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José Figueroa-O'Farrill
José Figueroa-O'Farrill
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pmoduli
pmoduli
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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.

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. Please edit this question if it is nonsensical. Thanks to everybody who responds.

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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pmoduli
pmoduli
  • 287
  • 2
  • 9

Elementary questions in arithmetic geometry

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. Please edit this question if it is nonsensical. Thanks to everybody who responds.