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I am currently self-studying elliptic curves using Silverman's AEC. I find his treatment of the background on varieties quite sloppy , and have so far kept going back and forth between AEC and Chapter 1 of Hartshorne, however having to translate between the 2, since their definitions are not identical. Is there an alternative?

E. Variste
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    I'm doing the same and I agree. However, I've always found Hartshorne impenetrable, so I'm looking forward to someone proposing an alternative. – rogerl Jun 01 '20 at 22:47
  • Elliptic curves are the toy problem of varieties, I think the game is to translate from $\Bbb{C}(x)$ (function field of Riemann sphere) to the complex torus (compact Riemann surface with obvious meaning of meromorphic functions and divisors and local ring) to the complex elliptic curve ($z\to(\wp_L(z),\wp_L'(z))$) then to algebraize the concepts so that they still apply to elliptic curves over $\Bbb{F_p,Q_p}$ and arbitrary fields. The degree of endomorphism, dual isogeny, invariant differential follow same idea, except for the non-separable degree and dual of Frobenius. @rogerl – reuns Jun 01 '20 at 23:12
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    My experience is that if you try to get a fully rigorous background on varieties first, then you will never get to the rest of the book in a reasonable time. Better to skim the background material on varieties, and take results from algebraic geometry for granted, to get to the actual material on elliptic curves. – Ted Jun 02 '20 at 02:16
  • @reuns I have a bountied question here that it sounds like you could answer, if you're interested. – rogerl Jun 02 '20 at 02:17
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    I think Fulton's curve book has all the necessary results on varieties, and is much friendlier than Hartshorne. I think you can just focus on Ch. 2 to get enough background to get started, and then return to it if you find you need more as you continue in Silverman. – Viktor Vaughn Jun 03 '20 at 05:06

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If it is just about varieties, you could give Mumford‘s Red Book a chance. There also is Garrity‘s Algebraic Geometry A problem solving approach a friend of mine is very fond of. Personally I can recommend Hulek‘s Elementary Algebraic Geometry

Jonas Linssen
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  • I was going to suggest the Red Book. It's not a substitute for all of Hartshorne, but the treatment of varieties is more careful. Be warned that the later, Latex'ed, edition introduced tons of annoying typos. – Anonymous Jun 02 '20 at 03:52
  • Both the Red book and Hartshorne only talk about their varieties over algebraically closed fields. I understand that in order to remove this restriction, one can go to schemes in full generality. However, is there a good treatment of varieties over general fields? Silverman talks about varieties, not schemes, over general fields. – E. Variste Jun 13 '20 at 20:22