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!



Roadmap to reach Arithmetic Geometry for a Physics Major
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.


