First, I apologize for my poor English.

I like number theory such as "when can prime $p$ be written as $x^2 + y^2$?" and "find the integer solutions of this equation." Because I've heard that these problems can be solved by arithmetic geometry, I want to study it.

So I've read Hartshorne's Algebraic Geometry and Neukirch's Algebraic Number Theory and so on. (I've heard that these are fundamental for arithmetic geometry.) And next I’m about to study Abelian varieties and etale cohomology for the same reason. But after reading list of contents of these books, I feel like that these are very abstract, and very distant from my first purpose.

So the question: what are applications of abelian variety and etale cohomology? I know the theory of etale cohomology solved the Weil's conjecture. But only this? I know this is the great achievement, but I want to know more applications. (and I do not know applications of the theory of abelian varieties at all.)

And the second question is: what is arithmetic geometry? What books should I read to study it's mainstream? I want, for example, books which tells me not only the definitions and fundamental properties of etale cohomology, but also interesting and elementary (like "find the integer solutions of this equation") applications. Or books which tells me many theories and techniques which are used in frontier researches.

Please help.

  • $\begingroup$ I'm a total novice, but did you read Silverman's books (there are 3 of them) about elliptic curves ? There might be many number-theory applications. $\endgroup$ – Nicolas Hemelsoet Apr 24 '18 at 19:55
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    $\begingroup$ Here's the most famous application: suppose that $a^p + b^p = c^p$ is a counterexample to Fermat's last theorem, with $p\ge 5$ a prime (plus some normalisations). Then there is an elliptic curve $E : y^2 = x(x-a^p)(x+b^p)$, with an associated Galois representation coming from the etale cohomology of $E$. By Wiles, this Galois representation comes from a modular form $f \in S_2(2)$, but this space is empty. Hence, $E$ cannot exist, and therefore FLT is true. Of course, after seeing that all this is part of a more general framework, it's easy to be distracted from the problems we started with. $\endgroup$ – Mathmo123 Apr 24 '18 at 20:12
  • $\begingroup$ Concerning Arithmetic Geometry, see the recommendations here. $\endgroup$ – Dietrich Burde Apr 24 '18 at 20:54
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    $\begingroup$ For modular forms with a view to FLT, the standard reference is Diamond and Shurman. You should see Prof. Emerton's comment to this blog post. A brief introduction to the Langlands program motivated b some of the questions you're interested in is here. If you haven't studied class field theory, look at Cox's "Primes of the form $x^2 + ny^2$" for an account motivated by exactly the question you asked about. $\endgroup$ – Mathmo123 Apr 25 '18 at 6:49
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    $\begingroup$ There are also tonnes of online resources, including a video lecture series by Kevin Buzzard and numerous records here. But at this point, it starts being worth seeking out someone in your department who can guide you through this. $\endgroup$ – Mathmo123 Apr 25 '18 at 6:53

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