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Im a doctorate student working on number theory and have hit a brickwall when it comes to scheme theory. I want to find a good book thats accessible to a graduate student so that I can understand results like the one I give below. Algebraic geometry is huge and I dont know where to look.

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[Edit] This post is similar and provides some really good suggestions, most notably: Hida's Geometric Modular Forms and Elliptic Curves, and Ravi Vakil's The Rising Sea. I found the latter extremely interesting since it is a self contained book covering all foundational tools from Algebraic Geometry and goes to great effort to explain the underlying intuition.

Guruprasad
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Camilo Gallardo
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    Ravi Vakil's Foundations of Algebraic Geometry is a great reference with great details. Hartshorne's Algbraic Geometry is a concise introduction to the algebraic geometry. Qing Liu's Algebraic Geometry and Arithmetic Curves was also helpful to me. I would recommend to learn some examples together, for example, elliptic curves, toric varieties, complex manifolds, moduli spaces. To me, scheme theory itself was very uninspiring and painful to learn. I think they are all available with scheme-free, classical level. You can compare the classical level with scheme theory applied one. – J1U Nov 27 '24 at 18:17
  • Could you suggest a book with applications to number theory and elliptic curves in particular? – Camilo Gallardo Nov 27 '24 at 20:13
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    This would be a good thing to talk to your advisor about, in addition to asking the internet. You may also wish to consult the general textbook recommendation post for AG over here. – KReiser Nov 27 '24 at 20:22
  • @KReiser Yes of course, but Im afraid they are not as familiar with deep algebraic geometry as some of you probably are – Camilo Gallardo Nov 27 '24 at 20:25
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    I am afraid that it is hard [at least hard for me] to find some existing text on elliptic curves via scheme theory. Based on the given picture, you need scheme theory as well as basics on etale cohomology theory. Qing Liu's book is already focused more on arithemtic side compared to geometric side. In addition you probably need to find some books on etale cohomology as well. Of course the Stacks Project have them all. – xbh Nov 28 '24 at 00:52
  • @CamiloGallardo Silverman's Arithmetic of Elliptic Curves might be a good choice. I am not an expert of elliptic curves, to be honest. For number theory, basic algebraic number theory is actually just algebraic geometry in disguise. But I can't think of a good book on algebraic number theory off the top of my head now. – J1U Nov 28 '24 at 03:47
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    For specifics on the result in the question, I suggest you just take this fact for granted and verify it for $X_0$ and $X_1$ (or prove it yourself using invariant theory -- the key is that the action has only $\pm I$ as stabilisers outside $0,1728,\infty$). Probably one day you'll come back to it and realise you know the words in the above proof. If you believe this fact, much of the literature on modular curves (so long as you're not classifying their rational points using fancy machines) tends to be quite accessible. – Mummy the turkey Nov 29 '24 at 17:43

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