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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.

k.j.
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  • 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. – Nicolas Hemelsoet Apr 24 '18 at 19:55
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    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. – Mathmo123 Apr 24 '18 at 20:12
  • Concerning Arithmetic Geometry, see the recommendations here. – Dietrich Burde Apr 24 '18 at 20:54
  • @NicolasHemelsoet Thank you for your comments. I've read Silverman, and these were very interesting. I want to study these topics more deeply. Are there good texts? (I'm reading Silverman's "Advanced ...", but I feel this is a little distant from my interesting, comparing his book AEC.) – k.j. Apr 25 '18 at 06:14
  • @Mathmo123 I'm interested in these topics very much. Could you tell me some texts which tells me those theories, such as the intersection of Galois representaion, etale cohomology and modular forms? I've heard this theory is called Langlands program, which I want to study so much. If there are some acknowledge for it, I'll try to study all of them. – k.j. Apr 25 '18 at 06:20
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    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. – Mathmo123 Apr 25 '18 at 06:49
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    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. – Mathmo123 Apr 25 '18 at 06:53
  • @Mathmo123 Thank you very much! These helps me so much! I determine to try Diamond-Shurman, Cornell-Silverman, and Deligne's, and Ribet's paper and etale cohomology by some texts. – k.j. Apr 25 '18 at 09:49
  • @k.j. More generally, étale cohomology of algebraic varieties provide Galois representations. If you take the first $\ell$-adic cohomology for abelian varieties, you will simply get the $\ell$-adic Tate module of the abelian variety. – Watson Apr 25 '18 at 13:10
  • As for abelian varieties, you may want to see how Faltings used them to solve Mordell's conjecture. – Watson Nov 22 '18 at 11:13

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