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I'm learning schemes via Hartshorne's book, and I need an ample stock of motivation to keep working through it. My primary interest is in applications to number theory, and I've learned a decent amount of arithmetic geometry without schemes before (e.g. Silverman's book on elliptic curves).

I'm looking for examples of results in number theory that either have no good solution without the language of schemes, or are significantly illuminated by using schemes.

Of course I know there are lots of major results that used schemes heavily and where there are probably no known methods without them, but ideally, I'd like something where the proof is more accessible than say, Mazur's theorem. Any ideas or resources are greatly appreciated!

stillconfused
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  • Does this answer your question? Do schemes help us understand elliptic curves?:"Schemes play an enormous role in all the modern theory of elliptic curves". Examples. Sato-Tate conjecture, BSD conjecture,... – Dietrich Burde Oct 13 '22 at 17:44
  • @DietrichBurde How is that a duplicate? The other question is specifically about elliptic curves and this one isn't. – David Lui Oct 13 '22 at 17:46
  • Even for elliptic curves, schemes are important. This answers a big part of the question (may be it is not completely a duplicate, but if it already applies for elliptic curves,...).Also, it is a bit unclear, what is excluded or not, e.g., Mazur's theorem. In this sense I believe the other post is a good answer. Also, this MO-post gives some examples, i.e., the study of number rings as schemes. They are "significantly illuminated" by this, I think. – Dietrich Burde Oct 13 '22 at 17:47
  • The examples given there are major theorems. As a student starting to learn schemes, it's not so easy for me to see how shcemes are an integral part of proving the Sato-Tate conjecture or working on BSD. Since I want this to motivate me to learn more about schemes, needing to know a huge amount about schemes to understand the motivation is a bit of an issue... – stillconfused Oct 13 '22 at 18:01
  • Yes, I understand. The MO post, for example, gives a good account, why the study of number rings as schemes is useful. However, not only schemes are used in number theory, but many other concepts from algebraic geometry, like motives, perfectoid spaces etc. Studying number theory means also to accept these concepts, and not only look for motivation. This will come also later automatically through the study. – Dietrich Burde Oct 13 '22 at 18:18

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