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In many theories there is a rough divide between elementary problems that can be solved with "one's hands", and "deep results that require powerful tools". For example, I am told that Hodge theory is a such a tool in the topology of complex algebraic varieties.

As somebody that is just beginning to learn about arithmetic questions, it is not clear to me what these divides are (if they exist) in all the fields under the heading of arithmetic geometry. So, I have a few naive questions.

  • Roughly, what are all of the subfields?

  • Generally, what type of problem is elementary?

  • What are some examples of elementary results?

  • What are the powerful tools, and how are they broadly used?

  • What are the overarching principles (e.g. Hasse principle; now calculate obstructions)?

If, like me, you know very little about this field, I can recommend this fantastic article by Jordan Ellenberg and these notes by Bjorn Poonen. Please edit this question if it is nonsensical. Thanks to everybody who responds.

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Your question isn't nonsensical, but we do try to discourage questions this broad. As a rule of thumb, if properly answering your question would require a book, you should probably narrow it down (or just ask for good books on it, which is in practice what you have done). –  Ben Webster Dec 14 '09 at 4:15

1 Answer 1

Lots of the answers are going to be of the form "read book X" so I'll start: Serre's Course in Arithmetic is the best conceivable answer to the question "where could I find a concise, clearly written compilation of results in arithmetic geometry that can be proved without advanced machinery?" It's really a beautiful book.

The kind words about my article are much appreciated.

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