Hi Everybody! I am physics major but I read mathematics for myself. my main fields of interest are number theory and geometry. it seems that due to the works of A.Grothendieck, algebraic geometry must be used for studying deepest problems of number theory which culminate in the field of Arithmetic Geometry or Arithmetic Algebraic Geometry (please correct me if this isn't so). Could someone help me for more elemntary raodmap to reach the subject! I know analysis and algebra at the level of A.W.Knapp books (Volume one of every field) and number theory at the level of "A Classical Introduction to Modern Number Theory" (now at chapter 19)! I found that "Lectures on algebraic Geometry" by G.Harder have subjects of the field like tate conjecture or etale cohomology. expected volume three of these lectures will be about topics like cohomology of arithmetic groups and Langlands program! Thanks!

1$\begingroup$ Take a look at the book "From Number Theory to Physics". Exerpt from the blurb : "The 14 chapters of this book are extended, selfcontained versions of expository lecture courses given at a school on "Number Theory and Physics" held at Les Houches for mathematicians and physicists. Most go as far as recent developments in the field. Some adapt an original pedagogical viewpoint." $\endgroup$– Laurent BergerJun 7, 2013 at 12:02

$\begingroup$ For the book of Günther Harder you will need more than the (beautiful) book of Ireland and Rosen. There was already a discussion about this type of question, see mathoverflow.net/questions/23385/…. $\endgroup$– Dietrich BurdeJun 7, 2013 at 12:26

3$\begingroup$ I must deliver some unfortunate (or maybe fortunate?) news for you. Even the subspecialty of Arithmetic Geometry is broad enough where I can't really tell from your post what you'd really like to learn. It's true that algebraic geometry is quite useful, but the sort of deep problem you're interested in can wildly alter the amount of algebraic geometry you need to master. For instance, Zhang's recent theorem on bounded gaps uses the Weil conjectures (now a deep theorem of Deligne) but it is not a prerequisite to completely understand Deligne's work in order to understand Zhang's result. $\endgroup$– stankewiczJun 7, 2013 at 15:25

2$\begingroup$ So what is it REALLY that you'd like to know about number theory? $\endgroup$– stankewiczJun 7, 2013 at 15:26

$\begingroup$ Dear stankewicz, Thanks for your attention! My longterm aims are followings 1) understanding Diophantine aspects of elliptic curves namely the problem of rational solution and integral points on the curve. specifically deep comprehension of MoredellWeil Theorem ,Siegel Theorem ,Faltings Theorem,Weil Theorem (STW Conjecture) and BirchSwinnerton Dyer Conjecture! 2) Arithmetic aspects of Langlands Program; 3) understanding the schemetheoretic approach to arithmetic problems (for this I found Q.Liu's "Algebraic Geometry and Arithmetic Curves" but it was a nightmare!) $\endgroup$– MathPhys LoverJun 7, 2013 at 22:06
4 Answers
I recommend Silverman's The Arithmetic of Elliptic Curves. Silverman takes the highbrow approach, but writes in such a way as to make his book friendly and accessible for newcomers.

$\begingroup$ Dear Frank Thorne: it seems that required prerequisite for this book is algebraic geometry which is beyond my knowledge! $\endgroup$ Jun 7, 2013 at 21:56

1$\begingroup$ @MathPhys Lover: "Arithmetic Geometry" the name itself suggests that there is going to be $\textbf{Algebraic Geometry}$ involved, even if you were to start from scratch. So I don't see anybody skipping Algebraic Geometry and learning Arithmetic Geometry. $\endgroup$– C.S.Jun 8, 2013 at 2:10

1$\begingroup$ The definition of "arithmetic geometry" is more or less "algebraic geometry applied to number theory". Silverman indeed uses algebraic geometry, but he gets a lot of mileage out of only a little bit of algebraic geometry, and the needed algebraic geometry is explained in two short introductory chapters. $\endgroup$ Jun 8, 2013 at 10:12

$\begingroup$ as the title of this topic suggests I require a roadmap! so my main question is "are the next step for someone like me with basic knowledge of Algebra & Analysis to comprehend basic arithmetic geometry for my aims explained in an answer to stankewicz is to use Silverman's book? I know that I should master Algebraic Geometry! $\endgroup$ Jun 8, 2013 at 12:27

$\begingroup$ Honestly based upon the answer you gave, Silverman's Arithmetic of Elliptic Curves is a pretty good starting point. As Frank says, the algebraic geometry prerequisites are covered in the first two chapters and he really dives into the subject in chapter 3. $\endgroup$ Jun 10, 2013 at 2:59
It seems to me a superhuman task to learn any substantial part of modern arithmetic geometry without any other guidance that a list of books. If you want to begin learning it seriously, the less hard way is to follow a math graduate course at your university or nearby on the subject (you say your a Phys major, so I assume you are currently affiliated in a university). Reading books by yourself is good, but will not take you very far (I don't say that to discourage you of reading books, on the contrary I say that in order that you understand it's normal if you get discouraged in trying to read more advanced books).
That said here is a list of relatively elementary books you might trade to read:
First, if you want to advance a little bit, you will need a solid background in algebra, more solid that you probably can get as a Physics Major Undergrad: good books are Lang's Algebra, or Jacobson's Algebra.
For algebraic number theory proper, I like algebraic theory of numbers, by Pierre Samuel.
Gouvêa's padic numbers: an introduction is also interesting.

$\begingroup$ Dear Joël, Thanks for your Comments and recommendations! as I said I read The Voume 1 of A.W.Knapp Book about algebra, "Basic Algebra". the second volume is related to number theory and algebraic geometry! it would be great if you can comment about the second volume! $\endgroup$ Jun 7, 2013 at 22:02
Learning number theory (or any thing for that matter) wouldn't be fun without solving a lot of problems. I'd recommend "Problems in analytic number theory" and "Problems in algebraic number theory" by Murty and others
"3) understanding the schemetheoretic approach to arithmetic problems (for this I found Q.Liu's "Algebraic Geometry and Arithmetic Curves" but it was a nightmare!)"
My suggestion for making Liu's excellent book more accessible is "The Geometry of Schemes" by EisenbudHarris. The latter is relatively short and gives away several of the basic "secrets"; in fact, the first half of this book would probably do so as to move on to Liu.
2) For arithmetic aspect of Langlands, a nice way to start is with BernsteinGelbart, "An Introduction to the Langlands Program"
1) for "deep comprehension" of arithmetic on curves, it would be nice to first go through most of Liu's book to understand many of the modern techniques. Silverman's books are also standards here.