Manageable project to learn some arithmetic geometry

OK, this may be an unusual question, since I'm asking you for advice, the way I would ask a supervisor were I a grad student.

I am a thirty-something maths teacher with an unresolved attraction to arithmetic geometry, and would like to know more about themes I find fascinating when I read texts about arithmetic geometry, such as:

• the local-global principle
• the way geometry influences the arithmetic behaviour of Diophantine equations, like in Faltings's theorem
• the way scheme theory helps number theory by making a lot of geometric constructions possible, starting from base change

One of the reasons this attraction is unresolved is that I never managed to seriously learn enough algebraic number theory, algebraic geometry, etc. in a "bottom-to-top" way: I have often started to read for instance R. Vakil's The rising sea, D. Eisenbud & J. Harris The geometry of schemes or Q. Liu's Algebraic geometry and arithmetic curves and find them to be very good texts, but I don't have the stamina to ingest hundreds of pages without a more specific goal.

Because of this there are a number of subjects I'm vaguely familiar with, but don't know in any depth. For instance, I know what a scheme is, I even think I understand in part their raison d'être, but am probably unable to pass an scheme theory exam (and the same goes for elliptic curves, derived categories, class field theory...).

To change this, I would like to try and understand a specific result and working my way from the top down to acquire the needed skills. The result has to be interesting enough to preserve my motivation in the long run, but "small" enough so that I could realistically understand it in a few months. I would love it to really mix number theory and geometry.

So the question is:

what theorem would constitute a reasonable-size project?

Personal facts that might be relevant: my PhD is in low-dimensional topology, so I'm quite comfortable with things like cohomology and Riemann surfaces, and I like it when geometric intuition is useful. Large doses of homological algebra or category theory don't frighten me.

Solution 1:

This is complementary to hm2020's answer, in that learning about the Hurwitz theorem and Riemann Roch are fantastic places to see some beautiful geometry. I think one more arithmetic direction this can go is to prove the (easier parts of) the Weil conjectures for smooth curves over finite fields. These conjectures motivated massive amounts of algebraic geometry, and are very down to earth, its all just counting solutions to equations over finite fields!

This is all done very well in Dino Lorenzini's book An Invitation to Arithmetic Geometry, and he goes out of his way to emphasise the arithmetic=geometry dictionary.

In the last chapter he even proves the hard part of the Weil conjectures, the Riemann Hypothesis for curves, which gives a nice perspective of why the integer version might be true (that the primes are as well distributed as possible).

Solution 2:

You could have a look at Lenstra's book Galois Theory for Schemes. It's only a bit over 100 pages, is very well-written, and contains tons of nice exercises. It builds up to the beautiful and useful result that any connected scheme $X$ has an etale fundamental group $\pi = \pi_1^{et}(X,x)$, and the category of finite etale covers of $X$ is equivalent to finite sets on which $\pi$ acts continuously. This is about as arithmetic geometric as I can imagine. Many examples are given. There is a deep relation to the topological fundamental group/algebraic topology, which is explained in the first chapter. The prerequesites are clearly laid out at the end of the introduction.

If this doesn't convince you, maybe Lenstra will, as the very first sentence of the book states, "One of the most pleasant ways to familiarize oneself with the basic language of abstract algebraic geometry is to study Galois theory for schemes."

Solution 3:

I recommend Silverman's The Arithmetic of Elliptic Curves, especially since you already have an understanding of compact Riemann surfaces.

(Or, phrased in terms of results: you should try to learn the complete proof of the Mordell-Weil theorem for Elliptic Curves.)

One caveat is that he avoids scheme language, and so some definitions are rather ad hoc in terms of Weierstrass equations. This is an advantage in that it makes the heart of the research on elliptic curves accessible more quickly, but a disadvantage for your particular use case of motivating scheme theory.

Solution 4:

If your interest is arithmetic geometry, a starting point could be to read sections IV.1 and IV.2 in Hartshorne (HH) on the Riemann-Roch theorem and the Hurwitz theorem. These sections are about morphisms of curves $v: C \rightarrow C'$ over any algebraically closed field $k$. In particular the results are valid for $k=\overline{\mathbb{F_q}}$ or $\overline{K}$ where $K$ is a number field. Fulton's book "Algebraic curves" is more elementary than HH and also works over any algebraically closed field. It requires some basic knowledge on commutative algebra. Milne's book "Etale cohomology" is self contained and in this book you will find a proof of the Weil conjectures and Riemann Hypothesis in general - in particular for a smooth projective curve over a finite field. Note also that exercise AppC.5.6+7 in HH proves the Weil conjectures for elliptic curves over a finite field. All these books are "elementary" in a sense - you need to know some commutative algebra.