#arithmetic geometry

I fear I'm actually enjoying Hartshorne...

I switched to Hartshorne from Liu as my main text because my advisor suggested that I do, and I think appreciate his writing style a lot more from the perspective of someone who has taken several runs at the subject at this point. A part of it might just be the immense amount of pride I feel being able to understand what used to be wholly impenetrable to me.

It feels so weird to read it and think that I not only understand what he's saying, but actively appreciate and enjoy the presentation.

I found a guide here that recommends Hartshorne II. 1-8, III. 1-5, 9, and IV. 1-3, which I realise is another reason I switched to Hartshorne: I couldn't find nearly as clear-cut a guide for any other book (I'm sure all of Liu/any other book is useful/important, but I want to get to the cool and fun stuff as soon as possible!!).

Afterwards, it recommends learning any one of the 'big three' (Weil Conjectures, Mordell, and Fermat), and I want to go above and beyond by studying all three. I'll be doing the Weil Conjectures in my reading course next semester, and I'll be doing a study on modular forms next year, and it seems like Fermat's Last Theorem will be a lot more manageable if I have those under my belt. That leaves the Mordell Conjecture. I've also read that Wiles's proof uses the finiteness theorems of Faltings, so it makes sense (kind of?) to look there first. I don't know if I actually will do that, both because I am unsure of how much free time I'll actually have next semester, and I have just seen that my university library doesn't actually have a copy of Cornell-Silverman's Arithmetic Geometry, and I've already expressed my grievances re: online copies. I could, I suppose, invest in a physical copy for mine self, but this would cost so much money...

I must confess that I'm sure that I am getting very far ahead of myself, but I also think it's understandable given how close I am (or at least feel) to the things that I used to read about as monumental achievements in the history of arithmetic geometry.

Today’s Book of the Day is Theorie des Topos et Cohomologie Etale des Schemas. Seminaire de Geometrie Algebrique du Bois-Marie 1963-1964 (SGA 4), curated by Michael Artin, A. Grothendieck, and J.L. Verdier in 1972 and published by Springer Verlag. I have chosen this book as I used some of its math to demonstrate to a colleague some topological errors he was making in creating a cognitive…

I went to see if I've talked about elliptic curves before, and I actually talked about this exact issue here. But now I've leveled up in Mathematica so I can do the visuals better.

Oversimplified, an elliptic curve is an equation that looks like y^2 = x^3+ax+b for some integers a and b. And they have lots of interesting number theoretic properties if you look for solutions that are rational numbers, none of which I want to talk about right now.

If you graph these equations over the real numbers you get pictures like one of these two (left: y^2=x^3+7x+1; right: y^2=x^3-7x+1):

And we like to imagine points at infinity in each direction, so the picture on the left looks like one giant circle (going through infinity at the top/bottom), and the picture on the right looks like two disconnected circles. And the difference is basically whether we get one or three solutions to the cubic when we set y equal to zero.

But over the complex numbers they should all look the same. The problem is, graphs of complex functions are four-dimensional and that's hard to display. But in that four-dimensional space, we should get a torus: a circle moved through a circular path (which still includes the point at infinity).

But we can make three-dimensional graphs, and then we can let them vary over time. So here is an animation of y^2=x^3+7x+1, with time controlling the imaginary coordinate of the output:

And here is y^2=x^3-7x+1:

You can see the moment in the middle where they "snap" into the real-plane-only version.

Sierpinski in the Sky with Diamonds.

Peter Scholze is a Fields Medalist. His work centers around the study of perfectoid spaces, his novel and breakthrough discovery.

Please know: I do not understand this in the slightest. I just know that p-adic numbers are astonishingly beautiful. 

Aside from Brian Conrad, another interesting comment is the one he replied to, who’s probably Peter Scholze (at least that’s what everyone’s guess in the math subreddit is). PS identified Corollary 3.12 as the main comprehension stumbling block for himself; BC replied that this was the same case for a bunch of other people too

Yeah, that does seem to be Scholze.  He’s getting yelled at (like, really mean-spirited “you should feel bad and you’re either lazy or incompetent” yelling) on Facebook.  (Hat tip to Peter Woit)

The only right thing to do is exactly the same as always: dedicate some time, study seriously, using all existing math texts, maybe forget that there are other mathematicians and their opinions: it is just you and the author. Then confirm if you don't find a mistake (in your terminology, "solid") or communicate to the author any mistake you found. If you can, try to simplify and develop further.

The weeks have been flying past, and I have been getting nothing done!

At least, it feels that way. As I alluded to in a previous post (on my main account, I think), I've decided to go and keep working through my foundations.

There are actually a lot of reasons for this. I've been hitting my head against more advanced books for probably the better part of the semester, and I don't think it's gotten me anywhere at all. I've also noticed that schemes don't really feel as concrete to me as they "should," I think. With a ring or a field, it feels like I can almost hold them in my hands, and feel how they move under morphisms, but schemes feel vaguely intangible. I can verify their properties and even visualise them if I try, but I can never hold them very tightly. Hopefully next semester's class on étale cohomology will help, but I don't want to spend forever just waiting.

Another, and probably the most important, reason, is that I kind of just need to get back into doing math, and feeling confident with math. I do feel confident, for example, teaching my friends abstract algebra or analysis, but it almost feels worse to be proud of explaining undergraduate level concepts when I know where I wish I were, and where I ought to be for the amount of time I've (claimed to) put into it.

With all that being said, I am not going back to either Hartshorne or Vakil. It would be my third time working through (the second and half the third chapter of) Hartshorne, and I wonder to what extent it will actually help. In contrast, with Vakil, I just really don't like his writing style I think. I really dislike the "three levels" approach he takes to schemes, and generally something about his writing style always leaves me more confused than before. I feel bad saying this because, in principle, it's the perfect book made freely available by a great (and really funny, I think) mathematician, but it just doesn't work for me personally. In fact, I think my year(ish)-long break from math was directly because of how Vakil's writing made me feel.

Instead, I will be working through Liu's Algebraic Geometry and Arithmetic Curves, and I've already spent the week working through the first chapter on commutative algebra, and I am just starting the chapter on general properties of schemes.

One issue with reviewing for the sake of building confidence is that, every time I don't immediately see a solution to a problem, my impostor syndrome gets exponentially stronger, until I figure it out (and then it doesn't reset, it just stops growing).

I’d probably start with William Stein’s open source [book on number theory and cryptography](http://wstein.org/ent/).  Chapter 3 is “public-key cryptography” an chapter six covers elliptic curve cryptography.  I think there are a few other books like that running around; a lot of number theory books do an intro to crypto.  

(By the way, everyone knows about AIM’s list of [Free and Open Source Textbooks](http://aimath.org/textbooks/approved-textbooks/), right?).  

I’m sure there’s a ton of other stuff out there, but I don’t know what it is; if any of my followers have suggestions please let me know.

For the second bit: I think it depends on your background, arithmetic geometry is a big field and can get pretty complex.  How much algebraic geometry do you have?  If you do too much arithmetic geometry people start talking about etale cohomology, and you can spend years just figuring out how that works.  (And then if you’re not careful they start throwing K-theory at you and then everything is terrifying).

Jordan Ellenberg wrote a nice overview of arithmetic geometry (PDF) and there appears to be an MIT Open Courseware page which collects some good links.  You also could look through the index of the Southwest Center for Arithmetic Geometry Winter Schools which has course materials and videos for each year’s program; some of those might be interesting or useful.