Conversely, I claim that there is some reasonable topology on the arithmetic geometry of modular curves whose compactification is the theory of Shimura curves. The reason is that in many cases there are several ways to establish a result about modular curves, and "the right one" generalizes to Shimura curves with little trouble. (For example, to define the rational canonical model for classical modular curves, one could use the theory of Fourier expansions at the cusps -- which won't generalize -- or the theory of moduli spaces -- which generalizes immediately. Better yet is to use Shimura's theory of special points, which nowadays you need to know anyway to study Heegner point constructions.) Most of the remainder concerns quaternion arithmetic, which, while technical, is nowadays well understood and worked out.
It is not hard to see that if $L/K$ is an extension of number fields, then the discriminant of $L/K$, which is an ideal of $K$, is a square in the ideal class group of $K$. Hecke's theorem lifts this fact to the different. (Recall that the discriminant is the norm of the different.)
If you recall that the inverse different $\mathcal D_{L/K}^{-1}$ is equal to $Hom_{\mathcal O_K}(\mathcal O_L,\mathcal O_K),$ you see that the inverse different is the relative dualizing sheaf of $\mathcal O_L$ over $\mathcal O_K$; it is analogous to the canonical bundle of a curve (which is the dualizing sheaf of the curve over the ground field). Saying that $\mathcal D_{L/K}$, or equivalently $\mathcal D_{L/K}^{-1}$, is a square is the same as saying that there is a rank 1 projective $\mathcal O_L$-module $\mathcal E$ such that $\mathcal E^{\otimes 2} \cong \mathcal D_{L/K}^{-1}$, i.e. it says that one can take a square root of the dualizing sheaf. In the case of curves, this is the existence of theta characteristics.
Thus, apart from anything else (and as indicated in the quotation given in the question), Hecke's theorem significantly strengthens the analogy between rings of integers in number fields and algebraic curves.
If you want to think more arithmetically, it is a kind of reciprocity law. It expresses in some way a condition on the ramification of an arbitrary extension of number fields: however the ramification occurs, overall it must be such that the different ramified primes balance out in some way in order to have $\prod_{\wp} \wp^{e_{\wp}}$ be trivial in the class group mod $2$ (where $\wp^{e_{\wp}}$ is the local different at a prime $\wp$). (And to go back to the analogy: this is supposed to be in analogy with the fact that if $\omega$ is any meromorphic differential on a curve, then the sum of the orders of all the divisors and poles of $\omega$ is even.) Note that Hecke proved his theorem as an application of quadratic reciprocity in an arbitrary number field.
Best Answer
Indeed Ramakrishnan and Valenza 's book is a pretty good reference.
Perhaps we could give more specific answers if you were more precise about exactly where your difficulties are?
EDIT: Since we've been given precisions in the comments below, I can confirm R&V's book will nicely do for the basics of the theory ; to get further, from the top of my head, you'll want to have a go with :