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!

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. 


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 


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. 


"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. 

