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So, my background in Number Theory is - Elementary Number Theory(that I did mainly from David Burton's text) and Serre's A course in Arithmetic(first half on algebraic methods that included p-adic numbers, Quadratic forms over $\mathbb{Q_p}$, Hasse-Minkowski's theorem etc.)
Having read this, I can say that I would certainly like to study some more of number theory. So, intending to do exactly that, I decided to some research on my own but that only helped me making me more confused as it turns out there are a zillion topics one can study in number theory with my background, like, number fields, modular forms etc.

My questions are-

  1. How should one decide what they want to study? IMO if I had some broad picture of what we are really trying to do ultimately in each of the branch, that could be of some help. So, can anyone here give me a rough break down of different branches of number theory?

  2. As of now, I don't really have any preference between analysis and algebra as I enjoy both of them(except maybe things related to groups, which I don't find very intuitive and hence not interesting) but people mainly classify number theory as these two branches. So, to decide what I want to study next in it, do I need to choose one of these two?

My background is: Algebra(Group theory, Rings and Modules, Field and Galois Theory), Analysis(Fourier and Complex Analysis from Stein and Shakarchi, Real Analysis, General Topology) and above mentioned Number Theory courses.

Thanks a lot in advance!

Shreya
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  • yeah..I know that I will have to study everything eventually if I plan to pursue research work in Number theory but there must be some order of topics that one must follow as a beginner. There are like so many ways topics one can study after elementaryNT. My question is, how does one decide what do they want to study next ans is there any order/sequence of topics one should follow? For ex: study number fields before modular forms. Like this. – Shreya May 02 '18 at 11:43
  • Perhaps the reason your question is not getting many responses is that it is extremely broad, mainly because number theory is extremely broad. For example, I feel qualified enough to point you in the direction of some algebraic number theory texts, especially if you are aiming to learn about modular forms/Galois representations, but not qualified at all to comment on other topics. – Mathmo123 May 02 '18 at 19:43
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    For your first question, this is highly relevant – Mathmo123 May 02 '18 at 19:47
  • Thanks! It would be great if you could explain about Algebraic Number Theory like what should be the starting point, where can one go from there. As much as I know, I think Number Fields should be the first topic I should study but am not really sure. Also, I don't have any idea what can I study after that. Can you start number fields before Modular forms or later? If you know and could point me to some source or some structured syllabus of first course in algebraic number theory, that would be great too! Thank you so much for replying when no one else was. – Shreya May 07 '18 at 10:38
  • Also, I have a billion questions about number theory and at my school, there aren't any graduate/post doc students working in it and only one or two faculties who don't seem to give too much time to annoying non-graduate students. I would really appreciate if someone here can help me to answer these questions I have about the field. – Shreya May 07 '18 at 13:40

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